aainstitutetext:  High Energy Physics, Cosmology & Astrophysics Theory Group
and The Laboratory for Quantum Gravity & Strings, Department of Mathematics
and Applied Mathematics,
University of Cape Town, South Africa bbinstitutetext: Mandelstam Institute
for Theoretical Physics, Witwatersrand University, Johannesburg, South
Africaccinstitutetext: Department of Physics, Pacific Lutheran University,
Tacoma, WA 98447

# Saturation of Thermal Complexity of Purification

S. Shajidul Haque b , Chandan Jana c , Bret Underwood<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We purify the thermal density matrix of a free harmonic oscillator as a two-
mode squeezed state, characterized by a squeezing parameter and squeezing
angle. While the squeezing parameter is fixed by the temperature and frequency
of the oscillator, the squeezing angle is otherwise undetermined, so that the
complexity of purification is obtained by minimizing the complexity of the
squeezed state over the squeezing angle. The resulting complexity of the
thermal state is minimized at non-zero values of the squeezing angle and
saturates to an order one number at high temperatures, indicating that there
is no additional operator cost required to build thermal states beyond a
certain temperature. We also review applications in which thermal density
matrices arise for quantum fields on curved spacetimes, including Hawking
radiation and a simple model of decoherence of cosmological density
perturbations in the early Universe. The complexity of purification for these
mixed states also saturates as a function of the effective temperature, which
may have interesting consequences for the quantum information stored in these
systems.

##  1 Introduction

The complexity of a quantum circuit, defined as the minimum number of unitary
operations that are needed to transform a given reference state into a
particular target state NL1 ; NL2 ; NL3 ; Jefferson , has many interesting
applications. There appear to be interesting connections to gravitational
holography, where the complexity of a field theory living on a boundary may
characterize features of its gravitational dual Susskind:2014rva ;
Stanford:2014jda ; Brown:2015bva . Quantum circuit complexity (hereafter
simply referred to as complexity) may also serve as a diagnostic for quantum
chaos Ali:2019zcj ; Bhattacharyya:2019txx ; Bhattacharyya:2020art , and the
complexity of quantum cosmological perturbations shows interesting behaviors
cosmology1 ; cosmology2 ; Lehners_2021 .

While pure states are simple to work with, many interesting systems and
phenomena are described by mixed states, including decoherence and thermal
states. A mixed state can be transformed into a pure state through
purification, in which the Hilbert space is enlarged to include ancillary
degrees of freedom such that the mixed state density matrix is recovered as
the trace of the pure state density matrix over the ancillary states. We will
follow Agn2019 ; MyersMixed ; DiGiulio:2020hlz ; Camargo:2020yfv  in defining
the _thermal complexity of purification_ as the minimum complexity of a set of
purifications of a thermal density matrix relative to the ground state.
Alternatively, a pure state can be associated to the density matrix through
the technique of operator-state mapping CHOI1975285 ; JAMIOLKOWSKI1972275 , in
such a way that the expectation values of observables are preserved. We will
consider both techniques in Section 2, finding that the extra freedom in
scanning over ancillary states for the thermal complexity of purification
leads to a smaller complexity (and thus a more optimal unitary operator) than
the unique pure state defined by the operator-state mapping procedure.

We will focus on the thermal density matrix of a free harmonic oscillator,
both for its simplicity and because it can be directly related to several
interesting applications. While the thermal complexity of purification for
this system has been studied before MyersMixed ; DiGiulio:2020hlz ;
Camargo:2020yfv  (see also Bhattacharyya:2018sbw ; Bhattacharyya:2019tsi  for
the entanglement of purification of similar systems), our analysis, based on
the approach of Jefferson ; me1 , will include a somewhat more general
purification as a two-mode squeezed state with non-zero squeezing angle
$\phi$. Squeezed states can play an important role in continuous variable
quantum computing TeleportQuantum ; Furusawa706 ; Braunstein_2005 ; qumode ;
Fukui_2018 , and show up naturally in descriptions of Hawking radiation and
cosmological perturbations Grishchuk ; Albrecht ; Martin1 ; Martin2 , among
other applications. We will find in Section 2.1 that the minimum thermal
complexity of purification relative to the ground state occurs at a non-zero
squeezing angle, and saturates at high temperature $T$ (or low frequencies
$\omega$) to a constant ${\mathcal{C}}_{\rm th}\approx\pi/(2\sqrt{2})$, as
opposed to the complexity for a vanishing squeezing angle that grows
logarithmically with temperature ${\mathcal{C}}_{\phi=0}\sim\ln(T/\omega)$. In
Section 2.2, we show that generalizing the purification further to also
include single-mode squeezing of the ancillary degrees of freedom introduces
additional degeneracy in the purification parameters for the minimized
complexity, but does not change the value of the minimized complexity. In
Section 2.3, we show how the pure state obtained from operator-state mapping
is a two-mode squeezed state with a vanishing squeezing angle, and thus does
not have a minimal complexity. Altogether, we find that the two-mode squeezed
complexity of purification with a non-zero squeezing angle produces the
simplest minimal complexity among these different purification techniques.

We apply our results on the thermal complexity of purification to two
interesting applications in quantum fields on curved spacetime where thermal
density matrices can arise: Hawking radiation and cosmological perturbation
theory. In Section 3.1, we review how Hawking radiation for a scalar field on
a curved spacetime with horizons, such as Rindler space or a black hole
spacetime, can be described as a thermal state by tracing over a two-mode
squeezed state in which modes on either side of the horizon are entangled with
each other. We then illustrate how the optimal purification of complexity of
this thermal state takes the form of a squeezed state with a different
squeezing angle and saturates at when the effective temperature of the horizon
is high, in contrast to the complexity of the original squeezed state which
grows with the effective temperature. While these are calculations of
complexity “in the bulk” (and are thus not directly related to holography),
they may have interesting implications for information-theoretic perspectives
of black holes. In Section 3.2, we review the application of the techniques of
complexity to quantum cosmological perturbations, such as those produced in
the very early universe during inflation. Considering a few simple models of
decoherence, we show how the Fourier modes of these perturbations can also be
described by a thermal density matrix, and illustrate how the resulting
saturation of the thermal complexity of purification of the Universe compares
before and after decoherence. We conclude in Section 4 with a summary of our
results and some speculation on their implications. Several appendices are
included for reference and review.

##  2 Complexity of a Thermal Density Matrix

As discussed in the Introduction, we are interested in the complexity of a
thermal state of a quantum harmonic oscillator,

$\displaystyle\hat{\rho}_{\rm th}=\frac{1}{Z}\sum_{n=0}^{\infty}e^{-\beta
E_{n}}|n\rangle\langle n|\,,$ (1)

where $E_{n}=\omega n$. In order to use established techniques Jefferson ; me1
to compute the complexity, we need to represent the thermal state (1) as a
pure state $|\Psi\rangle$. For any mixed state $\hat{\rho}_{\rm mix}$ on the
Hilbert space $\cal H$, we can construct a _purification_ of $\hat{\rho}_{\rm
mix}$ which consists of a pure state $|\Psi\rangle$ in an enlarged Hilbert
space ${\cal H}_{\rm pure}=\cal H\otimes\cal H_{\rm anc}$, where ${\cal
H}_{\rm anc}$ corresponds to an “ancillary” set of degrees of freedom. If the
trace of the density matrix of $|\Psi\rangle$ over the ancillary degrees of
freedom gives the original mixed state ${\rm Tr}_{\rm
anc}\left(|\Psi\rangle\langle\Psi|\right)=\hat{\rho}_{\rm mix}$, we say that
$|\Psi\rangle$ is a “purification” of $\hat{\rho}_{\rm mix}$. Note that
expectation values of operators acting in ${\cal H}$ are preserved under
purification, $\langle\hat{\cal O}\rangle={\rm Tr}_{\rm
anc}\left(\langle\Psi|\hat{\cal O}|\Psi\rangle\right)={\rm
Tr}\left(\hat{\rho}_{\rm mix}\hat{\cal O}\right)$, so that observables are
preserved by purification.

Clearly, the purification $|\Psi\rangle$ is not unique, since the choice of
the ancillary Hilbert space ${\cal H}_{\rm anc}$ is arbitrary as long as it
meets the purification requirement. For example, there may be a set of pure
states $\left\\{|\Psi\rangle_{\alpha,\beta,...}\right\\}$, parameterized by
$\alpha,\beta,...$, all of which satisfy the purification requirement. In
order to distinguish among the set of purifications, it is often helpful to
minimize a quantity of interest, such as the entanglement entropy or
complexity, with respect to the parameters. In this work, we are interested in
analyzing the complexity of the mixed thermal state (1), so we will minimize
the complexity of the set of purifications
$\left\\{|\Psi\rangle_{\alpha,\beta,...}\right\\}$ of $\hat{\rho}_{\rm th}$,
obtaining the thermal complexity of purification

$\displaystyle{\cal C}_{\rm th}(\beta)=\min_{\alpha,\beta,...}{\cal
C}\left(|\Psi\rangle_{\alpha,\beta,...},|\psi_{R}\rangle\right)\,,$ (2)

where we made explicit the dependence of the complexity of the pure state on
the reference state $|\psi_{R}\rangle$.

We will choose three explicit purifications of $\hat{\rho}_{\rm th}$ (1).
First, in Section 2.1 we will purify $\rho_{\rm th}$ as a two-mode squeezed
state, parameterized by a squeezing parameter $r$ and squeezing angle $\phi$,
in which the Hilbert spaces ${\cal H},{\cal H}_{\rm anc}$ are entangled in
such a way that a trace over the ancillary degree of freedom gives rise to a
thermal state. The corresponding complexity of purification _saturates_ as a
function of inverse temperature. Next, in Section 2.2 we generalize the two-
mode squeezed state purification to include additional squeezing of the
ancillary degree of freedom. We find that minimizing over the additional
parameters introduced by the extra squeezing provides more freedom in finding
a minimum of (2), but leaves the minimum value of the complexity of
purification unchanged. Finally, in Section 2.3 we construct a purification of
the thermal state through operator-state mapping.

###  2.1 Two Mode Squeezing as Purification

A straightforward purification of the generic thermal state (1) is the
thermofield double state,

$\displaystyle|{\rm TFD}\rangle=\frac{1}{\sqrt{Z}}\sum_{n=0}^{\infty}e^{-\beta
E_{n}/2}|n\rangle\otimes|n\rangle_{\rm anc}\,,$ (3)

where the ancillary Hilbert space ${\cal H}_{\rm anc}$ is taken to be a copy
of the original oscillator ${\cal H}$. However, the thermofield double state
(3) is by no means unique as a purification of (1); indeed, it is possible to
include an additional phase, so that the purification
$|\Psi\rangle_{\alpha,\beta...}$ becomes

$\displaystyle|\Psi\rangle_{\phi}=|{\rm
TFD}\rangle_{\phi}=\frac{1}{\sqrt{Z}}\sum_{n=0}^{\infty}(-1)^{n}e^{-2in\phi}e^{-n\beta\omega/2}|n\rangle\otimes|n\rangle_{\rm
anc}\,,$ (4)

where we took $E_{n}=\omega n$ for bosonic oscillators, and we introduced a
factor of $(-1)^{n}$ (which can be reabsorbed back into $\phi$) for
convenience. We recognize this as a two-mode squeezed vacuum state,

$\displaystyle|\Psi\rangle_{\phi}=\frac{1}{\cosh
r}\sum_{n=0}^{\infty}(-1)^{n}e^{-2in\phi}\tanh^{n}r|n\rangle\otimes|n\rangle_{\rm
anc}\equiv\hat{S}_{\rm sq}(r,\phi)|0\rangle\otimes|0\rangle_{\rm anc}\,,$ (5)

where the squeezing parameter $r$ is related to the oscillator frequency and
temperature through $\beta\omega=-\ln\tanh^{2}r$, and the squeezing angle
$\phi$ is a free parameter; as part of the purification of complexity process
(2), we will minimize the complexity with respect to $\phi$. The operator
$\hat{S}_{\rm sq}(r,\phi)$ is the two-mode squeezing operator, and is given in
terms of raising and lowering operators
$\\{\hat{a},\hat{a}^{\dagger}\\},\\{\hat{a}_{\rm anc},\hat{a}_{\rm
anc}^{\dagger}\\}$ on the physical and ancillary oscillator Hilbert spaces
${\cal H},{\cal H}_{\rm anc}$, respectively,

$\hat{S}_{\rm
sq}(r,\phi)=\exp\left[\frac{r}{2}\left(e^{-2i\phi}\hat{a}\hat{a}_{\rm
anc}-e^{2i\phi}\hat{a}^{\dagger}\hat{a}_{\rm anc}^{\dagger}\right)\right]\,.$
(6)

The purification of the thermal state (1) into (5) is thus obtained by acting
the squeeze operator $\hat{S}_{\rm sq}(r,\phi)$ on the two-mode vacuum
$|0\rangle\otimes|0\rangle_{\rm anc}$. The two-mode squeezing operator can be
interpreted as a type of _entanglement_ operator, as it mixes creation and
annihilation operators of the two Hilbert spaces ${\cal H},{\cal H}_{\rm anc}$

$\displaystyle\hat{S}^{\dagger}_{\rm sq}\ \hat{a}\ \hat{S}_{\rm sq}$
$\displaystyle=$ $\displaystyle\hat{a}\ \cosh r-\hat{a}_{\rm anc}^{\dagger}\
e^{2i\phi}\ \sinh r\,;$ (7) $\displaystyle\hat{S}^{\dagger}_{\rm sq}\
\hat{a}_{\rm anc}\ \hat{S}_{\rm sq}$ $\displaystyle=$
$\displaystyle\hat{a}_{\rm anc}\ \cosh r-\hat{a}^{\dagger}\ e^{2i\phi}\ \sinh
r\,.$ (8)

For the calculation of complexity111See Appendix A for details., we are
interested in the transformation of a reference state $|\psi_{R}\rangle$ into
a target state $|\psi_{T}\rangle$

$\displaystyle|\psi_{T}\rangle=\hat{\mathcal{U}}\ |\psi_{R}\rangle$ (9)

by a unitary operator $\hat{\mathcal{U}}$ representing the quantum circuit
connecting these two states. Following the geometric approach of NL1 ; NL2 ;
NL3 ; Jefferson , we decompose the unitary $\hat{\mathcal{U}}$ as a path-
ordered sequence of a set of fundamental operators
$\\{\hat{\mathcal{O}}_{I}\\}$

$\displaystyle\hat{U}(s)=\overleftarrow{\mathcal{P}}\ {\rm
exp}\left[-i\int_{0}^{s}\sum_{I}Y^{I}(s^{\prime})\hat{\mathcal{O}}_{I}\
ds^{\prime}\right]\,.$ (10)

We define complexity in a geometric way as the circuit depth along a minimal
path in the geometry generated by the algebra of the operators

$\displaystyle{\mathcal{C}}=\int_{0}^{1}\sqrt{\sum_{I}G_{IJ}Y^{I}Y^{J}}\
ds\,,$ (11)

where the $Y^{I}(s)$ are vectors that specify the path parameterized by $s$.
To calculate the complexity we follow Jefferson ; cosmology1  in transforming
our reference and target states into position-space wavefunctions by defining
the position variable
$\hat{q}=\frac{1}{\sqrt{2\omega}}(\hat{a}^{\dagger}+\hat{a})$ (with a similar
definition for the ancillary variable $\hat{q}_{\rm anc}$ in terms of the
$\hat{a}_{\rm anc},\hat{a}_{\rm anc}^{\dagger}$). Our reference state will be
the (purified) ground state $|\psi_{R}\rangle=|0\rangle\otimes|0\rangle_{\rm
anc}$

$\displaystyle\langle q,q_{\rm anc}|\psi_{R}\rangle={\mathcal{N}}_{R}\ {\rm
exp}\left[-\frac{1}{2}\omega(q^{2}+q_{\rm anc}^{2})\right]\,.$ (12)

Our target state is the purification (5),
$|\psi_{T}\rangle=|\Psi\rangle_{\phi}$, which also takes the form of a
Gaussian wavefunction

$\displaystyle\Psi_{\rm sq}\left(q,q_{\rm anc}\right)=\langle q,q_{\rm
anc}|\Psi\rangle_{\phi}={\cal
N}\exp\left\\{-\frac{\omega}{2}\,A(q^{2}+{q}_{\rm anc}^{2})-\omega B\ q\
q_{\rm anc}\right\\}$ (13)

where ${\cal N}$ is a normalization factor that will not be important here,
and the squeezed Gaussian parameters are

$\begin{split}&A=\frac{1+e^{-4i\phi}\tanh^{2}r}{1-e^{-4i\phi}\tanh^{2}r}\,,\qquad
B=\frac{2\tanh r\,e^{-2i\phi}}{1-e^{-4i\phi}\tanh^{2}r}\,.\end{split}$ (14)

Interestingly, the corresponding variances of the physical position $\hat{q}$
and momentum $\hat{p}=i\sqrt{\omega/2}(\hat{a}^{\dagger}-\hat{a})$ are
independent of the squeezing angle

$\displaystyle\langle q^{2}\rangle=\frac{1}{\omega}\cosh(2r)\,,\qquad\langle
p^{2}\rangle=\omega\cosh(2r)\,.$ (15)

A curious feature of the two-mode squeezed state is that the uncertainty of
the single oscillator grows as the squeezing is increased $\langle
q^{2}\rangle\langle p^{2}\rangle=\cosh^{2}(2r)$, in contrast to a one-mode
squeezed state in which the uncertainty in one direction of phase space is
reduced while the other grows, preserving the total uncertainty222Two mode
squeezed states do not preserve the total noise, but instead preserve the
difference in the total noises of the two modes. See Schumaker  for other
useful properties of two-mode squeezed states.. The growth in uncertainty of
the single oscillator of the two-mode squeezed state mirrors the expected
growth in uncertainty of a thermal state with temperature, and is a desired
property of our purification of the thermal mixed state.

The complexity (11) of the purified thermal state (13) thus becomes (see
Appendix A)

$\displaystyle\mathcal{C}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sqrt{\ln^{2}\left|\frac{1+e^{-2i\phi}\tanh
r}{1-e^{-2i\phi}\tanh r}\right|+\arctan^{2}(\sin 2\phi\sinh 2r)}$ (16)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sqrt{\ln^{2}\left|\frac{1+e^{-2i\phi}e^{-\beta\omega/2}}{1-e^{-2i\phi}e^{-\beta\omega/2}}\right|+\arctan^{2}\left(2\sin
2\phi\frac{e^{-\beta\omega/2}}{1-e^{-\beta\omega}}\right)}\,,$ (17)

where we substituted $\tanh r=e^{-\beta\omega/2}$ to make the temperature-
dependence explicit. The $\arctan$ contribution to the complexity is necessary
when the parameters $A,B$ of the Gaussian wavefunction (13) take on complex
values me1 , and will play an important role in the minimized complexity, as
we will see.

Figure 1: Complexity (16) of the purified thermal state as a squeezed state,
as a function of the squeezing parameter $r$ and the squeezing angle $\phi$.
The dashed line indicates the minimum of the complexity at $\phi=\pi/4$. The
complexity saturates at large $r$ (corresponding to high temperature) for
generic $\phi$ as in (19), but grows linearly with the squeezing $r$ for the
angles $\phi=0,\pi/2$.

Before we minimize the complexity (17) with respect to the squeezing angle
$\phi$ as part of the complexity of purification process (2), let us examine
some of its useful limits. The low-temperature limit
$\beta\omega\rightarrow\infty$ corresponds to vanishing squeezing
$r\rightarrow 0$; the system and ancillary degrees of freedom are no longer
entangled, as can be seen by the diagonalization of the wavefunction (13) for
$A\approx 1,B\approx 0$ in this limit. Correspondingly, the complexity (17)
vanishes as ${\cal C}_{\phi}\sim e^{-\beta\omega/2}$ at low temperatures
$\beta\omega\gg 1$, for all squeezing angles $\phi$.

Alternatively, the high-temperature limit $\beta\omega\ll 1$ corresponds to
large squeezing $r\gg 1$ in the squeezed-state language. For generic squeezing
angles (specifically, for $\phi\neq n\pi/2$) the Gaussian wavefunction
parameters (14) saturate to pure imaginary values at leading order

$\displaystyle A\approx-i\frac{\sin 4\phi}{1-\cos 4\phi}\,,\qquad
B\approx-2i\frac{\sin 2\phi}{1-\cos 4\phi}\,.$ (18)

Thus, at high temperature (large squeezing), the two-mode squeezed state
wavefunction (13) is an approximately pure phase entanglement between the
physical and ancillary degrees of freedom. Correspondingly, the position
variance diverges in the high-temperature limit $\omega\langle
q^{2}\rangle\sim(\beta\omega)^{-1}\gg 1$, consistent with the delocalization
that is expected of a thermal state. The saturation of the purified
wavefunction to a pure phase at high temperatures leads to a similar
saturation of the complexity (17)

$\displaystyle{\mathcal{C}}_{\phi}\approx\frac{1}{\sqrt{2}}\sqrt{\ln^{2}\left|\frac{1+\cos
2\phi}{1-\cos 2\phi}\right|+\left(\frac{\pi}{2}\right)^{2}}\,,$ (19)

which is independent of the temperature at leading order. The angle
$\phi=\pi/4$ is particularly interesting, since we see that the entangled
Gaussian wavefunction becomes purely off-diagonal in the high-temperature
limit $A\approx 0,B\approx i$, and the complexity is minimal
${\mathcal{C}}_{\phi/4}\approx\pi/2\sqrt{2}$.

The angles $\phi=n\pi/2$ are special cases, and lead to behaviors for the
wavefunction and complexity that are qualitatively different than the general
case. For example, for a vanishing squeezing angle $\phi=0$, the high-
temperature limit corresponds to a strongly entangled wavefunction with real
Gaussian parameters $A\sim B\sim(\beta\omega)^{-1}\gg 1$. The corresponding
complexity (17) is then dominated by the first term ${\cal C}_{\phi=0}\approx
r\approx\ln\left(\frac{1}{\beta\omega}\right)=\ln\left(\frac{T}{\omega}\right)$
and grows logarithmically with the temperature. The generic behavior of the
complexity, as well as the behavior for the special cases $\phi=n\pi/2$, can
be seen in Figure 1.

(a) Complexity of purification (17) as a function of temperature
$T=\beta^{-1}$ for $\phi=0$ and $\phi=\pi/4$.

(b) Thermal complexity of purification as a function of the frequency $\omega$
for different fixed temperatures $T$.

Figure 2: The complexity of purification is minimized by the squeezing angle
$\phi=\pi/4$ and has the unique feature that it saturates for high
temperatures to ${\mathcal{C}}_{\phi=\pi/4}\approx\pi/(2\sqrt{2})\approx 1.1$,
as seen in Figure 2(a). For a fixed temperature $T$, the minimal thermal
complexity of purification ${\mathcal{C}}_{\rm th}={\mathcal{C}}_{\phi=\pi/4}$
drops off sharply for high frequencies $\omega>T$, as seen in Figure 2(b).

We have purified the thermal state $\hat{\rho}_{\rm th}$ (1) into the two-mode
squeezed state (5), with corresponding complexity (17). The thermal complexity
of purification (2) is obtained by minimizing (17) over the squeezing angle
$\phi$, which we introduced as a free parameter. The squeezed state complexity
(17) is symmetric about $\pi/4$, and the complexity is minimized at
$\phi=\pi/4$ for all values of the squeezing parameter $r$ (or
correspondingly, for all values of the temperature $\beta\omega$) leading to

$\displaystyle\mathcal{C}_{\rm th}(\beta)$ $\displaystyle=$
$\displaystyle\mathcal{C}_{\phi}\big{|}_{\phi=\pi/4}=\frac{1}{\sqrt{2}}|\arctan(\sinh
2r)|=\frac{1}{\sqrt{2}}\left|\arctan\left(2\frac{e^{-\beta\omega/2}}{1-e^{-\beta\omega}}\right)\right|$
(20) $\displaystyle\approx$ $\displaystyle\begin{cases}\sqrt{2}\
e^{-\beta\omega/2}&\mbox{ low temperature limit }\beta\omega\gg
1\cr\frac{\pi}{2\sqrt{2}}&\mbox{ high temperature limit,
}\beta\omega\rightarrow 0\end{cases}$ (21)

with the corresponding purification as a two-mode squeezed state with
squeezing angle $\phi=\pi/4$

$\displaystyle|\Psi\rangle_{\rm
th,p}=|\Psi\rangle_{\phi=\pi/4}=\frac{1}{\sqrt{Z}}\sum_{n=0}^{\infty}(-i)^{n}\
e^{-n\beta\omega/2}|n\rangle\otimes|n\rangle_{\rm anc}\,.$ (22)

Interestingly, we find that the complexity of purification of the thermal
density matrix $\hat{\rho}_{\rm th}$ _saturates_ at high temperature, as can
also be seen in Figure 2. This saturation of the complexity of the purified
thermal density matrix with temperature is in contrast to previous results
MyersMixed , which did not include the squeezing angle in the process of
purification, and thus considered only real Gaussian wavefunctions. We see
that purifying the thermal density matrix to include a squeezing angle
qualitatively changes the behavior of the complexity of purification. As
discussed above, the physical origin of this saturation is that at high
temperature the Gaussian wavefunction parameters (18) become imaginary and
independent of the temperature at leading order. The corresponding circuit
depth saturates, even for arbitrarily large temperature.

It is interesting to compare the thermal complexity of purification to the
corresponding entanglement entropy of the thermal state Srednicki_1993 ;
demarie2012pedagogical

$\displaystyle\hat{S}_{\rm en}={\rm Tr}\left[\hat{\rho}_{\rm
th}\ln\hat{\rho}_{\rm
th}\right]=-\ln\left(1-e^{-\beta\omega}\right)-\frac{e^{-\beta\omega}}{1-e^{-\beta\omega}}\ln\left(e^{-\beta\omega}\right)\,.$
(23)

The entanglement entropy of the thermal state is independent of the
purification squeezing angle $\phi$, so that from the perspective of
entanglement entropy, all two-mode squeezed states that comprise the
purification state (5) are equivalent, for any squeezing angle. However, we
have seen that complexity for building such states is not equivalent – it is
in fact easier, by a factor $\sim|\ln(\beta\omega)|$, to build the optimal
purification state (22) ((5) with $\phi=\pi/4$) than it is to build the TFD
state (3) ((5) with $\phi=0$), even though both of these states give identical
reduced thermal density matrices and have identical entanglement entropies
(23). This is in line with other observations that complexity is a more
sensitive probe of a state than entanglement alone Camargo_2019 ; MyersMixed .

Another interesting perspective of the thermal complexity of purification (20)
is to consider its dependence on the oscillator frequency $\omega$ for fixed
temperature $\beta=1/T$, as in Figure 2(b). For fixed temperature, the
complexity drops off quickly as a function of frequency ${\mathcal{C}}_{\rm
th}\sim e^{-\beta\omega/2}$, so that high frequency modes with $\omega>T$
contribute less to the complexity than low frequency modes. The suppression of
the complexity for purified high frequencies of a thermal state characterizes
the extent to which these modes are washed out by the thermal background.

The thermal state of a single harmonic oscillator is easily generalizable to a
set of $N$ decoupled harmonic oscillators with Hamiltonian
$\hat{H}=\sum_{i=1}^{N}\left(\omega_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}+1/2\right)$,
each of which is in a thermal mixed state with individual thermal density
matrix given by $\rho_{\rm th}$ (1). The two-mode purification described above
simply becomes a direct product of the individual purifications, so that the
total thermal complexity of purification for all the modes is the sum

$\displaystyle{\mathcal{C}}_{\rm th}^{\rm
N}=\frac{1}{\sqrt{2}}\sqrt{\sum_{i=1}^{N}\arctan^{2}\left(2\frac{e^{-\beta\omega_{i}/2}}{1-e^{-\beta\omega_{i}}}\right)}\approx\frac{1}{\sqrt{2}}\sqrt{\sum_{\omega_{i}<T}\arctan^{2}\left(2\frac{e^{-\beta\omega_{i}/2}}{1-e^{-\beta\omega_{i}}}\right)}\,,$
(24)

where only the low frequencies $\omega_{i}<T$ contribute to the sum since the
$\arctan$ drops off at high frequencies. Unfortunately, while the
${\mathcal{F}}_{2}$ cost function used here for the complexity (see Appendix
A) has the advantage of being independent of basis choice, it is clearly not
extensive with respect to the number of independent oscillators. Instead, it
is more natural to use the Finsler ${\mathcal{F}}_{1}$ cost function to
include this behavior. Fortunately the minimum length geodesic is unchanged,
so that the (Finsler) thermal complexity of purification, minimized as a
function of the squeezing angle $\phi$, becomes

$\displaystyle{\mathcal{C}}_{1}^{\rm N}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\arctan\left(2\frac{e^{-\beta\omega_{i}/2}}{1-e^{-\beta\omega_{i}}}\right)\approx\sum_{\omega_{i}<T}\arctan\left(2\frac{e^{-\beta\omega_{i}/2}}{1-e^{-\beta\omega_{i}}}\right)$
(25) $\displaystyle\approx$ $\displaystyle\frac{\pi}{2\sqrt{2}}\ N_{T}\,,$
(26)

where $N_{T}$ is the number of oscillators with $\omega_{i}<T$.

It is straightforward to now take the continuum limit of (25) as the
complexity of the purification of Fourier modes of a free scalar field
$\varphi(x,t)$ in $(d+1)$-spacetime dimensions at constant temperature

$\displaystyle{\mathcal{C}}_{\rm th}^{\rm tot}={\rm Vol}_{d}\int
d^{d}k\arctan\left(2\frac{e^{-\beta\omega_{\vec{k}}/2}}{1-e^{-\beta\omega_{\vec{k}}}}\right)={\rm
Vol}_{d}\ {\cal I}_{d}\ \beta^{-d}\,,$ (27)

where ${\rm Vol}_{d}$ is the $d$-dimensional spatial volume,
$\omega_{\vec{k}}=\sqrt{\vec{k}^{2}+m^{2}}$ and

$\displaystyle{\cal I}_{d}=\int
d^{d}\ell\arctan\left(2\frac{e^{-\sqrt{\ell^{2}+m^{2}\beta^{2}}/2}}{1-e^{-\sqrt{\ell^{2}+m^{2}\beta^{2}}}}\right)$
(28)

is a numerical constant. For example, for $m=0$ and $d=1$, ${\cal
I}_{1}\approx 3.7$, the thermal complexity of purification grows linearly with
the temperature, while for $m=0$ and $d=3$, ${\cal I}_{3}\approx 398$, the
thermal complexity grows as the cube of the temperature. The temperature
dependence arises due to the density of states, since the complexity saturates
for each mode with frequency $k<T$ in a sphere with radius set by the
temperature.

We have considered the complexity of the purified target state (22) relative
to the ground state, but another natural candidate for a reference state is
the factorized Gaussian state Jefferson ; MyersMixed

$\displaystyle\langle q,q_{\rm
anc}|\psi_{f}\rangle={\mathcal{N}}_{f}\exp\left[-\frac{1}{2}\mu(q^{2}+q_{\rm
anc}^{2})\right]\,,$ (29)

where the reference frequency $\mu$ is the same for all oscillators and is not
equal to the ground state frequency $\omega$. For a set of oscillators, this
reference state is disentangled in the position basis, so the complexity of
the target state (22) relative to it is therefore a useful measure of the
difficulty in creating spatial entanglement. Minimizing the purified
complexity with respect to the squeezing angle, we now have an additional term
in the thermal complexity of purification

$\displaystyle{\mathcal{C}}_{\rm
th,\mu}(\beta)=\frac{1}{\sqrt{2}}\sqrt{\left(\ln\left|\frac{\omega}{\mu}\right|\right)^{2}+\arctan^{2}\left(2\frac{e^{-\beta\omega/2}}{1-e^{-\beta\omega}}\right)}\,.$
(30)

For fixed temperature, the additional term $\ln|\omega/\mu|$ modifies the
behavior of the complexity at both high $\omega\gg\mu$ and low frequencies
$\omega\ll\mu$, reflecting the additional complexity needed to establish
spatial correlations in the target state (22).

###  2.2 Purification with Additional Ancillary Squeezing

A number of different purifications can be used to construct the complexity of
purification (2) since the minimization procedure scans over the ancillary
parameters, while preserving the reduced density matrix. Ideally, we wish to
find the simplest purification that leads the smallest complexity of
purification of the thermal mixed state. We have already shown that extending
the purification of the thermal density matrix beyond that of a thermofield
double (3) to include a squeezing angle (4) qualitatively changes the value
and functional dependence of the complexity of purification on the
temperature. However, it is possible to generalize the purification further,
and it is worth investigating whether including additional ancillary
parameters further changes the result in a qualitative way.

The purification of the thermal density matrix as a pure state consisting of a
two-mode squeezed state (5) can be generalized by the action of a two-mode
rotation operator

$\displaystyle|\Psi\rangle_{\phi,\theta}=\hat{\mathcal{R}}(\theta)\,\hat{S}_{\rm
sq}(r,\phi)|0\rangle\otimes|0\rangle_{\rm anc}\,,$ (31)

where

$\displaystyle\hat{\mathcal{R}}(\theta)=\exp\left[-i\theta\left(\hat{a}^{\dagger}\hat{a}+\hat{a}^{\dagger}_{\rm
anc}\hat{a}_{\rm anc}\right)\right]\,.$ (32)

The rotation operator acts on the two-mode squeezed state as a shift
$\phi\rightarrow\phi+\theta$ and thus is a redundant purification, so we can
set $\theta=0$.

Because the ancillary degrees of freedom are traced out in the process of
obtaining the density matrix, we can perform additional transformations
operating purely in ${\cal H}_{\rm anc}$. In particular, we consider an
_additional_ single-mode squeezing of the ancillary oscillator by the
squeezing parameter and angle $r_{\rm anc},\phi_{\rm anc}$

$\displaystyle|\Psi\rangle_{\phi,r_{\rm anc},\phi_{\rm anc}}=\hat{S}_{\rm
anc}(r_{\rm anc},\phi_{\rm anc})\ \hat{S}_{\rm
sq}(r,\phi)|0\rangle\otimes|0\rangle_{\rm anc}\,,$ (33)

where

$\displaystyle\hat{S}_{\rm anc}(r_{\rm anc},\phi_{\rm
anc})=\exp\left[-\frac{r_{\rm anc}}{2}\left(e^{-2i\phi_{\rm anc}}\hat{a}_{\rm
anc}^{2}-e^{2i\phi_{\rm anc}}\hat{a}_{\rm anc}^{\dagger 2}\right)\right]\,.$
(34)

We have now introduced two additional free parameters $r_{\rm anc},\phi_{\rm
anc}$, which will need to include together with the two-mode squeezing angle
$\phi$ in the minimization of the complexity (2). This generalization is
similar to the purification considered in MyersMixed , which also considered
an additional squeezing of the ancillary degrees of freedom (again, with
vanishing squeezing angle). However, because our two-mode purification mixes
the physical and ancillary degrees of freedom through the two-mode squeezing
angle, in addition to having an additional parameter $\phi_{\rm anc}$, the
calculation of the wavefunction and the minimization of the complexity
corresponding to (33) is somewhat more involved.

We can construct the position-space wavefunction corresponding to (33) as

$\displaystyle\Psi(q,q_{\rm anc})_{\phi,r_{\rm anc},\phi_{\rm anc}}=\langle
q,q_{\rm anc}|\Psi\rangle={\cal S}_{\rm anc}(r_{\rm anc},\phi_{\rm
anc})\Psi_{\rm sq}(q,q_{\rm anc})\,,$ (35)

where the single-mode ancillary squeeze operator in position-space takes the
form

$\displaystyle{\cal S}_{\rm anc}=\exp\left\\{\frac{ir_{\rm
anc}}{2}\left[(q_{\rm anc}p_{\rm anc}+p_{\rm anc}q_{\rm anc})\cos(2\phi_{\rm
anc})+(\omega^{-1}p_{\rm anc}^{2}-\omega q_{\rm anc}^{2})\sin(2\phi_{\rm
anc})\right]\right\\}\,,\,$ (36)

with $p_{\rm anc}=-i\partial_{q_{\rm anc}}$. Using the auxiliary squeezing
variables

$\displaystyle v\equiv r_{\rm anc}\cos 2\phi_{\rm anc}\,,\qquad
u\equiv-\frac{ir_{\rm anc}}{2\omega}\sin 2\phi_{\rm anc}\,,$ (37)

and rewriting (36) using a BCH formula Van_Brunt_2015  as

$\displaystyle\mathcal{S}_{\rm anc}=e^{vq_{\rm anc}\partial_{q_{\rm
anc}}}\,e^{\frac{e^{v}-1}{v}u\partial_{q_{\rm
anc}}^{2}}\,e^{\frac{1-e^{-2v}}{2v}u\omega^{2}q_{\rm anc}^{2}}\,,$ (38)

the position-space wavefunction (35) takes the form

$\displaystyle\Psi(q,q_{\rm anc})_{\phi,r_{\rm anc},\phi_{\rm
anc}}=\tilde{\mathcal{N}}\exp\left\\{-\frac{\omega}{2}\,\tilde{A}q^{2}-\frac{\omega}{2}\,\tilde{A}_{\rm
anc}q_{\rm anc}^{2}-\omega\tilde{B}qq_{\rm anc}\right\\}\,.$ (39)

The Gaussian coefficients are

$\displaystyle\tilde{A}$ $\displaystyle\equiv$ $\displaystyle
A-\frac{B^{2}}{\left[A-(1-e^{-2v})\frac{u}{v}\right]+\left[2(e^{v}-1)\frac{u}{v}\right]^{-1}}\,,$
(40a) $\displaystyle\tilde{A}_{\rm anc}$ $\displaystyle\equiv$
$\displaystyle\frac{e^{2v}}{\left[A-(1-e^{-2v})\frac{u}{v}\right]^{-1}+\left[2(e^{v}-1)\frac{u}{v}\right]}\,,$
(40b) $\displaystyle\tilde{B}$ $\displaystyle\equiv$
$\displaystyle\frac{Be^{v}}{1+\left[A-(1-e^{-2v})\frac{u}{v}\right]\left[2(e^{v}-1)\frac{u}{v}\right]}\,.$
(40c)

where $A,B$ are as defined in (14). Notice that the wavefunction (39) has a
similar Gaussian form as (13), except now the coefficients of the physical
position $q^{2}$ and ancillary position $q_{\rm anc}^{2}$ terms are no longer
equal, as expected since we have performed an additional squeezing of the
ancillary oscillator. Notice that in the limit of vanishing ancillary
squeezing $r_{\rm anc}\rightarrow 0$, the parameters reduce to their two-mode
squeezing values $\tilde{A},\tilde{A}_{\rm anc}\rightarrow
A,\tilde{B}\rightarrow B$, while in the low-temperature limit $\beta\omega\gg
1$, $A\rightarrow 1,B\rightarrow 0$ as before, so that the physical and
ancillary degrees of freedom decouple.

The wavefunction (39) is again Gaussian in form, and can be written in terms
of the normal mode frequencies

$\Omega_{\pm}=\frac{1}{2}\left(\tilde{A}+\tilde{A}_{\rm
anc}\pm\sqrt{(\tilde{A}-\tilde{A}_{\rm anc})^{2}+4\tilde{B}^{2}}\right)\,.$
(41)

The corresponding complexity, including the ancillary squeezing, is then
Jefferson ; cosmology1

$\mathcal{C}_{\phi,r_{\rm anc},\phi_{\rm
anc}}=\frac{1}{2}\sqrt{\ln^{2}\left|\Omega_{+}\right|+\ln^{2}\left|\Omega_{-}\right|+\arctan^{2}\frac{\text{Im
}\Omega_{+}}{\text{Re }\Omega_{+}}+\arctan^{2}\frac{\text{Im
}\Omega_{-}}{\text{Re }\Omega_{-}}}\,.$ (42)

Figure 3: The purification complexity (42) as a function of the squeezing
angles $(\phi,\phi_{\rm anc})$ for $\beta\omega=-2\ln\tanh 50$ and ancillary
squeezing $r_{\rm anc}=1$. The thick dashed line corresponds to the minimum as
a function of $(\phi,\phi_{\rm anc})$, tracing out a one-parameter curve in
the valley of the complexity. In general, the minimized complexity of (42)
over the free purification parameters $(\phi,r_{\rm anc},\phi_{\rm anc})$ is
also a one-parameter curve, and approaches ${\cal C}_{\rm
th}\approx\frac{\pi}{2\sqrt{2}}$ in the high-temperature limit.

The thermal complexity of purification (2) is now the minimization of (42)
over all three parameters $\phi,r_{\rm anc},\phi_{\rm anc}$. Unfortunately,
the expression is sufficiently complex that it is difficult to do a purely
analytic minimization. Instead, we note that in the high-temperature limit
$\beta\omega\ll 1$ the Gaussian parameters $\tilde{A},\tilde{A}_{\rm
anc},\tilde{B}$ (40) and normal mode frequencies $\Omega_{\pm}$ (41) become
purely imaginary. Further, we note that, as with the minimization of the
complexity arising from two-mode squeezing (17), the minimum of (42) occurs
when the $\ln|\Omega_{\pm}|$ terms vanish, which together requires
$\Omega_{\pm}=\pm i$. This becomes a condition on the Gaussian parameters

$\tilde{A}+\tilde{A}_{\rm anc}=0\,,\qquad\frac{1}{4}(\tilde{A}-\tilde{A}_{\rm
anc})^{2}+\tilde{B}^{2}=-1\,.$ (43)

For example, the first condition implies that the minima are found as
solutions to the relation

$\tan 2\phi_{\rm anc}+\frac{r_{\rm anc}}{2}\left(1-e^{-2v}\right)\tan
2\phi_{\rm anc}+\frac{\cot 2\phi_{\rm anc}}{r_{\rm anc}(e^{v}-1)}=0\,.$ (44)

Together, the conditions (43) impose two constraints on our three free
parameters $\phi,r_{\rm anc},$ and $\phi_{\rm anc}$, implying that generically
there is a one-parameter set of minima. An example is shown in Figure 3, where
a valley of minimum complexity for $r_{\rm anc}=1$ is found as a curve in the
$(\phi,\phi_{\rm anc})$ parameter space. The corresponding complexity, in the
high-temperature limit, is the same as for the two-mode squeezing

$\displaystyle{\cal C}_{\rm th}=\min_{\phi,r_{\rm anc},\phi_{\rm
anc}}\mathcal{C}_{\phi,r_{\rm anc},\phi_{\rm anc}}=\frac{\pi}{2\sqrt{2}}\,.$
(45)

As discussed above, in the low-temperature limit $\beta\omega\gg 1$ the
physical and ancillary degrees of freedom decouple, so the extra squeezing of
the ancillary degrees of freedom becomes irrelevant.

Altogether, we have found that generalizing the purification of the thermal
density matrix to include an additional squeezing of the ancillary degrees of
freedom as in (33) leads to the _same_ minimized complexity of purification.
In particular, the minimized complexity of purification _saturates_ to a
constant value (45) at high temperature, and decays to zero at low
temperature. The additional degrees of freedom added lead to a one-parameter
set of such minima, as opposed to the single $\phi=\pi/4$ minimized complexity
of purification as with the pure two-mode squeezing. Because the additional
purification parameters $r_{\rm anc},\phi_{\rm anc}$ do not change the value
of the minimized complexity of purification, it seems that the additional
freedom (and difficult) arising from their addition is undesirable. However,
there may be situations where boundary conditions do not permit a squeezing
angle $\phi=\pi/4$, so that the additional freedom of (44) is useful.
Nevertheless, the conceptual picture that arises from both the two-mode
purification (13) and the additional ancillary squeezing (33) for the thermal
complexity of purification is the same: at low temperatures, the complexity
vanishes exponentially as a function of the inverse temperature, while it
saturates at high temperature to a constant value, becoming independent of the
temperature.

###  2.3 Complexity from Operator-State Mapping

An alternative approach to assigning a pure state to the thermal density
matrix $\hat{\rho}_{\rm th}$ (1) is the technique of _operator-state mapping_
(also known as _channel-state mapping_) CHOI1975285 ; JAMIOLKOWSKI1972275 .
Consider an operator on our physical oscillator Hilbert space ${\cal H}$ with
representation $\hat{\mathcal{O}}=\sum_{m,n}{\mathcal{O}}_{mn}|n\rangle\langle
m|$. In its simplest form, the mapping associates a state
$|{\mathcal{O}}\rangle$ to $\hat{\mathcal{O}}$ by flipping the bra to a ket,

$\displaystyle\hat{\mathcal{O}}=\sum_{m,n}{\mathcal{O}}_{mn}|n\rangle\langle
m|\hskip 14.45377pt\longleftrightarrow\hskip
14.45377pt|{\mathcal{O}}\rangle=\frac{1}{\sqrt{\text{Tr}[\mathcal{O}^{\dagger}\mathcal{O}]}}\sum_{m,n}\mathcal{O}_{mn}|m\rangle\otimes|n\rangle_{\rm
anc}\,.$ (46)

The state $|{\mathcal{O}}\rangle$ exists on the doubled Hilbert space
${\mathcal{H}}\otimes{\mathcal{H}}_{\rm anc}$, where again we denoted the
extra copy of ${\mathcal{H}}$ as ${\mathcal{H}}_{\rm anc}$ to distinguish it
from the original. The process of operator-state mapping is superficially
similar to purification studied in the previous subsection, in that both
associate to the operator a state on a doubled Hilbert space. One of the most
important differences, however, is that the state $|\mathcal{O}\rangle$ in
(46) associated to the operator $\hat{\mathcal{O}}$ is _unique_ – there are no
free parameters introduced in the mapping (indeed, this is essential for the
mapping to be an isomorphism) – as compared to the purification (4), which
introduces a squeezing angle $\phi$ that plays an important role in the
minimization of the complexity. The complexity associated with the operator-
state mapping, in contrast, does not require a minimization over parameters.

For the thermal density matrix (1) we can then directly associate to it the
two-mode squeezed state

$\displaystyle\hat{\rho}_{\rm
th}=\left(1-e^{-\beta\omega}\right)\sum_{n}e^{-n\beta\omega}|n\rangle\langle
n|\hskip 14.45377pt\longleftrightarrow\hskip 14.45377pt|\rho_{\rm
th}\rangle=\frac{1}{\cosh^{2}r}\sum_{n=0}^{\infty}(\tanh r)^{2n}\
|n\rangle\otimes|n\rangle_{\rm anc},\hskip 14.45377pt$ (47)

where $\tanh r=e^{-\beta\omega}$ is the two-mode squeezing parameter written
in terms of the temperature as before. It is now straightforward to write
$|\hat{\rho}_{\rm th}\rangle$ in position space

$\begin{split}\rho(q,q_{\rm anc})=\langle q,q_{\rm anc}|\rho_{\rm
th}\rangle=\frac{(\omega/\pi)^{1/2}}{\sqrt{\cosh
2r}}\exp\left(-\frac{\omega}{2}{\mathcal{A}}(r)\left(q^{2}+{q}_{\rm
anc}^{2}\right)+\omega{\mathcal{B}}(r)\,q\,q_{\rm anc}\right)\,,\end{split}$
(48)

where

${\mathcal{A}}(r)=\frac{1+\tanh^{4}r}{1-\tanh^{4}r}\,,\qquad{\mathcal{B}}(r)=\frac{2\tanh^{2}r}{1-\tanh^{4}r}\,.$
(49)

The corresponding operator-state thermal complexity of (47) relative to the
ground state now follows directly from Appendix A with a vanishing squeezing
angle $\phi=0$

$\displaystyle{\mathcal{C}}_{\rm os}\left[\hat{\rho}_{\rm
th}\right]=\frac{1}{\sqrt{2}}\ln\left(\frac{1+\tanh^{2}r}{1-\tanh^{2}r}\right)=\frac{1}{\sqrt{2}}\ln\left(\frac{1+e^{-\beta\omega}}{1-e^{-\beta\omega}}\right)\,.$
(50)

For low-temperature $\beta\omega\gg 1$,(50) vanishes as ${\mathcal{C}}_{\rm
os}[\hat{\rho}_{\rm th}]\sim e^{-\beta\omega}$, while for high temperatures
$\beta\omega\rightarrow 0$ the complexity grows as ${\mathcal{C}}_{\rm
os}[\hat{\rho}_{\rm th}]\sim 2\ln(T/\omega)$. The operator-state thermal
complexity (50) thus does not saturate at high temperatures, as compared to
the thermal complexity of purification (20). The reason is clear: the
operator-state mapping (47) does not allow for the inclusion of a squeezing
angle, so the state $|\rho_{\rm th}\rangle$ inherits a squeezing angle
$\phi=0$ from the thermal density matrix. The operator-state thermal
complexity is not identical to the $\phi=0$ complexity of purification (17),
however, since a factor of $\tanh^{2}r$ appears in (47) instead of a factor of
$\tanh r$, as in (5). This leads to a faster decay of the complexity
${\mathcal{C}}_{\rm os}\sim e^{-\beta\omega}$ at low-temperature as compared
to the thermal complexity of purification.

An alternative method of computing the complexity of a mixed state density
matrix such as $\rho_{\rm th}$ is to work with the square root of the thermal
density matrix $\hat{\rho}_{\rm th}$ as in haqueReducedDensityMatrix ,
assigning to it the state

$\displaystyle|\rho_{\rm th}^{1/2}\rangle=\sum_{n}\left(\rho_{\rm
th}^{1/2}\right)_{nn}|n\rangle\otimes|n\rangle_{\rm anc}\,,$ (51)

where $\left(\rho_{\rm th}^{1/2}\right)_{nn}=\langle n|\rho_{\rm
th}^{1/2}|n\rangle$ are the matrix elements of $\hat{\rho}_{\rm th}^{1/2}$.
The original operator is then obtained as a trace over the ancillary degrees
of freedom; for example,

$\displaystyle\hat{\rho}_{\rm th}$ $\displaystyle=$ $\displaystyle{\rm
Tr}_{\rm anc}\left[|\rho_{\rm th}^{1/2}\rangle\langle\rho_{\rm
th}^{1/2}|\right]={\rm Tr}_{\rm anc}\left[\sum_{m,n}\left(\rho_{\rm
th}^{1/2}\right)_{nn}\left(\rho_{\rm th}^{1/2}\right)_{mm}|n\rangle\langle
m|\otimes|n\rangle_{\rm anc}\langle m|_{\rm anc}\right]$ (52) $\displaystyle=$
$\displaystyle\sum_{n}\left(\rho_{\rm th}\right)_{nn}|n\rangle\langle n|\,.$
(53)

In order to calculate the complexity of the state (51), we again write it as a
Gaussian wavefunction in position-basis

$\displaystyle\langle q,q_{\rm anc}|\rho_{\rm th}^{1/2}\rangle=\rho_{\rm
th}^{1/2}(q,q_{\rm
anc})={\tilde{\mathcal{N}}}\exp\left(-\frac{\omega\alpha}{2}\left(q^{2}+{q}_{\rm
anc}^{2}\right)+\omega\gamma\,q\,q_{\rm anc}\right)\,.$ (54)

where now

$\displaystyle\alpha={\mathcal{A}}(r)+{\mathcal{B}}(r)\,,\qquad\gamma=\sqrt{2{\mathcal{B}}(r)({\mathcal{A}}(r)+{\mathcal{B}}(r))}\,,$
(55)

and ${\mathcal{A}}(r),{\mathcal{B}}(r)$ are the same functions of the
squeezing parameter given in (48). It is straightforward to see that
$\alpha,\gamma$ reduce to the purification quantities $A(r),B(r)$ with
$\phi=0$ from (14), so that the operator-state complexity corresponding to the
state $\hat{\rho}_{\rm th}^{1/2}$ relative to the ground state reduces to

$\mathcal{C}_{\rm os}\left[\hat{\rho}_{\rm
th}^{1/2}\right]=\sqrt{2}\,\text{arctanh}\left(e^{-\beta\omega/2}\right)\,.$
(56)

Thus, while the low-temperature behavior of $\mathcal{C}_{\rm
os}\left[\hat{\rho}_{\rm th}^{1/2}\right]\sim e^{-\beta\omega/2}$ matches that
of the minimal complexity of purification, the high-temperature behavior
$\mathcal{C}_{\rm os}\left[\hat{\rho}_{\rm th}^{1/2}\right]\sim\ln(T/\omega)$
grows with temperature, rather than saturating.

Figure 4 compares the results for the complexity of the thermal density matrix
$\hat{\rho}_{\rm th}$ for the different techniques we have used in this
section: the minimal complexity of purification ${\mathcal{C}}_{\rm th}$ (20),
the operator-state complexity associated with the density matrix
${\mathcal{C}}_{\rm os}[\hat{\rho}_{\rm th}]$ (50), and the operator-state
complexity associated with the square root of the thermal density matrix
${\mathcal{C}}_{\rm os}[\hat{\rho}_{\rm th}^{1/2}]$ (56). Each technique gives
a complexity that vanishes exponentially with the temperature at low
temperatures, while at high temperatures their behaviours differ: the minimal
complexity of purification saturates at high temperatures, while the operator-
state complexities grow as $\ln(T/\omega)$. Because of the latter behavior,
the operator-state complexity diverges in the infrared logarithmically with
the IR cutoff.

Figure 4: A complexity can be associated with a mixed thermal density matrix
$\hat{\rho}_{\rm th}$ through the complexity of purification
${\mathcal{C}}_{\rm th}$ (2), operator-state complexity of the density matrix
${\mathcal{C}}_{\rm os}[\hat{\rho}_{\rm th}]$ (50), and the operator-state
complexity of the square root of the density density matrix
${\mathcal{C}}_{\rm os}[\hat{\rho}_{\rm th}^{1/2}]$ (56). As also discussed in
the previous section, the complexity of purification saturates at high
temperatures, while the operator-state complexities grow logarithmically with
temperature $\sim\ln(T/\omega)$.

Qualitatively, each of the states $|\Psi\rangle_{\phi}$ (4), $|\rho_{\rm
th}\rangle$ (47) and $|\rho_{\rm th}^{1/2}\rangle$ reproduce the thermal
expectation values in the physical Hilbert space when traced over the
additional degrees of freedom, so which complexity should we associate with
the thermal density matrix $\hat{\rho}_{\rm th}$? Viewing complexity as the
minimization of the number of gates (“circuit depth”) needed to construct the
target state from a given reference state, the complexity of purification
allows us to construct a state, namely $|\Psi\rangle_{\phi=\pi/4}$, that
reproduces all of our physical observables with a smaller number of gate
resources than the other techniques. In contrast, because it maps to a unique
target state, the operator-state complexity for either $\hat{\rho}_{\rm th}$
or $\hat{\rho}_{\rm th}^{1/2}$ picks out a unique path in state space, which
has no particular reason to be minimal (and in fact we find it is not
minimal). Thus, from the perspective of resource management, the complexity of
purification allows us to simulate the mixed state as a pure state with a
smaller number of resources.

##  3 Thermal Complexity in Curved Spacetimes

In the previous section we have shown how the complexity for a thermal density
matrix of a harmonic oscillator can be obtained by calculating the complexity
of a corresponding two-mode squeezed state parameterized by a squeezing
parameter (fixed by the temperature) and a squeezing angle. Since the thermal
density matrix is independent of the squeezing angle, we minimized the
complexity with respect to the squeezing angle and found that the minimum
complexity corresponds to a non-zero squeezing angle.

The thermal density matrix of a harmonic oscillator arises as a simple model
in many different contexts; here, we would like to explore the complexity of
thermal density matrices that arise naturally when considering quantum fields
on curved spacetimes.

###  3.1 Complexity and Hawking Radiation

Thermal density matrices naturally arise when considering quantum fields on
spacetimes with horizons through the Unruh effect and the generation of
Hawking radiation, so these are ideal backgrounds to consider applications of
the thermal complexity of purification. See BirrellDavies ; Ford:1997hb ;
Jacobson:2003vx ; UnruhEffectReview ; krishnan2010quantum ; Polchinski_2016
for some useful reviews.

As an illustrative example that contains most of the relevant details, let us
begin by considering $(1+1)$-dimensional Minkowski space as seen by a
uniformly accelerating observer. The metric can be written

$\displaystyle ds^{2}=-dt^{2}+dx^{2}=-du\ dv=e^{2a\xi}(d\tau^{2}-d\xi^{2})\,,$
(57)

where $u=t-x,v=t+x$ are Minkowski light-cone coordinates and $(\tau,\xi)$
defined by

$\displaystyle t=\frac{1}{a}e^{a\xi}\sinh(a\tau)\,,\hskip
14.45377ptx=\frac{1}{a}e^{a\xi}\cosh(a\tau)$ (58)

are coordinates adapted to a uniformly accelerating observer with acceleration
$a$ along $\xi=0$. The coordinates $(\tau,\xi)$ in (58) only cover part of the
original Minkowski space, the so-called right $\rm R$ Rindler wedge $|x|>t$; a
similar definition is needed, with additional minus signs, for the $\rm L$
wedge. An interesting and important feature of Rindler coordinates is that
$u=0$ is a future horizon for an observer traveling along a constant $\xi$
line in the $\rm R$ wedge ($v=0$ is correspondingly a past horizon), so that
the $\rm R$ and $\rm L$ patches are causally disconnected from each other. See
Figure 5(a) for a spacetime diagram of Minkowski space and ${\rm R,L}$ Rindler
wedges.

A massless scalar field $\phi$ on this spacetime can be expanded in Minkowski
$(t,x)$ plane wave modes

$\displaystyle\hat{\phi}=\sum_{k}\left(u_{k}^{M}\ \hat{a}_{k}+u_{k}^{M*}\
\hat{a}_{k}^{\dagger}\right)$ (59)

where the Minkowski vacuum is defined by the annihilation operator
$\hat{a}_{k}|0\rangle_{M}=0$. We can also expand the scalar field in modes
adapted to Rindler $(\tau,\xi)$ coordinates

$\displaystyle\hat{\phi}=\sum_{k}\left(v_{k}^{\rm R}\ \hat{b}_{k}^{\rm
R}+v_{k}^{\rm L}\ \hat{b}_{k}^{\rm L}+v_{k}^{\rm R*}\ \hat{b}_{k}^{\rm
R\dagger}+v_{k}^{\rm L*}\ \hat{b}_{k}^{\rm L\dagger}\right)$ (60)

where $\hat{v}^{\rm R,L}_{k}$ are only non-zero in the ${\rm R,L}$ wedges,
respectively, and the Rindler vacuum can be written as the direct product of
vacuum states on the left and right wedges $|0\rangle_{\rm
R}\otimes|0\rangle_{\rm L}$, which are annihilated by the left- and right-
Rindler annihilation operators, $\hat{b}_{k}^{\rm R}|0\rangle_{\rm
R}=0=\hat{b}_{k}^{\rm L}|0\rangle_{\rm L}$.

(a) Rindler spacetime split into ${\rm R}$ and ${\rm L}$ wedges.

(b) Kruskal spacetime for a black hole.

Figure 5: Spacetime diagrams for Rindler and Kruskal spaces.

The Minkowski and Rindler vacuum states are not equivalent; instead, the
Minkowski vacuum of a single mode $k$ of the scalar field $\phi$ as seen in
the basis of Rindler modes (60) takes the form of an two-mode squeezed state
entangling the ${\rm L}$ and ${\rm R}$ modes UnruhEffectReview ;
krishnan2010quantum , which we will call the “Unruh” state

$\displaystyle|0\rangle_{M}=\left(1-e^{-2\pi
k/a}\right)^{1/2}\sum_{n_{k}=0}^{\infty}e^{-\frac{n\pi k}{a}}\
|n_{k}\rangle_{\rm R}\otimes|n_{k}\rangle_{\rm L}\,.$ (61)

Comparing (61) with (5), the squeezing parameter is $\tanh r=e^{-\pi k/a}$
with squeezing angle $\phi=\pi/2$. The corresponding complexity of this Unruh
state relative to a Rindler ground state $|0\rangle_{\rm
R}\otimes|0\rangle_{\rm L}$ is

$\displaystyle{\mathcal{C}}_{\rm
Unruh}(k)=\frac{1}{\sqrt{2}}\ln\left(\frac{1+e^{-\frac{\pi
k}{a}}}{1-e^{-\frac{\pi k}{a}}}\right)\,,$ (62)

and grows logarithmically ${\mathcal{C}}_{\rm Unruh}\sim r\sim\ln(a/k)$ for
large accelerations $a/k\gg 1$. Interestingly, the complexity (62) of the
Unruh two-mode squeezed state is _maximal_ with respect to the squeezing angle
– that is, for fixed acceleration, the squeezing angle of the squeezed state
(61) chooses the largest possible complexity. We will return to this
observation at the end of this section.

A Rindler observer in the $\rm R$-wedge sees a density matrix with the
unobservable $\rm L$ modes traced out

$\displaystyle\hat{\rho}_{\rm R}={\rm Tr}_{\rm L}\left[|0\rangle_{M}\langle
0|_{M}\right]=\left(1-e^{-2\pi k/a}\right)\sum_{n_{k}}e^{-2\pi n_{k}k/a}\
|n_{k}\rangle_{\rm R}\ \langle n_{k}|_{\rm R}\,,$ (63)

which is identified as a thermal density matrix of Hawking radiation with
temperature $T=a/2\pi$. The thermal complexity of purification from Section
2.1 assigns a minimal purification to (63)

$\displaystyle|\Psi\rangle_{\rm Hawk,p}=\left(1-e^{-2\pi
k/a}\right)^{1/2}\sum_{n_{k}=0}^{\infty}(-1)^{n_{k}}e^{-2in_{k}\phi}e^{-\frac{\pi
n_{k}k}{a}}\ |n_{k}\rangle_{\rm R}\ \otimes|n_{k}\rangle_{\rm L^{\prime}}\,,$
(64)

with the same squeezing parameter $r$ as with the Unruh state, but now with
squeezing angle $\phi=\pi/4$ as determined by the minimization of the
complexity of purification. Note that the purification includes an ancillary
Hilbert space, which we will denote as $\rm L^{\prime}$ due to its similarity
with the $\rm L$ wedge of Rindler space. Comparing the Unruh squeezed state
(61) and the purification of the Hawking radiation (64), these two
purifications differ only by the value of the squeezing angle $\phi$: The
Unruh squeezed state (61) has $\phi=\pi/2$, which as discussed in Section 2.1
maximizes the corresponding complexity for fixed acceleration, while the
minimization arising from the thermal complexity of purification selects
$\phi=\pi/4$. The minimized complexity of purification for the Hawking thermal
density matrix (63) is instead

$\displaystyle{\mathcal{C}}_{\rm
Hawk,p}(k)=\frac{1}{\sqrt{2}}\left|\arctan\left(2\frac{e^{-\pi
k/a}}{1-e^{-2\pi k/a}}\right)\right|\,,$ (65)

which saturates to ${\mathcal{C}}_{\rm Hawk,p}\sim\pi/(2\sqrt{2})$ for large
accelerations $a/k\gg 1$. The minimization imposed by the thermal complexity
of purification leads to a different qualitative behavior of the complexity of
the Hawking radiation as a function of the acceleration, compared to the
complexity of the full Unruh squeezed state. Nevertheless, because the
purification process preserves expectation values, both purifications (61),
(64) of the thermal density matrix (63) are equivalent to an observer in the
$R$ wedge. Thus, a $\rm R$ wedge observer can reconstruct thermal expectation
values by using the purification (64), which is easier to build (in that it
has a smaller complexity) compared to the Unruh squeezed state (61).

The role of the squeezing angle arises in the analytic continuation of
positive frequency mode functions from the $\rm R$ wedge to the $\rm L$ wedge.
To see this, we expand the global scalar field in a set of different positive
frequency mode functions in Minkowski space

$\displaystyle\hat{\phi}=\sum_{k}\left(\bar{U}_{k}^{I}\hat{c}_{k}^{I}+\bar{U}_{k}^{II}\hat{c}_{k}^{II}+\bar{U}_{k}^{I*}\hat{c}_{k}^{I\dagger}+\bar{U}_{k}^{II*}\hat{c}_{k}^{II\dagger}\right)\,$
(66)

where the modes $\bar{U}_{k}^{I},\bar{U}_{k}^{II}$ are defined as combinations
of the Rindler mode functions that have positive frequency in Minkowski
coordinates. This corresponds to a Bogoliubov transformation between the
Minkowski modes $\hat{c}_{k}^{I},\hat{c}_{k}^{II}$ and the Rindler modes
$\hat{b}_{k}^{\rm R},\hat{b}_{k}^{\rm L}$

$\displaystyle\hat{c}_{k}^{I}=\sqrt{\frac{e^{\frac{\pi
k}{a}}}{2\sinh(\frac{\pi k}{a})}}\left(\hat{b}_{k}^{\rm R}+e^{-\frac{\pi
k}{a}}\hat{b}_{k}^{\rm L\dagger}\right)\,;\hskip
14.45377pt\hat{c}_{k}^{II}=\sqrt{\frac{e^{\frac{\pi k}{a}}}{2\sinh(\frac{\pi
k}{a})}}\left(\hat{b}_{k}^{\rm L}+e^{-\frac{\pi k}{a}}\hat{b}_{k}^{\rm
R\dagger}\right)\,.$ (67)

The factors of $e^{-\frac{\pi\omega}{a}}$ arise from the analytic continuation
of the positive frequency mode functions from the $\rm R$ wedge to the $\rm L$
wedge, and give rise to the temperature as seen by an $\rm R$ wedge Rindler
observer.

As reviewed in Appendix B, the Bogoliubov transformations (67) can be seen as
a special case of a two-mode squeezing transformation

$\displaystyle\hat{c}_{k}^{I}=\sqrt{\frac{e^{\frac{\pi
k}{a}}}{2\sinh(\frac{\pi k}{a})}}\left(\hat{b}_{k}^{\rm
R}-e^{2i\phi}e^{-\frac{\pi k}{a}}\hat{b}_{k}^{\rm
L^{\prime}\dagger}\right)\,;\hskip
14.45377pt\hat{c}_{k}^{II}=\sqrt{\frac{e^{\frac{\pi k}{a}}}{2\sinh(\frac{\pi
k}{a})}}\left(\hat{b}_{k}^{\rm L^{\prime}}-e^{2i\phi}e^{-\frac{\pi
k}{a}}\hat{b}_{k}^{\rm R\dagger}\right)\,,$ (68)

with a squeezing angle $\phi=\pi/2$ that ensures continuity of the
$\bar{U}_{k}^{I},\bar{U}_{k}^{II}$ mode functions in Minkowski space across
the $u=0=v$ boundary between the $\rm R$ and $\rm L$ wedges. Interpreting the
$L^{\prime}$ wedge in a similar geometric way, an arbitrary squeezing angle
$\phi$ for the purification (64) would instead imply a phase discontinuity
when crossing the Rindler horizon, which can be absorbed in the Rindler mode
function behind the horizon, $v_{k}^{\rm L^{\prime}}\rightarrow v_{k}^{\rm
L^{\prime}}e^{2i\phi}$. The minimized squeezing angle $\phi=\pi/4$ arising
from the complexity of purification thus results in a purely imaginary phase.
It is interesting that this phase in the Bogoliubov transformations (68) has
qualitatively different effects on the complexity of the resulting squeezed
state, and it would be interesting to see whether this phase has any other
physical effects.

Other spacetimes with horizons also exhibit the Unruh effect, for similar
conceptual and technical reasons. For example, consider a black hole
spacetime, as shown in the Kruskal diagram of Figure 5(b). Ignoring the
angular directions and treating spacetime as $(1+1)$-dimensional, the metric
is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2GM}{r}\right)dt^{2}+\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}=-\left(1-\frac{2GM}{r}\right)du\
dv$ (69) $\displaystyle=$ $\displaystyle-\frac{16G^{2}M^{2}}{r}e^{-r/2GM}\ dU\
dV$

where we introduced the lightcone coordinates

$\displaystyle u=t-r_{*}=-4GM\ln(-U/2GM),\hskip
14.45377ptv=t+r_{*}=4GM\ln(V/2GM)$ (70)

where $r_{*}=r+2GM\ln(r-2GM)$. A massless scalar field can be expanded in
modes

$\displaystyle\hat{\phi}$ $\displaystyle=$ $\displaystyle\sum_{k}\left(u_{k}\
\hat{a}_{k}+u_{k}^{*}\hat{a}_{k}^{\dagger}\right)$ (71) $\displaystyle=$
$\displaystyle\sum_{k}\left(v_{k}^{\rm I}\ \hat{b}_{k}^{\rm I}+v_{k}^{\rm II}\
\hat{b}_{k}^{\rm II}+v_{k}^{\rm I*}\hat{b}_{k}^{\rm I\dagger}+v_{k}^{\rm
II*}\hat{b}_{k}^{\rm II\dagger}\right)$ (72)

where the $u_{k}$ modes are defined with respect to the global $(U,V)$
coordinates and define the vacuum in the asymptotic past
$\hat{a}_{k}|0\rangle_{\rm past}=0$, and the $v_{k}^{\rm I,II}$ modes are
defined with respect to the $(u,v)$ coordinates in the ${\rm I,II}$ patches
and define the vacuum seen by an observer outside the black hole in the
asymptotic future $\hat{b}_{k}^{\rm I}|0\rangle_{\rm I}=0$ and the internal
vacuum with respect to the modes inside the black hole $\hat{b}_{k}^{\rm
II}|0\rangle_{\rm II}=0$. As in Rindler space, these modes are mixed by a
Bogoliubov transformation (again, see BirrellDavies ; Ford:1997hb ;
Jacobson:2003vx ; UnruhEffectReview ; krishnan2010quantum ; Polchinski_2016
for reviews), so that the past vacuum can be written as a squeezed state with
$\phi=\pi/2$, entangling the external and internal modes with each other as an
Unruh state

$\displaystyle|0\rangle_{\rm past}$ $\displaystyle=$
$\displaystyle{\mathcal{N}}_{k}\ {\rm exp}\left(e^{-4\pi GMk}\hat{b}_{k}^{\rm
I\dagger}\hat{b}_{k}^{\rm II\dagger}\right)|0\rangle_{\rm
I}\otimes|0\rangle_{\rm II}$ (73) $\displaystyle=$
$\displaystyle{\mathcal{N}}_{k}\ \sum_{n_{k}}e^{-4\pi GMk\ n_{k}}\
|n_{k}\rangle_{\rm I}\otimes|n_{k}\rangle_{\rm II}\,,$

with corresponding complexity

$\displaystyle{\mathcal{C}}_{\rm
Unruh}(k)=\frac{1}{\sqrt{2}}\ln\left(\frac{1+e^{-4\pi GMk}}{1-e^{-4\pi
GMk}}\right)\,.$ (74)

As with Rindler space, the density matrix as seen by an external observer in
the asymptotic future is obtained from (73) by tracing out the modes inside
the horizon, leading to a thermal density matrix with temperature
$T_{BH}=(8\pi GM)^{-1}$ corresponding to Hawking radiation of the scalar field
$\phi$. The corresponding thermal purification of complexity associates a
squeezed state to this density matrix, entangling the external modes
associated with ${\rm I}$ with ancillary degrees of freedom on a space ${\rm
II^{\prime}}$ with squeezing angle $\phi=\pi/4$, leading to the purification
complexity

$\displaystyle{\mathcal{C}}_{\rm
Hawk,p}(k)=\frac{1}{\sqrt{2}}\left|\arctan\left(2\frac{e^{-4\pi
GMk}}{1-e^{-8\pi GMk}}\right)\right|\,,$ (75)

As before, the complexity (75) associated with the purification of the thermal
density matrix is qualitatively different from that of the global Unruh state
complexity (74), particularly at low-frequencies or small masses, saturating
instead of growing as $GMk\rightarrow 0$.

We have reviewed how quantum field theory on curved spacetimes with a horizon
naturally leads to a Unruh two-mode squeezed state entangling degrees of
freedom on either side of the horizon through the Unruh effect, with an
associated complexity. Interestingly, the complexity of this Unruh squeezed
state is maximal with respect to the squeezing angle. In particular, the
squeezing angle that naturally arises in the Bogoliubov transformation of the
Unruh state gives the largest complexity for a given squeezing, such that the
complexity grows with the squeezing ${\mathcal{C}}_{\rm Unruh}\sim r$, which
itself is an increasing function of the acceleration (for Rindler spacetimes)
or the black hole mass. For a black hole, this agrees with expectations from
other work that black holes are maximally chaotic quantum systems kitaev ;
FastScramblers ; Maldacena2016-mb , and perhaps similar statements apply to
Rindler space as well. It is therefore interesting to speculate that the
complexity for the Unruh black hole state is maximal because of some deeper
principle that applies more generally, which may take the form of a tendency
for the field configuration to maximize the complexity in a kind of second law
of complexity, similar to entropy. At least in our simple model of a scalar
field on curved backgrounds, this seems to be the case.

However, an observer outside the horizon sees a thermal density matrix of
Hawking radiation, obtained by tracing out over the internal modes. The
thermal complexity of purification from Section 2 then associates an ancillary
two-mode squeezed state description to the Hawking radiation. The purified
Hawking radiation state takes a similar form as the Unruh squeezed state, but
with a squeezing angle that instead minimizes the complexity at a constant
value ${\mathcal{C}}_{\rm Hawk,p}\sim\pi/(2\sqrt{2})$. This implies that for
an observer outside of the horizon, it is easier to build the Hawking
radiation as a two-mode squeezed state entangling the Hawking radiation with
ancillary degrees of freedom at a particular squeezing angle $\phi=\pi/4$ that
differs from the Unruh state. For a black hole of mass $M$, the difference
between the maximal Unruh complexity and the minimal purification complexity
for fixed frequency $k$ is only weakly dependent on the mass of the black
hole, ${\mathcal{C}}_{\rm Unruh}/{\mathcal{C}}_{\rm p}\sim\ln(GMk)$. However,
including many frequencies $k$ this effect can potentially become an important
effect.

Finally, as discussed in Section 2.1, note that the Unruh state (61) and the
purification of the Hawking radiation (64) have identical entanglement
entropies that only depend on the temperature and frequency through $\beta k$

$\displaystyle S_{\rm en}=\ln\left(1-e^{-\beta k}\right)-\frac{e^{-\beta
k}}{1-e^{-\beta k}}\ln\left(e^{-\beta k}\right)\,.$ (76)

Thus, while there are several ways that an observer outside the horizon can
construct a pure state representing the Hawking radiation with identical
entanglement entropies (parameterized by different values of the squeezing
angle $\phi$), the purification (64) is minimal with respect to the complexity
of building the state, by a factor $\sim\ln(GMk)$ compared to the usual Unruh
state (61) for black holes. It would be interesting to consider whether this
difference has an impact on the information contained in the Hawking radiation
in a more robust treatment. We leave these interesting questions for future
work.

###  3.2 Complexity and Cosmological Perturbations

Squeezed states and thermal density matrices also arise in the quantum
description of cosmological perturbations, in which the time-dependence of the
metric induces the pair creation of particles from the vacuum.

We will briefly review the description of cosmological perturbations as
squeezed states; see Mukhanov ; Grishchuk ; Albrecht ; Martin1 ; Martin2 ;
cosmology1 ; cosmology2  for more details. Our metric is the spatially flat
Friedmann-Lemaitre-Robertson-Walker (FLRW) metric

$ds^{2}=-dt^{2}+a(t)^{2}d\vec{x}^{2}=a(\eta)^{2}\left(-d\eta^{2}+d\vec{x}^{2}\right)\,.$
(77)

The Hubble expansion rate of this background is denoted by $H=\dot{a}/a$,
where a dot denotes a derivative with respect to cosmic time $t$. On this
background we will consider fluctuations of a scalar field, which combine with
fluctuations of the metric to form the gauge-invariant curvature perturbation
${\mathcal{R}}$. Written in terms of the Mukhanov-Sasaki variable $v\equiv
z{\mathcal{R}}$ where $z\equiv a\,\sqrt{2\epsilon}$, with
$\epsilon=-\dot{H}/H^{2}=1-{\mathcal{H}}^{\prime}/{\mathcal{H}}^{2}$, the
action takes the simple form

$\displaystyle S=\frac{1}{2}\int d\eta\,d^{3}x\left[v^{\prime
2}-(\partial_{i}v)^{2}+\left(\frac{z^{\prime}}{z}\right)^{2}v^{2}-2\frac{z^{\prime}}{z}v^{\prime}v\right]\,.$
(78)

where a prime denotes a derivative with respect to conformal time and
${\mathcal{H}}=a^{\prime}/a$. This action represents perturbations of a free
scalar field coupled to an external time-varying source. Promoting the
perturbation to a quantum field and expanding into creation and annihilation
modes

$\displaystyle\hat{v}(\eta,\vec{x})=\int\frac{d^{3}k}{(2\pi)^{3/2}}\frac{1}{\sqrt{2k}}\left(\hat{c}_{\vec{k}}e^{i\vec{k}\cdot\vec{x}}+\hat{c}_{-\vec{k}}^{\dagger}e^{i\vec{k}\cdot\vec{x}}\right)\,\,,$
(79)

the Hamiltonian can be written as

$\displaystyle\hat{H}=\int d^{3}k\,\hat{\mathcal{H}}_{\vec{k}}=\int
d^{3}k\left[k\left(\hat{c}_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}+\hat{c}_{-\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}\right)-i\frac{z^{\prime}}{z}\left(\hat{c}_{\vec{k}}\hat{c}_{-\vec{k}}-\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}\right)\right]\,.$
(80)

The momentum structure of the Hamiltonian indicates that the interaction with
the background leads to particle creation in pairs with opposite momenta.
Because of this, we are naturally led to consider our states as appearing in
two-mode pairs $(\vec{k},-\vec{k})$.

The ground state defined by $\hat{c}_{\vec{k}}|0_{\vec{k}}\rangle=0$ at early
times evolves at late times into the cosmological squeezed state Grishchuk ;
Albrecht

$\displaystyle|\Psi_{\rm cosmo}\rangle_{\vec{k},-\vec{k}}=\frac{1}{\cosh
r_{k}}\sum_{n=0}^{\infty}(-1)^{n}e^{-2in\varphi_{k}}\tanh^{n}r_{k}\,|n_{\vec{k}},n_{-\vec{k}}\rangle\,.$
(81)

The squeezing parameter $r_{k}$ and squeezing angle $\phi_{k}$ depend on time
through the time-dependence of the scale factor Albrecht ; cosmology1 ;
cosmology2

$\displaystyle\frac{dr_{k}}{da}$ $\displaystyle=$
$\displaystyle-\frac{1}{a}\cos(2\phi_{k})\,,$ (82)
$\displaystyle\frac{d\phi_{k}}{da}$ $\displaystyle=$
$\displaystyle\frac{k}{a{\mathcal{H}}}+\frac{1}{a}\coth(2r_{k})\sin(2\phi_{k})\,.$
(83)

The resulting complexity of this squeezed state relative to the ground state
cosmology1 ; cosmology2

$\displaystyle{\mathcal{C}}_{\rm
cosmo}=\frac{1}{\sqrt{2}}\sqrt{\left|\ln\left|\frac{1+e^{-2i\phi_{k}}\ \tanh
r_{k}}{1-e^{-2i\varphi_{k}}\ \tanh r_{k}}\right|\right|^{2}+\arctan\left(2\sin
2\varphi_{k}\,\sinh r_{k}\,\cosh r_{k}\right)^{2}}\,.$ (84)

also depends on time through the time-dependent squeezing parameter and
squeezing angle $r(\eta),\varphi(\eta)$. For example, following cosmology1  we
consider a simple model of the very early Universe in which the expanding
background starts as de Sitter (inflation) with Hubble constant $H_{dS}$, then
transitions to a radiation-dominated expansion. For a fixed comoving
wavelength $k$ starting inside the horizon at early times $k\gg aH_{dS}$, the
squeezing parameter is small $r_{k}(a)\approx aH_{dS}/k\ll 1$ and the
squeezing angle is constant $\varphi_{k}\approx-\pi/4$; the corresponding
complexity (84) is small, as the state (81) is still very nearly identical to
the ground state. As the accelerating universe expands, the mode eventually
exits the horizon. After horizon exit, the squeezing parameter begins to grow
as the log of the scale factor $r_{k}\approx\ln a$ and the squeezing angle
shifts to $\varphi\approx-\pi/2$, up to subleading corrections. The
corresponding complexity also grows as the log of the scale factor at this
time ${\mathcal{C}}_{\rm cosmo}\approx\ln(a)/\sqrt{2}$. As the Universe
transitions to radiation-domination, the squeezing parameter continues to grow
as the log of the scale factor, but the squeezing angle now is repelled away
from $\varphi\approx-\pi/2$ towards positive values. Curiously, even though
the squeezing parameter is growing, the growing squeezing angle causes the
complexity to _decrease_ during radiation domination333There is a slight delay
to the onset of de-complexification after radiation domination as the slope of
the squeezing angle gradually changes sign, as discussed in cosmology1 .,
leading to a period of _de-complexification_. At late times the mode re-enters
the horizon and the squeezing parameter “freezes in” to a constant
$r_{k}\approx r_{*}$ determined by horizon crossing. The squeezing angle when
the mode re-enters the horizon becomes large and $k$-dependent, so that
$e^{-2\varphi_{k}}$ oscillates rapidly. The resulting complexity also “freezes
in” to a constant value, up to oscillations, at late times. These behaviors
for the squeezing parameter $r_{k}$, squeezing angle $\varphi_{k}$ (through
$\cos\varphi_{k}$) are shown in Figure 6, and the corresponding complexity
${\mathcal{C}}_{\rm cosmo}$ is shown in Figure 7.

(a) Squeezing parameter $r_{k}$.

(b) Squeezing angle $\phi_{k}$.

Figure 6: The squeezing parameter and squeezing angle of a pure state
cosmological perturbation (81) with fixed $k$ are shown for a simple model of
the early Universe consisting of a period of de Sitter accelerated expansion,
followed by radiation domiation. The squeezing parameter $r_{k}$ grows while
the mode is outside the horizon, while $\phi_{k}\approx-\pi/2$ there. When the
mode eventually re-enters the horizon the squeezing parameter “freezes in”,
while the squeezing angle begins to grow, causing rapid oscillations in
$\cos(2\phi_{k})$. Figure 7: The complexity (84) of a pure squeezed
cosmological perturbation (81) grows while the mode is outside the horizon
during de Sitter accelerated expansion, but _decreases_ during radiation
domination, leading to a period of _de-complexification_. Once the mode re-
enters the horizon, rapid oscillations in $e^{-2i\phi_{k}}$ lead to rapid
oscillations in the complexity.

The complexity associated to the state (81) assumes that the cosmological
perturbations remain in a pure state throughout their evolution on the
classical expanding background, with density matrix

$\displaystyle\hat{\rho}_{\rm
pure}=|\Psi_{sq}\rangle\langle\Psi_{sq}|=\frac{1}{\cosh^{2}r_{k}}\sum_{n,m=0}^{\infty}(-1)^{n+m}e^{-2i(n-m)\varphi_{k}}\tanh^{n+m}r_{k}\,|n_{\vec{k}},n_{-\vec{k}}\rangle\langle
m_{\vec{k}},m_{-\vec{k}}|\,.\hskip 14.45377pt$ (85)

In general, however, we expect the perturbations to experience decoherence at
some point in their evolution, as the quantum excitations freeze-in as
classical density perturbations of the cosmic fluid. There are many
interesting models and descriptions of this process of cosmological
decoherence, see Brandenberger_1992 ; Brandenberger_1993 ; Burgess_2008 ;
Martin_2018 ; Shandera_2018 ; Gong_2019  and references therein for some
examples. Motivated by the rapid $k$-dependent growth of the squeezing angle
$\varphi_{k}$ when the mode re-enters the horizon during the radiation era,
with corresponding rapid oscillation of $e^{-2\varphi_{k}}$ as seen in Figure
6, a simple model for decoherence is to average the density matrix over the
squeezing angle Brandenberger_1992 ; Brandenberger_1993 ; Brahma:2020zpk . The
resulting reduced density matrix has only diagonal entries

$\displaystyle\hat{\rho}_{\rm
red}=\frac{1}{\cosh^{2}r_{k}}\sum_{n=0}^{\infty}\tanh^{2n}r_{k}\,|n_{\vec{k}},n_{-\vec{k}}\rangle\langle
n_{\vec{k}},n_{-\vec{k}}|\,.$ (86)

This reduced density matrix (86) has the form of a thermal density matrix
$\hat{\rho}_{\rm th}$ of a harmonic oscillator with temperature
$\beta=-\ln(\tanh^{2}r_{k})$. For modes inside the horizon the squeezing
parameter is approximately constant $r_{k}\approx r_{*}$, so the resulting
temperature of modes in this mixed state is also constant,
$\beta\approx-\ln(\tanh^{2}r_{*})$. Note, however, that this temperature is
different from the Gibbons-Hawking temperature of de Sitter space $T_{GH}\sim
H/2\pi$ Gibbons:1977mu , as it depends on the evolution history of the mode
and the amount of growth it experiences when it is outside the horizon.

Figure 8: The complexity for cosmological perturbations in the simple
cosmological background of Figures 6,7 with a simple model of decoherence in
which the perturbations are described by the pure state (81) until they re-
enter the horizon in the radiation era, after which they are described by the
mixed state (86) with corresponding complexity (88). We see that all $k$-modes
saturate to ${\mathcal{C}}_{\rm cosmo,p}\approx\pi/(2\sqrt{2})$ upon horizon
re-entry.

Since the reduced density matrix (86) resulting from our simple model of
decoherence is thermal, we can calculate the associated thermal complexity of
purification of the cosmological perturbations. To do this, we expand our
Hilbert space to include an ancillary copy (suppressing the $-\vec{k}$ mode
index), ${\mathcal{H}_{\vec{k}}}\otimes{\mathcal{H}^{\prime}_{\vec{k},{\rm
anc}}}$, resulting in the purification

$\displaystyle|\Psi_{\rm cosmo,p}\rangle_{\vec{k}}=\frac{1}{\cosh
r_{k}}\sum_{n=0}^{\infty}(-1)^{n}e^{-2in\phi}\tanh^{n}r_{k}\,|n_{\vec{k}}\rangle\otimes|n^{\prime}_{\vec{k}}\rangle\,.$
(87)

The purification (87) and the original cosmological squeezed state (81) look
nearly identical. The primary difference is that in the original squeezed
state (81) the squeezing parameter and squeezing angle $\varphi_{k}$ are
determined by the dynamics of the expanding background, while for the
purification (87) the squeezing parameter is still dynamical, in principle,
but the purification squeezing angle $\phi$ is fixed to $\phi=\pi/4$ by the
minimization of the complexity. This modest difference modifies the behavior
of the cosmological complexity in qualitative ways. In particular, the
complexity of the purified state (87) for the minimized value $\phi=\pi/4$
becomes

$\displaystyle{\mathcal{C}}_{\rm
cosmo,p}(k)=\frac{1}{\sqrt{2}}\arctan(\sinh(2r_{k}))\,.$ (88)

As noted before, this complexity has universal behavior as a function of the
squeezing parameter $r_{k}$. In particular, for large squeezing $r_{k}\gg 1$
the complexity saturates ${\mathcal{C}}_{\rm cosmo,p}\approx\pi/(2\sqrt{2})$.
More generally, since this complexity is independent of the dynamical
squeezing angle $\varphi_{k}$ it can have qualitatively different behavior
than the pure state complexity (84) when the reduced density matrix (86) is an
appropriate description.

Figure 9: The complexity for cosmological perturbations with an alternative
model of decoherence in which the perturbations evolve into the mixed state
(86) at the transition between the de Sitter and radiation phases. The
complexity jumps discontinuously to the saturated value ${\mathcal{C}}_{\rm
cosmo,p}\approx\pi/(2\sqrt{2})$ at the moment of decoherence.

Let’s return to the simple model described earlier in which the Universe
begins in a phase of accelerated de Sitter expansion, followed by a period of
radiation domination. A simple model of decoherence of the density
perturbations in this background is to take the density perturbation modes to
be described by the pure state (81) up until horizon re-entry in the radiation
phase, after which the squeezing angle becomes large and $k$-dependent,
triggering the averaging leading to the thermal density matrix (86). The
resulting complexity is shown in Figure 8 for several different wavelengths
$k$, and has the following qualitative features: while the mode is described
by the pure state (81), the complexity follows (84) as seen in Figure 7. Since
each $k$-mode has a slightly different horizon-exit time, the corresponding
total growth in the complexity is different, while the slope is universal (see
cosmology2  for more discussion of the slope of complexity for different
cosmological backgrounds). Upon the transition to radiation-domination the
complexity for all of the $k$-modes begins to decay. As each mode re-enters
the horizon, the corresponding squeezing angle begins to grow rapidly,
triggering our model of decoherence (86),(87). The corresponding complexity
(88) saturates to ${\mathcal{C}}_{\rm cosmo,p}\approx\pi/(2\sqrt{2})$ for each
mode inside the horizon, smoothing out the rapidly oscillating complexity seen
in Figure 7. This simple model suggests that the complexity of each
cosmological wavelength is universal and is equal to the saturated value after
decoherence, once the mode re-enters the horizon.

While the above model of phase decoherence (or “re-entry decoherence”) seems
quite natural, let us briefly consider what the effects on complexity would be
for decoherence of the form (86) occurring at different times. If decoherence
occurs at an earlier time than horizon re-entry, the complexity of the state
will abruptly jump from its current value to the constant ${\mathcal{C}}_{\rm
cosmo,p}\approx\pi/(2\sqrt{2})$. For example, if decoherence of inflationary
density perturbations occurs at the transition between the de Sitter and
radiation phases, the complexity for a given superhorizon mode will jump
discontinuously, as in Figure 9. Naturally, if the decoherence occurs at some
other moment in the cosmological history we expect a corresponding
discontinuous jump in the complexity.

##  4 Discussion

We constructed a purification of a thermal density matrix of a harmonic
oscillator as a generic two-mode squeezed state, where the squeezing is fixed
by the temperature and the squeezing angle is a free parameter. Minimizing the
complexity relative to the ground state with respect to the squeezing angle,
we found that the complexity _saturates_ at high temperatures $\beta\omega\ll
1$ to ${\mathcal{C}}_{\rm th}\rightarrow\frac{\pi}{2\sqrt{2}}\approx 1.1$ with
a non-zero squeezing angle $\phi_{\rm min}=\pi/4$, in contrast to previous
results MyersMixed ; DiGiulio:2020hlz ; Camargo:2020yfv . Since all
purifications of the thermal density matrix result in identical expectation
values, this implies a bound on the complexity of a thermal state
${\mathcal{C}}_{\rm th}\leq{\mathcal{O}}(1)$ for all temperatures and
frequencies. While this geometrized notion of complexity is not identical to a
counting of the number of discrete unitary operators needed to build up the
target state from the reference state, the bound indicates that the number of
such operators needed to construct a minimal purification of a thermal state
is similarly bounded. We considered further purifications, including an
additional squeezing of the ancillary degree of freedom, and demonstrated that
the minimum complexity remains the same and is similarly bounded, although the
additional squeezing allows more freedom and degeneracy in the minimized
purification. We further showed how operator-state mapping, which also
associates a pure state to the thermal density matrix, does not minimize the
complexity because it does not scan over all allowed squeezing angles. The
simplest purification of a thermal state with a minimized complexity is thus a
two-mode squeezed state with squeezing angle $\phi_{\rm min}=\pi/4$.

More generally, by analyzing the complexity of a two-mode squeezed state as a
function of both the squeezing angle and squeezing parameter, we found that
the complexity generically saturates for a non-zero squeezing angle. In
contrast, for squeezing angles $\phi=n\pi/2$ the complexity is maximized and
grows linearly with the squeezing parameter
${\mathcal{C}}_{\phi=n\pi/2}\approx r$ for large $r$. Interestingly, several
solutions of quantum fields on curved spacetimes can also be described as two-
mode squeezed states with particular values of the squeezing angle. For
example, vacuum states of a scalar field on a curved background with a horizon
in one basis, such as Rindler space or a black hole spacetime, can naturally
be described as a two-mode squeezed vacuum state (“Unruh” state) entangling
modes on either side of the horizon. The squeezing parameter is set by the
inverse temperature of the horizon, while the squeezing angle is fixed at the
maximal value $\phi=\pi/2$. The complexity of these Unruh states, therefore,
is also maximal and grows logarithmically with temperature ${\mathcal{C}}_{\rm
Unruh}\sim\ln(T/\omega)$ at large temperature. In a different context,
cosmological curvature perturbations are also described as a two-mode squeezed
state, entangling modes with opposite momenta. For accelerating backgrounds,
such as de Sitter, these modes are dynamically driven on superhorizon scales
to high squeezing and squeezing angle $\phi\rightarrow-\pi/2$.
Correspondingly, the complexity for these modes continues to grow during the
accelerating phase as the number of e-folds since horizon exit
${\mathcal{C}}_{\rm cosmo}\sim\ln(a(t)/a_{\rm exit})$. Without the perspective
of complexity, the dynamical preference of these backgrounds for the squeezing
angle $\phi=n\pi/2$ would be opaque; note, for example, that the entanglement
entropy of these squeezed states is independent of the squeezing angle. Other
quantities, such as Hawking radiation for backgrounds with horizons and the
power spectrum for cosmological perturbations, are also independent of the
squeezing angle. However, complexity444Other measures of complexity of two-
mode squeezed states, particularly those based on the correlation matrix
DiGiulio:2020hlz , appear to be independent of the squeezing angle. appears to
be unique in being sensitive to the squeezing angle. Moreover, the squeezing
angle dynamically selected maximizes the complexity of the full state,
suggesting that these backgrounds may be seeking to find a field configuration
that maximizes the complexity in a form of a second law of complexity
Brown:2017jil . It would be interesting to study this intriguing possibility
further.

We applied our minimal purification to thermal density matrices that arise
from these interesting curved space examples under certain conditions. For
example, tracing out the modes beyond the horizon, an observer in a Rindler or
black hole spacetime sees a thermal distribution of Hawking radiation. While
the global solution includes modes on both sides of the horizon and maximizes
the complexity, an observer can purify the thermal density matrix of Hawking
radiation with a smaller complexity by choosing a squeezing angle $\phi_{\rm
min}=\pi/4$ instead. This purified two mode squeezed state resembles the
global solution, but on a space in which the observable modes are entangled
with ancillary modes on a “twisted” wedge beyond the horizon. The reduced
complexity of the minimized purification seems to suggest that it is possible
to model the radiation external to a black hole with fewer operators than the
global solution would have suggested. It would be interesting to see if other
information-theoretic quantities of the black hole depend sensitively on the
squeezing angle, and thus might demonstrate some preference for some squeezing
angles over others.

For cosmological perturbations, we implemented a simple model of decoherence
where the squeezing angle is averaged over, leading to a thermal density
matrix with a temperature determined by the squeezing of the mode.
Implementing this model of decoherence when modes re-enter the horizon during
the radiation-dominated expansion of the Universe, the resulting complexity
smoothly matches to the preceding period of de-complexification, saturating at
${\mathcal{C}}_{\rm cosmo}\approx\pi/(2\sqrt{2})$, see Figure 8.
Alternatively, implementing this model of decoherence at other times, such as
at the transition between de Sitter- and radiation-domination, leads to a
sharp drop in the purified complexity. This suggests that complexity might be
a sensitive measure of different models and implementations of decoherence.

## Acknowledgements

We would like to thank Arpan Bhattacharyya and Sayura Das for helpful
conversations and discussions. S.H. would like to thank the University of Cape
Town for funding this project.

##  Appendix A Complexity of Two-Mode Squeezing

In this section, we review the calculation of the quantum circuit complexity
for two-mode squeezing, following NL1 ; Jefferson ; me1 . A reference state
$|\psi_{R}\rangle$ is transformed into a target state $|\psi_{T}\rangle$

$\displaystyle|\psi_{T}\rangle=\hat{\mathcal{U}}\ |\psi_{R}\rangle$ (89)

by a unitary operator $\hat{\mathcal{U}}$ representing the quantum circuit
connecting these two states. The unitary $\hat{\mathcal{U}}$ can be written as
a path-ordered exponential constructed as a sequence of gates

$\displaystyle\hat{U}(s)=\overleftarrow{\mathcal{P}}\ {\rm
exp}\left[-i\int_{0}^{s}\sum_{I}Y^{I}(s^{\prime})\hat{\mathcal{O}}_{I}\
ds^{\prime}\right]$ (90)

where $s$ parameterizes a path the circuit takes in the space of operators,
$\hat{\mathcal{O}}_{I}$ are a set of operators that will act as our gates, and
the $Y^{I}(s)$ are vectors that specify the path in the space of operators.
The path is parameterized with the boundary conditions such that $s=0$
corresponds to the identity operator $\hat{U}(0)=\mathds{1}$ while $s=1$
corresponds to the final state $\hat{U}(1)=\hat{\mathcal{U}}$.

The _quantum circuit complexity_ (denoted as “complexity”) for this
construction is defined as the minimum circuit depth

$\displaystyle{\mathcal{C}}=\int_{0}^{1}{\mathcal{F}}(\hat{U})\ ds\,,$ (91)

where ${\mathcal{F}}(\hat{U})$ is a _cost function_ , which must be chosen by
hand. The minimum complexity then amounts to finding the path $Y^{I}(s)$ that
generates the unitary $\hat{U}(s)$ subject to the boundary conditions, and
minimizes the cost function ${\mathcal{F}}(Y)$. Several different choices of
cost functions exist; some common choices are NL1 ; NL2 ; NL3 ; Jefferson

$\displaystyle{\mathcal{F}}_{1}(Y)$ $\displaystyle=$
$\displaystyle\sum_{I}|Y^{I}|\,,\hskip
14.45377pt{\mathcal{F}}_{2}=\sqrt{\sum_{I}G_{IJ}Y^{I}Y^{J}}\,,\hskip
14.45377pt{\mathcal{F}}_{\kappa}=\sum_{I}|Y^{I}|^{\kappa}\,.$ (92)

The ${\mathcal{F}}_{1}$ cost function is useful in that it directly counts the
number of gates, and is clearly extensive, but the result depends on the basis
chosen for the operators. The ${\mathcal{F}}_{2}$ cost function, on the other
hand, is clearly the total length of a geodesic, but is not extensive. The
metric $G_{IJ}$ represents a cost or penalty factor for certain operators. We
will primarily work with the ${\mathcal{F}}_{2}$ cost function, and will
choose a flat Riemannian metric with no penalty factors $G_{IJ}=\delta_{IJ}$
for simplicity.

We consider two-mode states in which the Hilbert space
${\mathcal{H}}\otimes{\mathcal{H}}_{\rm anc}$ consists of two copies of a
quantum harmonic oscillator. We will take as our reference state the ground
state of a harmonic oscillator
$|\psi_{R}\rangle=|0\rangle\otimes|0\rangle_{\rm anc}$ with (real) reference
frequency $\omega_{R}$

$\displaystyle\langle q,q_{\rm anc}|\psi_{R}\rangle={\mathcal{N}}_{R}\ {\rm
exp}\left[-\frac{1}{2}\omega_{R}(q^{2}+q_{\rm anc}^{2})\right]\,.$ (93)

Our target state is the two-mode squeezed state
$|\psi_{T}\rangle=|\Psi\rangle_{\phi}$ (5)

$\displaystyle\Psi_{\rm sq}\left(q,q_{\rm anc}\right)=\langle q,q_{\rm
anc}|\Psi\rangle_{\phi}={\cal N}_{T}\
\exp\left\\{-\frac{\omega}{2}\,A(q^{2}+{q}_{\rm anc}^{2})-\omega B\ q\ q_{\rm
anc}\right\\}$ (94)

where

$\begin{split}&A=\frac{1+e^{-4i\phi}\tanh^{2}r}{1-e^{-4i\phi}\tanh^{2}r}\,,\qquad
B=\frac{2\tanh r\,e^{-2i\phi}}{1-e^{-4i\phi}\tanh^{2}r}\,.\end{split}$ (95)

The wavefunctions (93),(94) can be diagonalized in terms of rotated
coordinates $\tilde{q}_{1},\tilde{q}_{2}$

$\displaystyle{\rm
exp}\left[-\frac{1}{2}\tilde{q}_{a}A_{ab}\tilde{q}_{b}\right],\hskip
14.45377pt\vec{\tilde{q}}=\begin{pmatrix}\tilde{q}_{1}\cr\tilde{q}_{2}\end{pmatrix}$
(96)

so that the symmetric square matrices representing the reference and target
states take the form

$\displaystyle A_{R}=\omega_{R}\mathds{1},\hskip
14.45377ptA_{T}=\omega\begin{pmatrix}\Omega_{1}&0\cr
0&\Omega_{2}\end{pmatrix}\,$ (97)

where $\Omega_{1}=-2A+B,\Omega_{2}=-2A-B$.

The unitary operator (90) acts on the reference matrix as
$A(s)=\hat{U}(s)A_{R}\hat{U}^{T}(s)$. Since the entries (95) can be complex,
we will use the diagonal elements of $GL(N,\mathbb{C})$ as our set of gate
operators $\hat{\mathcal{O}}_{I}$ (90)

$\displaystyle\hat{U}(s)={\rm exp}\left[y^{a}(s)M_{a}^{\rm diag}\right]={\rm
exp}\left[\begin{pmatrix}y^{1}(s)&0\cr 0&y^{2}(s)\end{pmatrix}\right]$ (98)

where the $y^{a}(s)=\alpha^{a}(s)+i\beta^{a}(s)$ are complex. The resulting
metric on the reduced space of operators becomes me1

$\displaystyle
ds^{2}=G_{IJ}dY^{I}dY^{J}=\sum_{a=1}^{2}|y^{a}|^{2}=\sum_{a=1}^{2}\left[(\alpha^{a})^{2}+(\beta^{a})^{2}\right]\,.$
(99)

The resulting complexity

$\displaystyle{\mathcal{C}}=\int_{0}^{1}\sqrt{G_{IJ}dY^{I}dY^{J}}\
ds=\int_{0}^{1}\sqrt{\sum_{a=1}^{2}\left[(\alpha^{a})^{2}+(\beta^{a})^{2}\right]}\
ds,$ (100)

is minimized, subject to the boundary conditions, by the straight-line
geodesic

$\displaystyle\alpha^{a}(s)$ $\displaystyle=$
$\displaystyle\ln\left|\Omega_{a}\frac{\omega}{\omega_{R}}\right|\ s,\hskip
21.68121pt\beta^{a}(s)=\arctan\left[\frac{{\rm Im}(\Omega_{a})}{{\rm
Re}(\Omega_{a})}\right]\ s\ .$ (101)

Choosing the reference frequency to be the ground state frequency of the
oscillator $\omega_{R}=\omega$ leads to the minimized complexity used in the
main text

$\displaystyle{\mathcal{C}}=\frac{1}{2}\sqrt{\sum_{a=1}^{2}\left[\left(\ln|\Omega_{a}|\right)^{2}+\left(\arctan\left[\frac{{\rm
Im}(\Omega_{a})}{{\rm Re}(\Omega_{a})}\right]\right)^{2}\right]}\,.$ (102)

##  Appendix B Bogoliubov Transformations and Squeezed States

We start with a massless scalar field expanded in modes

$\displaystyle\hat{\phi}=\sum_{i}\left(u_{i}\hat{a}_{i}+u_{i}^{*}\hat{a}_{i}^{\dagger}\right)$
(103)

where $\hat{a}_{i},\hat{a}_{i}^{\dagger}$ are creation and annihilation
operators in the basis $(u_{i},u_{i}^{*})$. The vacuum state for this basis is
defined by $\hat{a}_{i}|0\rangle_{a}=0$ for all $\hat{a}_{i}$. In curved
spacetimes or quench models, it is often possible to expand $\hat{\phi}$ in
another basis

$\displaystyle\hat{\phi}=\sum_{j}\left(v_{j}\hat{b}_{j}+v_{j}^{*}\hat{b}_{j}^{\dagger}\right)\,,$
(104)

with its own distinct vacuum state defined by $\hat{b}_{j}|0\rangle_{b}=0$.
Since both sets are complete, we can expand the annihilation operators in the
original basis in terms of the new basis through a Bogoluibovtransformation

$\displaystyle\hat{a}_{i}=\sum_{j}\left[\alpha_{ij}\hat{b}_{j}+\beta^{*}_{ij}\hat{b}_{j}^{\dagger}\right]\,,$
(105)

where the Bogoluibov coefficients must satisfy the conditions

$\displaystyle\sum_{k}\left(\alpha_{ik}\alpha^{*}_{jk}-\beta_{ik}\beta^{*}_{jk}\right)=\delta_{ij}\,.$
(106)

We are primarily interested in mixings between only two modes, which we will
call ${\rm R}$ and ${\rm L}$, in the following way555In the examples discussed
in the main text, spacetime naturally splits into wedges which we will call
${\rm R}$ and ${\rm L}$, which is the motivation for this choice.

$\displaystyle\hat{a}_{\rm R}$ $\displaystyle=$
$\displaystyle\alpha\,\hat{b}_{\rm R}+\beta^{*}\,\hat{b}^{\dagger}_{\rm L}\,;$
(107) $\displaystyle\hat{a}_{\rm L}$ $\displaystyle=$
$\displaystyle\alpha\,\hat{b}_{\rm L}+\beta^{*}\,\hat{b}^{\dagger}_{\rm R}\,.$
(108)

The vacuum state of the $\hat{b}$ basis is populated with particles when
viewed from the $\hat{a}$ basis, since

$\displaystyle\hat{a}_{\rm
R}|0\rangle_{b}=\beta^{*}\hat{b}^{\dagger}_{L}|0\rangle_{b}\neq 0$ (109)

so that

$\displaystyle{}_{b}\langle 0|\hat{a}^{\dagger}_{\rm R}\hat{a}_{\rm
R}|0\rangle_{b}=|\beta|^{2}\,.$ (110)

The normalization condition (106) becomes $|\alpha|^{2}-|\beta|^{2}=1$, so a
natural parameterization is

$\displaystyle\alpha=e^{-i\theta}\cosh r,\hskip
14.45377pt\beta=-e^{-i(\theta+2\phi)}\sinh r\,.$ (111)

This choice of the parameterization of the Bogoluibov coefficients allows us
to easily make contact with the squeezed state language, since (107),(108)
with (111) can also be written as the transformation

$\displaystyle\hat{a}_{\rm R}=\hat{\mathcal{U}}^{\dagger}\,\hat{b}_{\rm
R}\,\hat{\mathcal{U}}\,,$ (112)

where $\hat{\mathcal{U}}$

$\displaystyle\hat{\mathcal{U}}=\hat{\mathcal{R}}(\theta)\,\hat{\mathcal{S}}(r,\phi)$
(113)

is constructed from combination of the two-mode squeeze and rotation
operators,

$\displaystyle\hat{\mathcal{S}}(r,\phi)$ $\displaystyle=$
$\displaystyle\exp\left[\frac{r}{2}\left(e^{-2i\phi}\hat{a}_{\rm
R}\hat{a}_{\rm L}-e^{-2i\phi}\hat{a}_{\rm R}^{\dagger}\hat{a}_{\rm
L}^{\dagger}\right)\right]\,;$ (114) $\displaystyle\hat{\mathcal{R}}(\theta)$
$\displaystyle=$ $\displaystyle\exp\left[-i\theta\left(\hat{a}_{\rm
R}^{\dagger}\hat{a}_{\rm R}+\hat{a}_{\rm L}^{\dagger}\hat{a}_{\rm
L}\right)\right]\,.$ (115)

Applying (107) to the vacuum state $|0\rangle_{a}$

$\displaystyle\left(\alpha\,\hat{b}_{\rm R}+\beta^{*}\,\hat{b}^{\dagger}_{\rm
L}\right)|0\rangle_{a}=0$ (116)

leads to the two-mode squeezed state solution

$\displaystyle|0\rangle_{a}={\mathcal{N}}\exp\left[-\left(\frac{\beta^{*}}{\alpha}\right)\hat{b}_{\rm
R}^{\dagger}\hat{b}_{\rm
L}^{\dagger}\right]|0\rangle_{b}={\mathcal{N}}\sum_{n=0}^{\infty}(-1)^{n}\left(\frac{\beta^{*}}{\alpha}\right)^{n}|n\rangle_{\rm
R}\otimes|n\rangle_{\rm L}\,,$ (117)

so that an alternative way of viewing the Bogoluibov transformations
(107),(108) is that the vacuum state in one basis is a squeezed vacuum state
(with non-vanishing particle number) in another basis. We note in passing that
the relations (111) allow us to extract the squeezing parameter $r$, and
squeeze and rotation angles $\phi,\theta$ from the coefficients
$\alpha,\beta$. A thermal density matrix $\hat{\rho}_{\rm th}$ naturally
arises from (117) upon tracing out over the ${\rm L}$-modes of the $\hat{b}$
basis:

$\displaystyle\hat{\rho}$ $\displaystyle=$ $\displaystyle{\rm Tr}_{\rm
L}\left[|0\rangle_{a}\langle 0|_{a}\right]={\rm Tr}_{\rm
L}\left[|{\mathcal{N}}|^{2}\sum_{mn}(-1)^{m+n}\left(\frac{\beta}{\alpha^{*}}\right)^{n}\left(\frac{\beta^{*}}{\alpha}\right)^{m}|n\rangle_{\rm
R}\langle m|_{\rm R}\,\otimes|n\rangle_{\rm L}\langle m|_{\rm L}\right]$ (118)
$\displaystyle=$
$\displaystyle|{\mathcal{N}}|^{2}\sum_{n}\left|\frac{\beta}{\alpha^{*}}\right|^{2n}|n\rangle_{\rm
R}\langle n|_{\rm R}\,.$ (119)

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