# Complete Information Balance in Quantum Measurement

Seung-Woo Lee<EMAIL_ADDRESS>Center for Quantum Information, Korea
Institute of Science and Technology, Seoul, 02792, Korea  Quantum Universe
Center, Korea Institute for Advanced Study, Seoul 02455, Korea  Jaewan Kim
School of Computational Sciences, Korea Institute for Advanced Study, Seoul
02455, Korea  Hyunchul Nha  Department of Physics, Texas A&M University at
Qatar, Education City, P.O.Box 23874, Doha, Qatar

###### Abstract

Quantum measurement is a basic tool to manifest intrinsic quantum effects from
fundamental tests to quantum information applications. While a measurement is
typically performed to gain information on a quantum state, its role in
quantum technology is indeed manifold. For instance, quantum measurement is a
crucial process element in measurement-based quantum computation. It is also
used to detect and correct errors thereby protecting quantum information in
error-correcting frameworks. It is therefore important to fully characterize
the roles of quantum measurement encompassing information gain, state
disturbance and reversibility, together with their fundamental relations.
Numerous efforts have been made to obtain the trade-off between information
gain and state disturbance, which becomes a practical basis for secure
information processing. However, a complete information balance is necessary
to include the reversibility of quantum measurement, which constitutes an
integral part of practical quantum information processing. We here establish
all pairs of trade-off relations involving information gain, disturbance, and
reversibility, and crucially the one among all of them together. By doing so,
we show that the reversibility plays a vital role in completing the
information balance. Remarkably, our result can be interpreted as an
information-conservation law of quantum measurement in a nontrivial form. We
completely identify the conditions for optimal measurements that satisfy the
conservation for each tradeoff relation with their potential applications. Our
work can provide a useful guideline for designing a quantum measurement in
accordance with the aims of quantum information processors.

##  1 Introduction

Since the early days of quantum mechanics, quantum measurement has been one of
the central topics in quantum theory [1]. Unlike the measurement in classical
world that only passively provides information on a physical state, quantum
measurement has a critical role to reveal quantum phenomena from fundamental
tests to quantum technology applications. For instance, incompatible quantum
measurements are essential to demonstrate quantum nonlocality [2, 3]. Quantum
measurement is also a crucial tool in quantum information protocols for not
only extracting information but also processing computation in one-way quantum
computation [4], protecting quantum information by detecting and correcting
errors in quantum error-correcting protocols, etc. [5, 6, 7, 8, 9, 10, 11,
12]. That is, the role of quantum measurement is manifold in quantum
technology, which requires a thorough characterization encompassing all
relevant aspects of measurement.

A well-known effect of quantum measurement is its inevitable disturbance on
quantum states, which constitutes a profound basis for secure quantum
information processing, e.g. quantum cryptography [13, 14]. It has been a
topic of fundamental and practical importance to identify information balance
in quantum measurement, particularly the trade-off relation between
information gain and state disturbance [15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26]. On the other hand, the usual notion that quantum measurement is
irreversible has been substantially reassessed in recent years [27, 28, 29,
30, 43, 44, 40, 31, 32, 33, 35, 34, 36, 37, 38, 39, 41, 42]. It turns out that
a quantum measurement weakly interacting with a system can be reversed even to
faithfully recover the input state with a non-zero success probability [27,
28]. This issue of reversibility thus becomes an integral part to consider in
characterizing a quantum measurement in its entirety [27, 28, 29, 30]. The
measurement reversibility has also widely applied to range of applications,
e.g. quantum error corrections [31], gate operations [32, 33] and decoherence
suppression [35, 34, 36]. Reversing the disturbance of quantum measurement was
experimentally demonstrated with superconducting [37], trapped-ion [38] and
photonic qubits [35, 36, 39]. Along this line, a trade-off of the
reversibility against information gain in quantum measurement was rigorously
assessed [40, 41, 42]. Trade-offs among information gain, state disturbance
and reversibility were also quantitatively studied at the level of a single
measurement outcome [43, 44].

However, there does not exist a unified framework yet to deal with all the
information contents together as universal quantities in general quantum
measurement [26], which is essential to draw a complete picture on information
balance in quantum measurement. This is very important for a broad
applicability, e.g. designing a quantum measurement to be adapted to the aim
of quantum information processing at hand. Here we establish such a complete
information balance in quantum measurement. We derive the full trade-off
relations among those three information contents, information gain $\cal G$,
disturbance $\cal D$, and reversibility $\cal R$. We particularly show that
the reversibility $\cal R$ plays a crucial role in information balance
accounting for the gap between $\cal G$ and $\cal D$. That is, we find that
the global trade-off relation ($\cal G$-$\cal D$-$\cal R$) tightens the trade-
off between the gain and disturbance ($\cal G$-$\cal D$) only [20]. On the
other hand, the trade-off between disturbance and reversibility ($\cal
D$-$\cal R$) compensates the gain and reversibility relation ($\cal G$-$\cal
R$). Consequently, our result shows that the total information is balanced in
an ideal quantum measurement process, which can be interpreted as a
conservation of information contents in quantum measurement.

We fully obtain the conditions to saturate all of the trade-off relations and
thereby define an optimal quantum measurement providing maximal information
contents quantum mechanics fundamentally allows. Our framework can offer
useful guidelines for designing measurement-based quantum information
protocols in quantum computation [9], teleportation [10], quantum metrology
[11], and quantum error corrections [12], etc.. As all the information
contents defined here are directly measurable, our results are readily
testable and applicable to any quantum information platforms. Our work may
contribute to deepening our fundamental understanding of quantum measurement
and provide a rigorous practical benchmark to optimize measurement-based
quantum information protocols.

##  2 General framework

We begin with the general framework for addressing the information changes by
a quantum measurement $\bf M$. Assume that an arbitrary quantum state $\rho$
is prepared to convey information. We perform a quantum measurement $\bf M$ to
extract the information, which disturbs and changes the input state $\rho$ to
another state. We then apply a subsequent reversing operation $\bf R$ to
characterize the reversibility of $\bf M$ (see Fig. 1(a)). The reversing
operation is assumed to be chosen to recover the input state $\rho$, i.e.,
$({\bf R}\circ{\bf M})(\rho)\propto\rho$.

Without loss of generality, we assume here that the input information is
encoded onto pure states $\rho=|\psi\rangle\langle\psi|$ in $d$-dimensional
Hilbert space, but the results derived in what follows are valid for any mixed
input states. Consider a quantum measurement described by a set of operators
${\bf M}=\\{\hat{M}_{r}|r=1,\ldots,n\\}$, satisfying the completeness relation
$\sum_{r}\hat{M}^{{\dagger}}_{r}\hat{M}_{r}=\hat{\mathbbm{1}}$ [6, 7]. When
the measurement outcome is $r$, the input state is changed to
$|\psi_{r}\rangle=\hat{M}_{r}|\psi\rangle/{\sqrt{p(r,\psi)}}$ where
$p(r,\psi)=\langle\psi|\hat{M}_{r}^{\dagger}\hat{M}_{r}|\psi\rangle$. For each
outcome $r$, we may estimate the input state as $|\widetilde{\psi}_{r}\rangle$
by a certain estimation strategy. We then apply a subsequent reversing
operation described by a set of operators ${\bf
R}=\\{\hat{R}_{r,l}|l=1,\ldots,m\\}$, satisfying the completeness relation
$\sum_{l}\hat{R}^{{\dagger}}_{r,l}\hat{R}_{r,l}=\hat{\mathbbm{1}}$ for each
$r$. The final output state is then given by
$|\psi_{r,l}\rangle=\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle/{\sqrt{p(r,l,\psi)}}$,
where
$p(r,l,\psi)=\langle\psi|\hat{M}_{r}^{{\dagger}}\hat{R}^{\dagger}_{r,l}\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle$.
The reversing operation $\bf R$ here is assumed to be appropriately chosen
according to the measurement outcome $r$. We note that the whole process
cannot be described by a unitary operation as it can be probabilistic and
conditional on the result of $\bf M$ and observer’s choice of $\bf R$. Note
that our framework differs from the Petz recovery map [45, 46].

Figure 1: (a) Quantum measurement ${\bf M}$ and reversal process $\bf R$. (b)
The input, the post-measurement, the estimated and the output state in a
single trial, when the outcomes of the measurement and the reversal are $r$
and $l$, respectively, are denoted by $|\psi\rangle$, $|\psi_{r}\rangle$,
$|\widetilde{\psi}_{r}\rangle$, and $|\psi_{r,l}\rangle$, respectively. The
closeness of those states on average determines the information contents of a
given quantum measurement ${\bf M}$.

##  3 Information contents

Let us define the information contents within the above general framework. For
a given quantum measurement, the changes of information due to the measurement
$\bf M$ and the reversing operation $\bf R$ can be defined in terms of the
closeness between the input $|\psi\rangle$, the post-measurement
$|\psi_{r}\rangle$, the estimated $|\widetilde{\psi}_{r}\rangle$, and the
output $|\psi_{r,l}\rangle$ states. See Fig. 1(b) together with the details of
the definitions in Appendix A.

(i) Information Gain: The amount of information obtained by $\bf M$ can be
quantified based on the overlap between the input $|\psi\rangle$ and the
estimated state $|\widetilde{\psi}_{r}\rangle$. The estimation fidelity is
obtained by averaging $|\langle\widetilde{\psi}_{r}|\psi\rangle|^{2}$ over all
input states and possible measurement outcomes, i.e., $\int
d\psi\sum_{r=1}^{n}p(r,\psi)|\langle\widetilde{\psi}_{r}|\psi\rangle|^{2}$,
which has different values depending on the estimation strategies. We define
the information gain as the maximum estimation fidelity over all possible
strategies, which can be obtained as (see Appendix A.2),

${\cal
G}=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}(\lambda_{0}^{r})^{2}\bigg{]},$ (1)

where $\lambda_{0}^{r}$ is the largest singular value of $\hat{M}_{r}$ in the
singular value decomposition (Appendix A.1) [20]. The information gain thus
lies in the range $1/d\leq{\cal G}\leq 2/(d+1)$, where the upper bound is
reached by a von Neumann measurement and the lower bound by a unitary
operation or a random guess.

(ii) Disturbance: In order to quantify the amount of disturbance by $\bf M$,
we consider the operation fidelity first. The operation fidelity of $\bf M$ is
given by averaging the overlap between the input $|\psi\rangle$ and the post-
measurement state $|\psi_{r}\rangle$, i.e., $\int
d\psi\sum_{r=1}^{n}p(r,\psi)|\langle\psi_{r}|\psi\rangle|^{2}$. We can then
evaluate the maximum of the operation fidelity (see Appendix A.3), resulting
in

${\cal
F}=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}\Big{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\Big{)}^{2}\bigg{]}.$
(2)

In this case, all the singular values $\lambda_{i}^{r}$ of $\hat{M}_{r}$ are
involved in determining the average fidelity of the disturbed states. We may
then define the disturbance induced by $\bf M$ as the minimum operation
infidelity given by

${\cal D}=1-{\cal F},$ (3)

scaled in the range $0\leq{\cal D}\leq(d-1)/(d+1)$.

Similarly, the operation fidelity of the overall process ${\bf R}\circ{\bf M}$
without postselection can be obtained by evaluating the average fidelity
between the input $|\psi\rangle$ and output state $|\psi_{r,l}\rangle$, i.e.,
$\int
d\psi\sum^{n}_{r=1}\sum^{m}_{l=1}P(r,l,\psi)|\langle\psi_{r,l}|\psi\rangle|^{2}$.
Its maximum can be obtained as (see Appendix A.4)

${\cal F}({\bf R}\circ{\bf
M})=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}\sum^{m}_{l=1}\Big{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\lambda_{i}^{r,l}\Big{)}^{2}\bigg{]},$
(4)

where the singular values $\lambda_{i}^{r,l}$ of $\hat{R}_{r,l}$ are
additionally involved in determining the output fidelity. The operation
fidelity lies in the range $2/(d+1)\leq{\cal F}\leq 1$, where the upper bound
is reached by a unitary operation and the lower bound by a von Neumann
measurement.

(iii) Reversibility: We consider the success event of the reversing operation
${\bf R}$, which faithfully recovers the input state, i.e.,
$|\psi_{r,l}\rangle\propto|\psi\rangle$, to evaluate the reversibility. Assume
that the operators $\hat{R}_{r,l}$ for $l=1,\ldots,s<m$ are associated with
the success events s.t.
$\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle=\eta_{r,l}|\psi\rangle$, where
$|\eta_{r,l}|^{2}$ is the success probability of the reversing operation when
the outcome is $r$. The reversibility is then obtained as the maximum overall
success probability,

${\cal R}=\max_{\\{\hat{R}_{r,l}\\}}\int
d\psi\sum^{n}_{r=1}\sum^{s<m}_{l=1}|\langle\psi|\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle|^{2}=\sum_{r=1}^{n}(\lambda^{r}_{d-1})^{2},$
(5)

where $\lambda^{r}_{d-1}$ is the smallest singular value of $\hat{M}_{r}$
[40]. The reversibility is scaled as $0\leq{\cal R}\leq 1$.

The optimal reversing operator can be defined in the following context. Assume
that the measurement operator $\hat{M}_{r}$ is represented in the singular
value decomposition as $\hat{M}_{r}=\hat{V}_{r}\hat{D}_{r}$ with a unitary
operator $\hat{V}_{r}$ and a diagonal matrix
$\hat{D}_{r}=\sum_{i=0}^{d-1}\lambda^{r}_{i}|i\rangle\langle i|$ (see Appendix
A.1). Its optimal reversing operator can then be written by
$\hat{R}_{r,1}=\lambda^{r}_{d-1}\hat{D}^{-1}_{r}\hat{V}^{{\dagger}}_{r}$,
where $\hat{D}^{-1}_{r}=\sum_{i=0}^{d-1}(\lambda^{r}_{i})^{-1}|i\rangle\langle
i|$ with nonzero $\lambda^{r}_{i}$ (Here, we set $s=1$ without loss of
generality). For example, a quantum measurement described by
$\hat{M}_{1}=\sqrt{\eta}|1\rangle\langle 1|$ and $\hat{M}_{2}=|0\rangle\langle
0|+\sqrt{1-\eta}|1\rangle\langle 1|$ with measurement strength $0\leq\eta\leq
1$, can be optimally reversed by $\hat{R}_{2,1}=\sqrt{1-\eta}|0\rangle\langle
0|+|1\rangle\langle 1|$ and $\hat{R}_{2,2}=\sqrt{\eta}|0\rangle\langle 0|$ for
the measurement outcome $r=2$ and the reversal process $l=1$, respectively. In
this case, the reversibility is given by ${\cal R}=1-\eta$ from Eq. (5). Note
that a unitary operation (no measurement) is deterministically reversible
${\cal R}=1$ while a von Neumann measurement is completely irreversible ${\cal
R}=0$.

We now have three information contents characterizing a quantum measurement,
i.e., information gain ${\cal G}$, disturbance ${\cal D}$, and reversibility
${\cal R}$. These are universal quantities averaged over all input states of a
given dimension $d$ and have a clear operational meaning in terms of quantum
fidelity, fulfilling the requirements for the information contents in quantum
measurement [26]. Note that the information contents evaluated here with pure
input states $\rho=|\psi\rangle\langle\psi|$ and the accompanying consequences
in information balance are generally valid for arbitrary mixed input states
$\rho$, since the maximum averaged in the space of pure states must represent
the maximum in the space of convex combination of pure states, i.e. mixed
states.

##  4 Information balance in quantum measurement

In this section, we now derive the trade-off relations among the information
contents of quantum measurement. Before presenting those trade-off relations,
we first introduce and prove a useful inequality on the relation between the
reversibility $\cal R$ and the overall operation fidelity ${\cal F}({\bf
R}\circ{\bf M})$ after the measurement followed by the reversing operation
defined in the previous section:

Lemma 1. For any quantum measurement ${\bf M}$ and subsequent reversing
operation ${\bf R}$, the reversibility ${\cal R}$ and the operation fidelity
of the overall process ${\cal F}({\bf R}\circ{\bf M})$ always satisfy an
inequality

$2+(d-1){\cal R}\leq(d+1){\cal F}({\bf R}\circ{\bf M}),$ (6)

in arbitrary finite dimension $d\geq 2$.

Detailed proof of Lemma 1 is given in Appendix B. We define a vector
$\vec{u}^{l}_{i}=(\lambda^{r=1}_{i}\lambda^{r=1,l}_{i},\ldots,\lambda^{r=n}_{i}\lambda^{r=n,l}_{i})$
for $i=0,\ldots,d-1$ in terms of the singular values $\lambda^{r}_{i}$ of the
measurement operators $\\{\hat{M}_{r}\\}$ and $\lambda^{r,l}_{i}$ of the
reversing operators $\\{\hat{R}_{r,l}\\}$. The equality in (6) holds if and
only if the quantum measurement satisfies

$\vec{u}^{l}_{i}\cdot\vec{u}^{l}_{j}=\delta_{ij}|\vec{u}^{l}_{i}|^{2},\leavevmode\nobreak\
\forall l\neq 1.$ (7)

We will use Lemma 1 to derive the trade-off relations presented in what
follows.

Before presenting our main results, we introduce two previously known trade-
off relations between information contents, denoted by $\cal G$-$\cal D$ [20]
and $\cal G$-$\cal R$ [40] as below.

($\cal G$-$\cal D$: Information gain and Disturbance trade-off) The trade-off
relation between the information gain ${\cal G}$ and disturbance ${\cal
D}=1-F$ was derived in Ref. [20] as

$\sqrt{{\cal F}-\frac{1}{d+1}}\leq\sqrt{{\cal
G}-\frac{1}{d+1}}+\sqrt{(d-1)\bigg{(}\frac{2}{d+1}-{\cal G}\bigg{)}}.$ (8)

This is the quantitative proof of the heuristic knowledge ‘the more
information a quantum measurement obtains, the more disturbed the quantum
state becomes.’ It gives the lower bound of diturbance for a given amount of
information that a quantum measurement extracts.

($\cal G$-$\cal R$: Information gain and Reversibility trade-off) The trade-
off relation between the information gain $\cal G$ and the reversibility $\cal
R$ was derived in Ref. [40] as

$d(d+1){\cal G}+(d-1){\cal R}\leq 2d,$ (9)

which was the first information-theoretic approach introducing the role of the
reversibility in quantum measurement. It captures the idea that ‘the more
information a quantum measurement obtains, the less reversible the quantum
measurement is.’

However, a global trade-off relation including all three information contents
has been missing so far. The full quantitative links among the three
information contents have also not been completed (see Fig. 2). Let us now
derive this global trade-off relation, including all the information contents,
$\cal G$, $\cal D$, and $\cal R$, aiming to complete the total information
balance as follows.

Theorem 1. ($\cal G$-$\cal D$-$\cal R$: Global trade-off relation) The
information gain ${\cal G}$, the disturbance ${\cal D}$ and the reversibility
${\cal R}$ of quantum measurement always satisfy an inequality

$\displaystyle\sqrt{{\cal F}-\frac{1}{d+1}}$ $\displaystyle\leq\sqrt{{\cal
G}-\frac{1}{d+1}}+\sqrt{\frac{\cal R}{d(d+1)}}$ (10) $\displaystyle\hskip
8.53581pt+\sqrt{(d-2)\bigg{(}\frac{2}{d+1}-{\cal G}-\frac{\cal
R}{d(d+1)}\bigg{)}},$

with ${\cal F}=1-{\cal D}$, in an arbitrary dimension $d\geq 2$.

The inequality in (10) determines the quantitative relation among the three
information contents, $\cal G$, $\cal D$, and $\cal R$ (see Appendix C for the
details of the proof). This relation can be interpreted in various ways. It
draws the upper bound of $\cal F$ or equivalently the lower bound of $\cal D$
with respect to both ${\cal G}$ and ${\cal R}$. Or, it indicates the maximum
possible reversibility for a given pair of the information amounts ${\cal G}$
and $\cal D$.

When $d=2$, the inequality in (10) is equivalent to $\cal G$-$\cal D$ in (8).
This is due to the relation between the information gain ${\cal G}$ and the
reversibility ${\cal R}$ for $d=2$, i.e. $\frac{2}{3}-{\cal G}=\frac{{\cal
R}}{6}$. It is given by the completeness condition
$\sum_{r}\hat{M}^{{\dagger}}_{r}\hat{M}_{r}=\hat{\mathbbm{1}}$ leading to
$\sum_{r=1}^{n}(\lambda^{r}_{0})^{2}+\sum_{r=1}^{n}(\lambda^{r}_{1})^{2}=d=2$
together with Eqs. (1) and (5).

On the other hand, for $d>2$ in general, the $\cal G$-$\cal D$-$\cal R$
inequality in (10) is fundamentally different from $\cal G$-$\cal D$ in (8)
and $\cal G$-$\cal R$ in (9). Note that $\cal G$-$\cal D$-$\cal R$ provides a
tighter bound than $\cal G$-$\cal D$ in the relation between the information
gain and the disturbance. This becomes clearer with its saturation condition
described below.

($\cal G$-$\cal D$-$\cal R$ saturation condition) The $\cal G$-$\cal D$-$\cal
R$ inequality (10) is saturated if and only if the quantum measurement
satisfies following conditions: all $\vec{v}_{i}$ for $i=0,\cdots,d-1$ are
collinear and

$|\vec{v}_{1}|=\cdots=|\vec{v}_{d-2}|,$ (11)

where $\vec{v}_{i}=(\lambda^{r=1}_{i},\ldots,\lambda^{r=n}_{i})$ is a vector
defined with the singular values of $\hat{M}_{r}$.

See Appendix C for the detailed description of the condition. We can see that
a quantum measurement satisfying the $\cal G$-$\cal D$ saturation condition
(i.e., all $\vec{v}_{i}$ are collinear and
$|\vec{v}_{1}|=\cdots=|\vec{v}_{d-1}|$ [20]) also satisfies the saturation
condition of $\cal G$-$\cal D$-$\cal R$ (Table 1), but the converse is not
always true. It indicates that the information balance can be more tightly
characterized by $\cal G$-$\cal D$-$\cal R$ in a broader set of quantum
measurements than $\cal G$-$\cal D$. In section 5, we will further discuss on
the different sets of quantum measurements classified based on the saturation
conditions of the trade-off relations. Besides, we also analytically show that
the right-hand side of the inequality (10) is always lower than or equal to
the right-hand side of the inequality (8) (see Appendix D), which guarantees
that $\cal G$-$\cal D$-$\cal R$ tightens $\cal G$-$\cal D$.

We here introduce and prove another useful inequality on the operation
fidelity by a reversing operation as below:

Lemma 2. The overall operation fidelity covering all output states without
postselection by the quantum measurement and the reversal ${\bf R}\circ{\bf
M}$ is upper bounded by the operation fidelity of the quantum measurement $\bf
M$, i.e.,

${\cal F}({\bf R}\circ{\bf M})\leq{\cal F}({\bf M}).$ (12)

See Appendix E for the details of the proof. Note that the equality in (12)
holds when $\bf M$ is a unitary operation or a von Neumann measurement. The
inequality (12) implies that ‘the disturbance in quantum measurement never
decreases by any subsequent reversing operation.’ We can further generalize
this for arbitrary $k$-times sequential quantum measurements, s.t. ${\cal
F}({\bf M}_{k}\circ\cdots{\bf M}_{2}\circ{\bf M}_{1})\leq\cdots\leq{\cal
F}({\bf M}_{2}\circ{\bf M}_{1})\leq{\cal F}({\bf M}_{1})$.

The inequality in (12) and its extension mentioned above are intuitively
plausible by the second law of thermodynamics. These indicates the non-
increasing of the average fidelity between the input and output states by
reversing operations so that it differs from, but may be fundamentally related
to, the data processing inequality [47]. We use (12) to derive a trade-off
relation in what follows.

Figure 2: Three information contents of quantum measurement, i.e., information
gain ${\cal G}$, disturbance ${\cal D}=1-{\cal F}$, and reversibility ${\cal
R}$, and their quantitative links: ($\cal G$-$\cal D$) in Eq. (8) [20], ($\cal
G$-$\cal R$) in Eq. (9) [40], ($\cal G$-$\cal D$-$\cal R$) Theorem 1 in Eq.
(10), ($\cal D$-$\cal R$) Theorem 2 in Eq. (13).

Let us now derive the quantitative relation between the disturbance and the
reversibility of quantum measurement using Lemma 1 and Lemma 2.

Theorem 2. ($\cal D$-$\cal R$: Disturbance and Reversibility trade-off) The
disturbance ${\cal D}$ and the reversibility $\cal{R}$ of quantum measurements
always satisfy

$(d-1){\cal R}+(d+1){\cal D}\leq d-1,$ (13)

in an arbitrary dimension $d$.

Proof– From Lemma 1 and Lemma 2, $(d-1){\cal R}\leq(d+1){\cal F}({\bf
R}\circ{\bf M})-2\leq(d+1){\cal F}-2$. By ${\cal D}=1-{\cal F}$, we obtain the
inequality (13). $\square$

The inequality in (13) complements other trade-off relations, determining the
upper bound of $\cal{R}$ by ${\cal D}$. It implies that ‘the more disturbing a
quantum measurement is, the less reversible it becomes.’ Here the equality
holds for von Neumann measurements or unitary operations.

Figure 3: Left: Comparison of the upper bounds by $\cal G$-$\cal D$-$\cal R$
and $\cal G$-$\cal D$ relations for the measurement in the main text. The
bound by $\cal G$-$\cal D$-$\cal R$ is tighter than the one by $\cal G$-$\cal
D$. Right: The gap is due to the reversibility $\cal R$, which is also in
trade-off relation with the information gain $\cal G$.

We have now completed the full quantitative links among the three information
contents, i.e., information gain $\cal G$, disturbance $\cal D$, and
reversibility $\cal R$ of quantum measurement, as illustrated in Fig. 2. These
clearly show how the total information is balanced during the measurement and
the reversal process. Suppose that we choose and modify a quantum measurement,
intending to extract information maximally from the input quantum state. Such
an optimization inevitably increases the disturbance $\cal D$ (i.e., decreases
$\cal F$) from the relation $\cal G$-$\cal D$. The operation fidelity $\cal F$
can be interpreted here as the remaining information in the post-measurement
state as a part of the initially encoded information into the input state. On
the other hand, there exists a hidden part of information, by which we can
recover the input state using a subsequent reversing operation on the output
state, quantified by $\cal R$. It turns out that $\cal R$ fills the gap
between $\cal G$ and $\cal D$ and tightens further the trade-off relations
beyond $\cal G$-$\cal D$ as proved in this section.

As an example, let us consider a quantum measurement with operators
$\hat{M}_{i}=\sqrt{p}|i\rangle\langle i|+\sqrt{(1-p)(3-p)/3}|i+1\rangle\langle
i+1|+\sqrt{p(1-p)/3}|i+2\rangle\langle i+2|$ ($i=0,1,2$) for $0.458619\leq
p\leq 1$. The basis states are $|i\rangle\equiv|i\mod 3\rangle$ in the Hilbert
space of $d=3$. Using Eqs. (1),(2) and (5), we can obtain the information gain
${\cal G}=(1+p)/4$, the operation fidelity ${\cal
F}=1/4+(\sqrt{p}+\sqrt{(1-p)(3-p)/3}+\sqrt{p(1-p)/3})^{2}/4$, and the
reversibility ${\cal R}=p(1-p)$. These turn out to saturate the global $\cal
G$-$\cal D$-$\cal R$ tradeoff in Eq. (10) but not the $\cal G$-$\cal D$ one in
Eq. (8). In Fig. 3 (left), we show the curve (black solid) representing the
values of the pair ${\cal G}$ and ${\cal F}$ corresponding to the considered
measurement, together with the curve (blue dashed) representing the bound from
the $\cal G$-$\cal D$ relation in Eq. (8). We clearly see that the bound by
$\cal G$-$\cal D$-$\cal R$ is tighter than the one by $\cal G$-$\cal D$.

In view of the $\cal G$-$\cal D$ tradeoff, the measurement is not optimal in
the sense that it does not maximize the output fidelity ${\cal F}$ for a given
degree of ${\cal G}$. However, in fact, it is an optimal measurement
saturating the $\cal G$-$\cal D$-$\cal R$ tradeoff in Eq. (10). The gap
between the solid and the dashed curves is due to the reversibility $\cal R$,
which is also in trade-off with $\cal G$ as plotted in Fig. 3 (right). It thus
illustrates the importance of including all information contents $\cal
G$-$\cal D$-$\cal R$ to characterize a quantum measurement as optimal or non-
optimal.

We note that there would exist some missing parts of the information accounted
for by none of $\cal G$, $\cal F$, or $\cal R$. Such a missing part should be
due to the non-optimality of quantum measurement or ignorance in the
estimation of the input state based on the measurement outcomes. For instance,
if we take into account the effect of errors in quantum measurement leading to
imperfect reversal, the final output state may not be the same as the input
state, i.e., $({\bf R}\circ{\bf M})(\rho)\propto\rho^{\prime}\neq\rho$.
Specifically, let us consider the case when the reversing operation succeeds
but yields the output state $\rho(\psi,\epsilon)$ as

$\hat{R}_{r,1}\hat{M}_{r}|\psi\rangle\langle\psi|\hat{M}_{r}^{\dagger}\hat{R}_{r,1}^{\dagger}=|\eta_{r,1}|^{2}\rho(\psi,\epsilon)$
(14)

which deviates from the original input state due to an error parameterized by
$0\leq\epsilon\leq 1$. The reversibility can then be evaluated as (see
Appendix F)

${\cal R}=\sum_{r}(\lambda^{r}_{d-1})^{2}F_{s}.$ (15)

Here $F_{s}=\int d\psi\int^{1}_{0}d\epsilon\leavevmode\nobreak\
p(\epsilon)\langle\psi|\rho(\psi,\epsilon)|\psi\rangle$ is the average
fidelity between the input state and the output state of the successful
reversal, where $p(\epsilon)$ denotes the error probability density. It shows
that errors tend to decrease the reversibility in proportion to $F_{s}$. Note
that when $F_{s}=1$ without errors the reversibility can reach the maximum
value in Eq. (5). Similarly, the amount of information gain can be degraded
from the maximum $\cal G$ in Eq. (1) in the presence of errors. Therefore,
imperfection and errors result in a missing part of the total information. As
a result, the information contents $\cal G$, $\cal D$ and $\cal R$ of the
quantum measurement with errors satisfy but cannot saturate the information
trade-off relations.

In this context, we may define the optimal quantum measurement as the one that
reaches the upper bounds of trade-off relations without any missing part of
the total information. This further generalizes the definition of optimal
quantum measurement to reach the bound of $\cal G$-$\cal D$ with minimal
disturbance in [23, 48, 41]. In a sense, the optimal measurement is the one
that conserves the total information in a nontrivial form obeying the trade-
off relations. We present how optimal quantum measurements can be classified
into different sets according to the considered trade-off relations in the
next section.

Figure 4: A Venn diagram for quantum measurements based on the optimality to reach the bounds of $\cal G$, $\cal D$, and $\cal R$ by the trade-off relations. The quantum measurements (i)-(iv) presented in the main text are the representative examples of the different types of optimal quantum measurements in each divided region, while the measurement in the region (v) is non-optimal.  Table 1:  Saturation condition on the trade-off relations. | Saturation condition
---|---
$\cal G$-$\cal D$-$\cal R$ | all $\vec{v}_{i}$ are collinear111$\vec{v}_{i}=(\lambda^{r=1}_{i},\ldots,\lambda^{r=n}_{i})$ & $|\vec{v}_{1}|=\cdots=|\vec{v}_{d-2}|$
$\cal G$-$\cal D$ | all $\vec{v}_{i}$ are collinear & $|\vec{v}_{1}|=\cdots=|\vec{v}_{d-1}|$
$\cal G$-$\cal R$ |  $\hat{M}^{{\dagger}}_{r}\hat{M}_{r}=a_{r}|j_{r}\rangle\langle j_{r}|+b_{r}\hat{\mathbbm{1}}$222$j_{r}\in\\{0,\ldots,d-1\\}$ with $a_{r},b_{r}\geq 0$
$\cal D$-$\cal R$ | von Neumann meas. or unitary op.

##  5 Classifying optimal quantum measurements

The saturation condition on each trade-off relation determines the criterion
for classifying optimal quantum measurements. Let $\cal S$ be the set of all
possible quantum measurements. We then denote by $\cal{S}_{\cal
G\text{-}D\text{-}R}$, $\cal{S}_{\cal G\text{-}D}$, $\cal{S}_{\cal
G\text{-}R}$, and $\cal{S}_{\cal D\text{-}R}$ those subsets of quantum
measurements saturating the trade-off relations $\cal G$-$\cal D$-$\cal R$,
$\cal G$-$\cal D$, $\cal G$-$\cal R$, and $\cal D$-$\cal R$, respectively. The
saturation condition of $\cal G$-$\cal R$ is
$\hat{M}^{{\dagger}}_{r}\hat{M}_{r}=a_{r}|j_{r}\rangle\langle
j_{r}|+b_{r}\hat{\mathbbm{1}}$ where $j_{r}\in\\{0,\cdots,d-1\\}$ and $a_{r}$
and $b_{r}$ are non-negative parameters [40]. It was shown in Ref. [41] that a
quantum measurement saturating $\cal G$-$\cal D$ always satisfies the
condition saturating $\cal G$-$\cal R$, while the converse is not true, s.t.
$\cal{S}_{\cal G\text{-}R}\supset\cal{S}_{\cal G\text{-}D}$. In the previous
section from Theorem 1, we have observed that $\cal{S}_{\cal
G\text{-}D\text{-}R}\supset\cal{S}_{\cal G\text{-}D}$, i.e, a quantum
measurement saturating $\cal G$-$\cal D$ always saturates $\cal G$-$\cal
D$-$\cal R$, but the converse is not true. We further find that $\cal{S}_{\cal
G\text{-}D\text{-}R}\cap\cal{S}_{\cal G\text{-}R}=\cal{S}_{\cal G\text{-}D}$
from the saturation conditions of $\cal G$-$\cal D$-$\cal R$ and $\cal
G$-$\cal R$ (Table 1). The detailed proof is given in Appendix G. The elements
of $\cal{S}_{\cal D\text{-}R}$ are either unitary operators or von Neumann
measurements by Theorem 2. Now, the Venn diagram of optimal quantum
measurement sets can be constructed as illustrated in Fig. 4, classified based
on the optimality of quantum measurement to reach the upper bounds of
information contents by the trade-off relations.

Let us consider some examples of different types of quantum measurements.

(i) Assume that a von Neumann measurement $\hat{P}=|i\rangle\langle i|$ is
performed on arbitrary $d$-dimensional quantum states. It allows one to obtain
the maximum information ${\cal G}=2/(d+1)$. Therefore, no significant
information remains on the post-measurement state ${\cal F}=2/(d+1)$ nor is
recoverable ${\cal R}=0$ (we remind that the range of information gain,
operation fidelity, and reversibility are $1/d\leq{\cal G}\leq 2/(d+1)$,
$2/(d+1)\leq{\cal F}\leq 1$, and $0\leq{\cal R}\leq 1$, respectively). It is
straightforward to see that these quantities saturate all the trade-off
relations (Fig. 4).

(ii) Consider a quantum measurement $\hat{M}_{i}=\sqrt{p}|i\rangle\langle
i|+\sqrt{(1-p)/2}(\hat{\mathbbm{1}}-|i\rangle\langle i|)$ ($i=0,1,2$) for
$1/3\leq p\leq 1$. It becomes a von Neumann measurement when $p=1$ and a
unitary operator when $p=1/3$. It causes a partial collapse of the input state
and can be reversed for $1/3<p<1$. For each outcome $i$, the optimal reversing
operation is defined by $\hat{R}_{i,1}=\sqrt{(1-p)/2p}|i\rangle\langle
i|+(\hat{\mathbbm{1}}-|i\rangle\langle i|)$ (success) and
$\hat{R}_{i,2}=\sqrt{(3p-1)/2p}|i\rangle\langle i|$ (non-success). We obtain
the information contents as ${\cal G}=(1+p)/4$, ${\cal
F}=(3-p+2\sqrt{2p(1-p)})/4$, and ${\cal R}=3(1-p)/2$, which are saturating
$\cal G$-$\cal D$, $\cal G$-$\cal D$-$\cal R$, and $\cal G$-$\cal R$
relations, i.e., $\\{\hat{M}_{r}\\}\in S_{\cal G\text{-}D\text{-}R}$,
$\\{\hat{M}_{r}\\}\in S_{\cal G\text{-}D}$ and $\\{\hat{M}_{r}\\}\in S_{\cal
G\text{-}R}$.

(iii) Consider a quantum measurement $\hat{M}_{i}=\sqrt{p}|i\rangle\langle
i|+\sqrt{2(1-p)/3}|i+1\rangle\langle i+1|+\sqrt{(1-p)/3}|i+2\rangle\langle
i+2|$ ($i=0,1,2$) for $2/5\leq p\leq 1$ with $|i\rangle\equiv|i\mod 3\rangle$
in 3-dimensional Hilbert space. It can be optimally reversed by
$\hat{R}_{i,1}=\sqrt{(1-p)/3p}|i\rangle\langle i|+\sqrt{1/2}|i+1\rangle\langle
i+1|+|i+2\rangle\langle i+2|$ (success) and
$\hat{R}_{i,2}=\sqrt{(4p-1)/3p}|i\rangle\langle
i|+\sqrt{1/2}|i+1\rangle\langle i+1|$ (non-success). We obtain ${\cal
G}=(1+p)/4$, ${\cal F}=(3+\sqrt{2}(1-p)+(\sqrt{3}+\sqrt{6})\sqrt{p(1-p)})/6$,
and ${\cal R}=1-p$. These saturate $\cal G$-$\cal D$-$\cal R$ but do not
saturate $\cal G$-$\cal D$ and $\cal G$-$\cal R$, i.e., $\\{\hat{M}_{r}\\}\in
S_{\cal G\text{-}D\text{-}R}$, $\\{\hat{M}_{r}\\}\notin S_{\cal G\text{-}D}$
and $\\{\hat{M}_{r}\\}\notin S_{\cal G\text{-}R}$.

(iv) Consider a quantum measurement $\hat{M}_{i}=\sqrt{1/3}|i\rangle\langle
i|+\sqrt{p/6}(|i+1\rangle\langle i+1|+|i+2\rangle\langle i+2|)$ ($i=0,1$) and
$\hat{M}_{2}=\sqrt{(3-p)/3}|2\rangle\langle 2|+\sqrt{(4-p)/6}(|0\rangle\langle
0|+|1\rangle\langle 1|)$ for $0\leq p\leq 1$. It can be optimally reversed by
$\hat{R}_{i,1}=\sqrt{p/2}|i\rangle\langle i|+|i+1\rangle\langle
i+1|+|i+2\rangle\langle i+2|$ (success) and
$\hat{R}_{i,2}=\sqrt{(1-p)/2}|i\rangle\langle i|$ (non-success) for $i=0,1$,
and $\hat{R}_{2,1}=\sqrt{(4-p)/2(3-p)}|2\rangle\langle 2|+|0\rangle\langle
0|+|1\rangle\langle 1|$ (success) and
$\hat{R}_{2,2}=\sqrt{(2-p)/2(3-p)}|2\rangle\langle 2|$ (non-success). We
obtain ${\cal G}=(14-p)/36$, ${\cal
F}=(32-p+4\sqrt{2p}+2\sqrt{6-2p}\sqrt{4-p})/72$, and ${\cal R}=(4+p)/6$.
Notably, these saturate $\cal G$-$\cal R$ but do not saturate $\cal G$-$\cal
D$-$\cal R$ and $\cal G$-$\cal D$, i.e., $\\{\hat{M}_{r}\\}\in S_{\cal
G\text{-}R}$, $\\{\hat{M}_{r}\\}\notin S_{\cal G\text{-}D\text{-}R}$ and
$\\{\hat{M}_{r}\\}\notin S_{\cal G\text{-}D}$.

(v) Consider a weak quantum measurement with operators
$\hat{M}_{1}=\sqrt{p}|1\rangle\langle 1|$ and $\hat{M}_{2}=|0\rangle\langle
0|+\sqrt{1-p}|1\rangle\langle 1|+|2\rangle\langle 2|$, performed on an
arbitrary quantum state $|\psi\rangle$ in 3-dimensional Hilbert space. While
the input state $|\psi\rangle$ is completely collapsed on $|1\rangle$ when the
outcome $r=1$, it is partially collapsed when $r=2$ so that the measurement is
reversible for $p<1$. The optimal reversing operators are given by
$\hat{R}_{2,1}=\sqrt{1-p}|0\rangle\langle 0|+|1\rangle\langle
1|+\sqrt{1-p}|2\rangle\langle 2|$ (success) and
$\hat{R}_{2,2}=\sqrt{p}|0\rangle\langle 0|+\sqrt{p}|2\rangle\langle 2|$ (non-
success). The amount of obtained information is ${\cal G}=(4+p)/12$, while the
remaining and the reversible ones are ${\cal F}=(2+\sqrt{1-p})/3$ and ${\cal
R}=1-p$, respectively. While these quantities satisfy all the trade-off
relations, they do not saturate any of them.

We have observed that all of the above examples satisfy the trade-off
inequalities. The examples (i) - (iv) are optimal quantum measurements that
saturate at least one of the trade-off relations. In an optimal quantum
measurement, the total information is divided into $\cal G$, $\cal D$, and
$\cal R$ and balanced by the change of the parameter $p$. On the other hand,
the example (v) is non-optimal which satisfies all the trade-off inequalities
but does not saturate any of them. Each of the examples (i) - (v) given above
represents each divided region in the diagram of Fig. 4.

##  6 Discussion

We have established the complete information balance in quantum measurement by
deriving the full quantitative trade-off relations among information gain,
disturbance, and reversibility. Under a quantum measurement, the initial
information contained in the input state is divided into i) the obtained
information $\cal G$, ii) the remaining information $\cal F$ in the post-
measurement state, and iii) the reversible information $\cal R$. Our result
clearly shows that the three quantities are balanced by the trade-off
relations. The reversibility $\cal R$ turns out to play an essential role in
completing the information balance, filling the gap between the information
gain $\cal G$ and disturbance ${\cal D}=1-{\cal F}$. Note that the three
information contents are defined to be universal for a given dimension by
covering all possible input states and have clear operational meanings,
fulfilling a general requirement as an information content in quantum
measurement [26].

While all quantum measurements, including noisy or weak measurements, must
satisfy the trade-off relations, the conditions to saturate them define
optimal quantum measurements. Those optimal measurements may be said to
conserve the total information in a nontrivial form according to the trade-off
relation. We have classified all quantum measurements into different sets
based on their optimality to reach the upper bounds of $\cal G$, $\cal D$, and
$\cal R$. We then thoroughly analyzed their relations resulting in the Venn
diagram in Fig. 4. These may provide useful guidelines for designing a quantum
measurement according to its aim in quantum information protocol. For example,
a maximum information gain with minimal disturbance is desirable for
estimating or discriminating quantum information [41], whereas a maximum
reversibility suits the aim of transmitting [10] or protecting [12] quantum
information.

Our results imply that total information does not increase in quantum
measurement and reversal process and may hint at the extension of the
information conservation law. It should be obviously the case with a unitarity
process in line with the second law of thermodynamics [49, 50]. Extending to
the non-unitary quantum measurement and selective processes, our optimal
measurements saturating trade-off relations may be considered as defining the
information conservation.

Our results can also be interpreted as a quantitative refinement of the no-
cloning theorem [51] in the context of quantum measurement. As a direct
application, we can consider a universal cloning machine, which has been
analyzed so far either as a deterministic process with a lower fidelity [52,
53, 54, 55, 56, 57] or a probabilistic one for an exact cloning [58, 59]. By
contrast, our results allow us to optimize further the protocol by
compromising the output fidelity and the success probability of the cloning.
For example, consider a $1\rightarrow N+1$ asymmetric, probabilistic, cloning
machine yielding

$|\psi\rangle\rightarrow\eta_{r,l}|\tilde{\psi}_{r}\rangle^{\otimes
N}|\psi\rangle,$ (16)

where $|\tilde{\psi}_{r}\rangle^{\otimes N}$ are the approximate $N$ copies
and $|\psi\rangle$ a perfect copy. Employing the optimal measurement and the
reversal process can realize this cloning machine when $N\rightarrow\infty$,
as illustrated in Fig. 5, with $|\eta_{r,l}|^{2}$ the success probability, and
$r$ and $l$ the outcomes of the measurement and the reversal process,
respectively. Compared to the deterministic version [55, 56, 57], our results
provide a method to optimize the cloning process further with enhanced
fidelities at the expense of success probability. Note that the trade-off
relations among $\cal G$, $\cal D$, and $\cal R$ determine the quantitative
upper bound of the performance of the cloning machine.

Figure 5: The state estimation by a quantum measurement $\\{\hat{M}_{r}\\}$
followed by a reversal process is equivalent to an $1\rightarrow N+1$
asymmetric probabilistic cloning machine as $N\rightarrow\infty$. The output
at the bottom yields exactly the same state as the input state with the
probability given by $\cal R$, while the upper estimation part yields an
infinite number of approximated copies with the fidelity given by $\cal G$.

An important path for further study is exploring practical applications, e.g.
measurement-based quantum processor, teleportation, metrology, and quantum
error correction. Characterizing the information flow in sequential quantum
measurements would also be interesting, in which the uncertainty relation
between incompatible measurements may be crucial. It may also be valuable to
translate our results into the context of quantum thermodynamics [60, 61]. As
the information contents $\cal G$, $\cal D$, and $\cal R$ are defined as
measurable quantities, the derived trade-off relations are ready to be tested
in any quantum information platform, e.g., superconducting [37], ion trap
[38], and photonic qubits [35, 36, 39, 41, 42]. Our work establishes
fundamental criteria for characterizing quantum measurement and offers useful
guidelines for designing optimal measurement-based quantum processors.

This work was supported in part by the KIAS Advanced Research Program
(QP029902 and CG014604), the National Research Foundation of Korea (NRF)
(2020M3E4A1079939), and the KIST institutional program (2E31021).

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##  Appendix A Definitions

###  A.1 Singular value decomposition

By a singular value decomposition, a measurement operator $\hat{M}_{r}$ can be
represented as $\hat{M}_{r}=\hat{V}_{r}\hat{D}_{r}\hat{W}_{r}$ in terms of
unitary operators $\hat{W}_{r}$ and $\hat{V}_{r}$ and a diagonal matrix
$\hat{D}_{r}=\sum_{i=0}^{d-1}\lambda^{r}_{i}|i\rangle\langle i|$. Without loss
of generality, we set $\hat{W}_{r}=\hat{\mathbbm{1}}$ and also assume that the
singular values are defined here in decreasing order,
$\lambda^{r}_{0}\geq\lambda^{r}_{1}\geq\ldots\geq\lambda^{r}_{d-1}\geq 0$.
Note that the singular values satisfy the completeness relation

$\sum_{r}\sum_{i}(\lambda^{r}_{i})^{2}=d.$ (17)

Similarly, the operator of reversing operation can be represented as
$\hat{R}_{r,l}=\hat{V}_{r,l}\hat{D}_{r,l}\hat{W}_{r,l}$ with unitary operators
$\hat{W}_{r,l}$ and $\hat{V}_{r,l}$ and a diagonal matrix
$\hat{D}_{r,l}=\sum_{i=0}^{d-1}\lambda^{r,l}_{i}|i\rangle\langle i|$. From the
completeness relation
$\sum_{l}\hat{R}_{r,l}^{{\dagger}}\hat{R}_{r,l}=\hat{\mathbbm{1}}$, the
singular values for the reversing operators satisfy

$\sum_{l}\sum_{i}(\lambda^{r,l}_{i})^{2}=d.$ (18)

Without loss of generality, we can set $\hat{V}_{r,l}=\hat{\mathbbm{1}}$ so
that the completeness relation can be written by
$\sum_{l}(\lambda^{r,l}_{i})^{2}=1$.

###  A.2 Information Gain

For a given ${\bf M}=\\{\hat{M}_{r}\\}$, the fidelity between the input and
the estimated states $|\langle\psi|\widetilde{\psi}_{r}\rangle|^{2}$ can be
averaged over all possible input states and the measurement outcomes $r$ as

${\cal\bar{G}}({\bf M})=\int
d\psi\sum_{r=1}^{n}p(r,\psi)|\langle\widetilde{\psi}_{r}|\psi\rangle|^{2}.$
(19)

Let us introduce the Schur’s lemma, which can be used for any operator
$\hat{O}$ in $d\times d$ Hilbert space;

$\displaystyle\int_{G}$ $\displaystyle
dg\Big{(}\hat{U}^{{\dagger}}_{g}\otimes\hat{U}^{{\dagger}}_{g}\Big{)}\hat{O}\left[\hat{U}_{g}\otimes\hat{U}_{g}\right]=\alpha_{1}\hat{\mathbbm{1}}\otimes\hat{\mathbbm{1}}+\alpha_{2}\hat{S},$
(20)
$\displaystyle\alpha_{1}=\frac{d^{2}\mathrm{Tr}[\hat{O}]-d\mathrm{Tr}[\hat{O}\hat{S}]}{d^{2}(d^{2}-1)},\leavevmode\nobreak\
\alpha_{2}=\frac{d^{2}\mathrm{Tr}[\hat{O}\hat{S}]-d\mathrm{Tr}[\hat{O}]}{d^{2}(d^{2}-1)},$

where $\hat{U}_{g}$ is a unitary representation of $d$-dimensional unitary
group $g\in G=\mbox{U}(d)$ such that $\int_{G}dg=1$ and $\hat{S}$ is a swap
operator defined as
$\hat{S}|i\rangle\otimes|j\rangle=|j\rangle\otimes|i\rangle$. Then, Eq. (19)
is written as

$\sum_{r=1}^{n}\int
d\psi\langle\psi|\otimes\langle\psi|\Big{(}\hat{M}_{r}^{\dagger}\hat{M}_{r}\otimes|\widetilde{\psi}_{r}\rangle\langle\widetilde{\psi}_{r}|\Big{)}|\psi\rangle\otimes|\psi\rangle,$

which can be recast by the Schur’s lemma into

${\cal\bar{G}}({\bf
M})=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}\langle\widetilde{\psi}_{r}|\hat{M}_{r}^{\dagger}\hat{M}_{r}|\widetilde{\psi}_{r}\rangle\bigg{]}.$
(21)

The second term can be written again as
$\langle\widetilde{\psi}_{r}|\hat{M}_{r}^{\dagger}\hat{M}_{r}|\widetilde{\psi}_{r}\rangle=\langle\widetilde{\psi}_{r}|\hat{D}^{\dagger}_{r}\hat{V}^{\dagger}_{r}\hat{V}_{r}\hat{D}_{r}|\widetilde{\psi}_{r}\rangle=\sum_{i}\langle\widetilde{\psi}_{r}|\hat{D}^{\dagger}_{r}|i\rangle\langle
i|\hat{D}_{r}|\widetilde{\psi}_{r}\rangle=\sum_{i}(\lambda^{r}_{i})^{2}|\langle
i|\widetilde{\psi}_{r}\rangle|^{2}$. Then, the information gain can be defined
as its maximum value, obtained when the estimate state is
$|\widetilde{\psi}_{r}\rangle=|0\rangle$ for outcome $r$ so that

${\cal G}\equiv{\cal\bar{G}}_{\max}({\bf
M})=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}(\lambda_{0}^{r})^{2}\bigg{]}.$
(22)

Note that this is valid for arbitrary input states $\rho$, since the maximum
evaluated in the space of pure states is also the maximum evaluated in the
space of mixed states due to the convex structure of quantum states.

###  A.3 Disturbance

The fidelity between the input and the output states
$|\langle\psi_{r}|\psi\rangle|^{2}$, for a given quantum measurement ${\bf
M}=\\{\hat{M}_{r}\\}$, can be averaged over all possible input states and
measurement outcomes $r$ as

${\cal\bar{F}}({\bf M})=\int
d\psi\sum_{r=1}^{n}p(r,\psi)|\langle\psi_{r}|\psi\rangle|^{2}=\int
d\psi\sum^{n}_{r=1}P(r,\psi)\frac{\langle\psi|\hat{M}_{r}|\psi\rangle\langle\psi|\hat{M}^{\dagger}_{r}|\psi\rangle}{P(r,\psi)}=\int
d\psi\sum^{n}_{r=1}\Big{|}\langle\psi|\hat{M}_{r}|\psi\rangle\Big{|}^{2}.$
(23)

By the Schur’s lemma, we can rewrite it as

${\cal\bar{F}}({\bf
M})=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}\big{|}\mathrm{Tr}\hat{M}_{r}\big{|}^{2}\bigg{]}.$
(24)

Since $|\mathrm{Tr}\hat{M}_{r}|=|\sum_{i}\langle
i|\hat{V}_{r}\hat{D}_{r}|i\rangle|=|\sum_{i}\langle
i|\hat{V}_{r}\hat{D}_{r}|i\rangle|=|\sum_{i}\lambda_{i}^{r}\langle
i|\hat{V}_{r}|i\rangle|\leq\sum_{i}\lambda_{i}^{r}|\langle
i|\hat{V}_{r}|i\rangle|\leq\sum_{i}\lambda_{i}^{r}$, the maximum operation
fidelity is then given as

${\cal F}\equiv{\cal\bar{F}}_{max}({\bf
M})=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}\Big{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\Big{)}^{2}\bigg{]},$
(25)

from which the disturbance can also be defined as the minimum operation
infidelity ${\cal D}=1-{\cal F}$.

###  A.4 Maximum Operation Fidelity after Reversal

For given ${\bf M}=\\{\hat{M}_{r}\\}$ and ${\bf R}=\\{\hat{R}_{r,l}\\}$, the
operation fidelity including all output states without postselection is

${\cal\bar{F}}({\bf R}\circ{\bf M})=\int
d\psi\sum^{n}_{r=1}\sum^{m}_{l=1}P(r,l,\psi)|\langle\psi_{r,l}|\psi\rangle|^{2},$
(26)

where the right hand side can be written as

$\int
d\psi\sum^{n}_{r=1}\sum^{m}_{l=1}P(r,l,\psi)\frac{\langle\psi|\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle\langle\psi|\hat{M}^{\dagger}_{r}\hat{R}^{\dagger}_{r,l}|\psi\rangle}{P(r,l,\psi)}=\int
d\psi\sum^{n}_{r=1}\sum^{m}_{l=1}\Big{|}\langle\psi|\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle\Big{|}^{2}.$
(27)

From Eq. (24), it can be written again as

${\cal\bar{F}}({\bf R}\circ{\bf
M})=\frac{1}{d(d+1)}\bigg{[}d+\sum^{n}_{r=1}\sum^{m}_{l=1}\Big{|}\mathrm{Tr}\hat{R}_{r,l}\hat{M}_{r}\Big{|}^{2}\bigg{]}.$
(28)

Since

$\Big{|}\mathrm{Tr}\hat{R}_{r,l}\hat{M}_{r}\Big{|}=\Big{|}\sum_{i}\langle
i|\hat{D}_{r,l}\hat{W}_{r,l}\hat{V}_{r}\hat{D}_{r}|i\rangle\Big{|}=\Big{|}\sum_{i}\lambda_{i}^{r}\lambda_{i}^{r,l}\langle
i|\hat{W}_{r,l}\hat{V}_{r}|i\rangle\Big{|}\leq\sum_{i}\lambda_{i}^{r}\lambda_{i}^{r,l}\Big{|}\langle
i|\hat{W}_{r,l}\hat{V}_{r}|i\rangle\Big{|},$ (29)

the maximum of ${\cal\bar{F}}({\bf R}\circ{\bf M})$ is obtained when
$\hat{W}_{r,l}\hat{V}_{r}=\hat{\mathbbm{1}}$. Therefore,

${\cal\bar{F}}_{\max}({\bf R}\circ{\bf
M})=\frac{1}{d(d+1)}\bigg{[}d+\sum_{r=1}^{n}\sum^{m}_{l=1}\Big{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\lambda_{i}^{r,l}\Big{)}^{2}\bigg{]}.$
(30)

###  A.5 Reversibility

We assume that $\hat{R}_{r,l}$ for $l=1,2,...,s$ are associated with the
success reversal events such that

$\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle=\eta_{r,l}|\psi\rangle,$ (31)

with a complex variable $\eta_{r,l}$. Since
$\hat{\mathbbm{1}}-\sum^{s}_{l=1}\hat{R}^{\dagger}_{r,l}\hat{R}_{r,l}$ is
positive definite from the completeness relation,

$\sup_{|\phi\rangle}\langle\phi|\sum^{s}_{l=1}\hat{R}^{\dagger}_{r,l}\hat{R}_{r,l}|\phi\rangle\leq
1$ (32)

is satisfied for arbitrary quantum state $|\phi\rangle$. Then,

$\sup_{|\phi\rangle}\langle\phi|\sum^{s}_{l=1}\hat{R}^{\dagger}_{r,l}\hat{R}_{r,l}|\phi\rangle\geq\sup_{|\psi_{r}\rangle}\langle\psi_{r}|\sum^{s}_{l=1}\hat{R}^{\dagger}_{r,l}\hat{R}_{r,l}|\psi_{r}\rangle=\sup_{|\psi\rangle}\frac{\langle\psi|\hat{M}^{{\dagger}}_{r}\sum^{s}_{l=1}\hat{R}^{\dagger}_{r,l}\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle}{p(r,\psi)}=\frac{\sum^{s}_{l=1}|\eta_{r,l}|^{2}}{\inf_{|\psi\rangle}p(r,\psi)},$
(33)

so that $\sum^{s}_{l=1}|\eta_{r,l}|^{2}\leq\inf_{|\psi\rangle}p(r,\psi)$ is
satisfied. For an arbitrary input state
$|\psi\rangle=\sum_{i=0}^{d-1}\alpha_{i}|i\rangle$ where
$\sum_{i=0}^{d-1}|\alpha_{i}|^{2}=1$, $\inf_{|\psi\rangle}p(r,\psi)$ is
obtained when $\alpha_{d-1}=1$ and all other $\alpha_{i}$ are zero, because
the singular values of $\hat{M}_{r}$ are assumed to be defined in decreasing
order, so that

$\sum^{s}_{l=1}|\eta_{r,l}|^{2}\leq\inf_{|\psi\rangle}p(r,|\psi\rangle)=(\lambda^{r}_{d-1})^{2}.$
(34)

Therefore, the reversibility can be defined as its maximum

${\cal R}\equiv\max_{\\{\hat{R}_{r,l}\\}}\int
d\psi\sum^{n}_{r=1}\sum^{s}_{l=1}|\langle\psi|\hat{R}_{r,l}\hat{M}_{r}|\psi\rangle|^{2}=\sum_{r=1}^{n}(\lambda^{r}_{d-1})^{2}.$
(35)

##  Appendix B Proof of Lemma 1

For an optimally chosen ${\bf R}=\\{\hat{R}_{r,l}\\}$ to reverse ${\bf
M}=\\{\hat{M}_{r}\\}$, the singular values are given by
$\lambda_{i}^{r,l=1}=\lambda_{d-1}^{r}/\lambda_{i}^{r}$. If we define
$\vec{u}_{i}^{l}=(\lambda_{i}^{r=1}\lambda_{i}^{r=1,l},\ldots,\lambda_{i}^{r=n}\lambda_{i}^{r=n,l})$,
the second term of ${\cal F}({\bf R}\circ{\bf M})$ in Eq. (4)

$\sum_{r=1}^{n}\sum^{m}_{l=1}\Big{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\lambda_{i}^{r,l}\Big{)}^{2}=\sum_{i,j}\sum_{l}\vec{u}_{i}^{l}\ldots\vec{u}_{j}^{l}=\sum_{i,j}(\vec{u}_{i}^{l=1}\cdot\vec{u}_{j}^{l=1}+\sum_{l\neq
1}\vec{u}_{i}^{l}\cdot\vec{u}_{j}^{l})=d^{2}\sum_{r}(\lambda^{r}_{d-1})^{2}+\sum_{i,j}\sum_{l\neq
1}\vec{u}_{i}^{l}\cdot\vec{u}_{j}^{l},$ (36)

and $\sum_{i,j}\sum_{l\neq
1}\vec{u}_{i}^{l}\cdot\vec{u}_{j}^{l}\geq\sum_{i}\sum_{l\neq
1}|\vec{u}_{i}^{l}|^{2}$, where the equality is reached with a condition
$\vec{u}_{i}^{l\neq 1}\cdot\vec{u}_{j}^{l\neq
1}=\delta_{i,j}|\vec{u}_{i}^{l}|^{2}$. From the completeness relation of the
overall measurements,

$\displaystyle\sum_{r=1}^{n}\sum^{m}_{l=1}\sum_{i=0}^{d-1}(\lambda_{i}^{r}\lambda_{i}^{r,l})^{2}=d\sum_{r}(\lambda^{r}_{d-1})^{2}+\sum_{i}\sum_{l\neq
1}|\vec{u}_{i}^{l}|^{2}=d.$

Therefore,

$\displaystyle{\cal F}({\bf R}\circ{\bf M})$
$\displaystyle=\frac{1}{d(d+1)}\bigg{[}d+d^{2}\sum_{r}(\lambda^{r}_{d-1})^{2}+\sum_{i,j}\sum_{l\neq
1}\vec{u}_{i}^{l}\cdot\vec{u}_{j}^{l}\bigg{]}$
$\displaystyle\geq\frac{1}{d(d+1)}\big{[}2d+(d^{2}-d){\cal R}\big{]},$ (37)

where ${\cal R}=\sum_{r}(\lambda^{r}_{d-1})^{2}$. As a result, we obtain

$2+(d-1){\cal R}\leq(d+1){\cal F}({\bf R}\circ{\bf M}),$ (38)

with the equality condition $\vec{u}_{i}^{l\neq 1}\cdot\vec{u}_{j}^{l\neq
1}=\delta_{i,j}|\vec{u}_{i}^{l}|^{2}$. $\square$

##  Appendix C Proof of Theorem 1

Let us first define
$\vec{v}_{i}=(\lambda^{r=1}_{i},\ldots,\lambda^{r=n}_{i})$. Then, $g\equiv
d(d+1){\cal G}-d=\sum_{r}(\lambda_{0}^{r})^{2}=|\vec{v}_{0}|^{2}$, $f\equiv
d(d+1){\cal
F}-d=\sum_{r}(\sum_{i}\lambda_{i}^{r})^{2}=\sum_{i,j}\vec{v}_{i}\cdot\vec{v}_{j}$,
and ${\cal R}=\sum_{r}(\lambda^{r}_{d-1})^{2}=|\vec{v}_{d-1}|^{2}$. The
completeness relation in Eq. (17) can be written as
$\sum_{r}\sum_{i}(\lambda^{r}_{i})^{2}=\sum_{i}|\vec{v}_{i}|^{2}=d$.

From the Schwarz inequality,

$f\leq\sum^{d-1}_{i,j=0}|\vec{v}_{i}||\vec{v}_{j}|=\Big{(}\sum_{i}|\vec{v}_{i}|\Big{)}^{2}=\bigg{(}\sqrt{g}+\sqrt{{\cal
R}}+\sum^{d-2}_{i=1}|\vec{v}_{i}|\bigg{)}^{2},$ (39)

where the equality can be reached when all the vectors $\vec{v}_{i}$ are
collinear. Then, from the inequality of arithmetic and quadratic means,

$\sum^{d-2}_{i=1}|\vec{v}_{i}|\leq\sqrt{(d-2)\sum^{d-2}_{i=1}|\vec{v}_{i}|^{2}},$
(40)

where the equality can be reached when $|\vec{v}_{1}|=\cdots=|\vec{v}_{d-2}|$.
Here, the right hand side can be rewritten by the completeness relation as
$\sqrt{(d-2)(d-g-{\cal R})}$. Therefore, we obtain

$\sqrt{f}\leq\sqrt{g}+\sqrt{{\cal R}}+\sqrt{(d-2)(d-g-{\cal R})},$ (41)

and equivalently

$\sqrt{{\cal F}-\frac{1}{d+1}}\leq\sqrt{{\cal
G}-\frac{1}{d+1}}+\sqrt{\frac{{\cal
R}}{d(d+1)}}+\sqrt{(d-2)\bigg{(}\frac{2}{d+1}-{\cal G}-\frac{{\cal
R}}{d(d+1)}\bigg{)}}.$ (42)

$\square$

##  Appendix D Proof that $\cal G$-$\cal D$-$\cal R$ tightens $\cal G$-$\cal
D$

Assume that we have $d-1$ non-negative real numbers $x_{i}$ where
$i=1,\cdots,d-1$. From the inequality between arithmetic and quadratic mean,

$\bigg{(}\frac{1}{d-1}\sum^{d-1}_{i=1}x_{i}\bigg{)}^{2}\leq\frac{1}{d-1}\sum^{d-1}_{i=1}x^{2}_{i}.$
(43)

where the equality holds if and only if $x_{1}=x_{2}=\cdots=x_{d-1}$. By
letting $x_{1}=\sqrt{{\cal R}}$ and $x_{2}=\cdots=x_{d-1}=\sqrt{(d-g-{\cal
R})/(d-2)}$ with $g\equiv d(d+1){\cal G}-d$, we obtain

$\sqrt{{\cal R}}+\sqrt{(d-2)(d-g-{\cal R})}\leq\sqrt{(d-1)(d-g)},$ (44)

which indicates that $\cal G$-$\cal D$-$\cal R$ is tighter than $\cal G$-$\cal
D$. $\square$

##  Appendix E Proof of Lemma 2

If we define a vector
$\vec{w}_{i}^{r}=(\lambda_{i}^{r}\lambda_{i}^{r,l=1},\ldots,\lambda_{i}^{r}\lambda_{i}^{r,l=m})$,
the second term of ${\cal F}({\bf R}\circ{\bf M})$ in Eq. (4) can be written
by

$\sum_{r=1}^{n}\sum^{m}_{l=1}\bigg{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\lambda_{i}^{r,l}\bigg{)}^{2}=\sum_{r=1}^{n}\sum_{i,j}\vec{w}_{i}^{r}\cdot\vec{w}_{j}^{r}.$
(45)

Then, by the Schwarz inequality,

$\sum_{r=1}^{n}\sum_{i,j}\vec{w}_{i}^{r}\cdot\vec{w}_{j}^{r}\leq\sum_{r=1}^{n}\sum_{i,j}\big{|}\vec{w}_{i}^{r}\big{|}\big{|}\vec{w}_{j}^{r}\big{|}=\sum_{r=1}^{n}\Big{(}\sum_{i}\big{|}\vec{w}_{i}^{r}\big{|}\Big{)}^{2}=\sum_{r=1}^{n}\bigg{(}\sum_{i}\lambda^{r}_{i}\sqrt{\sum_{l}(\lambda^{r,l}_{i})^{2}}\bigg{)}^{2},$
(46)

where the equality holds if all the vectors $\vec{w}_{i}^{r}$ are collinear.
From the completeness relation for the reversing operators in Eq. (18), we
arrive at

$\sum_{r=1}^{n}\sum^{m}_{l=1}\bigg{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\lambda_{i}^{r,l}\bigg{)}^{2}\leq\sum_{r=1}^{n}\bigg{(}\sum_{i=0}^{d-1}\lambda_{i}^{r}\bigg{)}^{2},$
(47)

and, from Eq. (2),

${\cal F}({\bf R}\circ{\bf M})\leq{\cal F}({\bf M}).$ (48)

$\square$

##  Appendix F Reversibility with errors

Using a non-optimal quantum measurement or operation, the reversing process
may not be able to exactly reverse the quantum measurement due to errors so
that the output state is not exactly the same as the input state, s.t. $({\bf
R}\circ{\bf M})(\rho)\propto\rho^{\prime}\neq\rho$. Let us consider the case
that the reversing operation succeeds after measurement as

$\hat{R}_{r,1}\hat{M}_{r}|\psi\rangle\langle\psi|\hat{M}_{r}^{\dagger}\hat{R}_{r,1}^{\dagger}=|\eta_{r,1}|^{2}\rho(\psi,\epsilon)$
(49)

but the resulting output state $\rho(\psi,\epsilon)$ deviates from the
original input state. Here, we consider an arbitrary error model parameterized
by $0\leq\epsilon\leq 1$ and the error density function $p(\epsilon)$, s.t.
$\int^{1}_{0}p(\epsilon)d\epsilon=1$. We assume that the error occurs
independently of the measurement outcome for simplicity.

From the definition of the reversibility in Eq. (5), we can evaluate

$\displaystyle{\cal R}(\epsilon)$
$\displaystyle=\max_{\\{\hat{R}_{r,1}\\}}\int
d\psi\sum_{r}\langle\psi|\hat{R}_{r,1}\hat{M}_{r}|\psi\rangle\langle\psi|\hat{M}_{r}^{\dagger}\hat{R}_{r,1}^{\dagger}|\psi\rangle$
(50) $\displaystyle=\max_{\\{\hat{R}_{r,1}\\}}\int
d\psi\sum_{r}|\eta_{r,1}|^{2}\langle\psi|\rho(\psi,\epsilon)|\psi\rangle$
$\displaystyle=\sum_{r}(\lambda^{r}_{d-1})^{2}\int
d\psi\langle\psi|\rho(\psi,\epsilon)|\psi\rangle.$

By averaging over the whole error region, we obtain

${\cal R}=\int^{1}_{0}d\epsilon\leavevmode\nobreak\ p(\epsilon){\cal
R}(\epsilon)=\sum_{r}(\lambda^{r}_{d-1})^{2}F_{s},$ (51)

where $F_{s}=\int d\psi\int^{1}_{0}d\epsilon\leavevmode\nobreak\
p(\epsilon)\langle\psi|\rho(\psi,\epsilon)|\psi\rangle$ is defined as the
average fidelity between the input state and the output state when the
reversing operation succeeds.

##  Appendix G Proof of $\cal{S}_{\cal G\text{-}D\text{-}R}\cap\cal{S}_{\cal
G\text{-}R}=S_{\cal G\text{-}D}$

We can prove that a quantum measurement satisfying both saturation conditions
of $\cal G$-$\cal D$-$\cal R$ and $\cal G$-$\cal R$ always saturates $\cal
G$-$\cal D$. From the condition to saturate $\cal G$-$\cal D$-$\cal R$
described in Eq. (11), the singular values satisfy
$\lambda^{r}_{0}\geq\lambda^{r}_{1}=\cdots=\lambda^{r}_{d-2}\geq\lambda^{r}_{d-1}$.
The measurement operator is then written by
$\hat{M}_{r}=\lambda^{r}_{0}|0\rangle\langle
0|+\lambda^{r}_{1}(|1\rangle\langle 1|+\cdots+|d-2\rangle\langle
d-2|)+\lambda^{r}_{d-1}|d-1\rangle\langle d-1|$ so that

$\hat{M}^{{\dagger}}_{r}\hat{M}_{r}=\big{\\{}(\lambda^{r}_{0})^{2}-(\lambda^{r}_{d-1})^{2}\big{\\}}|0\rangle\langle
0|+\big{\\{}(\lambda^{r}_{1})^{2}-(\lambda^{r}_{d-1})^{2}\big{\\}}\big{(}|1\rangle\langle
1|+\cdots+|d-2\rangle\langle
d-2|\big{)}+(\lambda^{r}_{d-1})^{2}\hat{\mathbbm{1}}.$ (52)

If it satisfies the saturation condition of $\cal G$-$\cal R$, i.e.,
$\hat{M}^{{\dagger}}_{r}\hat{M}_{r}=a_{r}|j_{r}\rangle\langle
j_{r}|+b_{r}\hat{\mathbbm{1}}$ with non-negative $a_{r}$ and $b_{r}$ where
$j_{r}\in\\{0,\ldots,d-1\\}$ [40], we find that

$\lambda^{r}_{1}=\lambda^{r}_{d-1}.$ (53)

As a result, the quantum measurement satisfies the saturation condition of
$\cal G$-$\cal D$, i.e., all $\vec{v}_{i}$ are collinear and
$|\vec{v}_{1}|=\cdots=|\vec{v}_{d-1}|$, where
$\vec{v}_{i}=(\lambda^{r=1}_{i},\ldots,\lambda^{r=n}_{i})$ [20]. $\square$