# Completely entangled subspaces of entanglement depth $k$

Maciej Demianowicz  Institute of Physics and Applied Computer Science, Faculty
of Applied Physics and Mathematics, Gdańsk University of Technology,
Narutowicza 11/12, 80-233 Gdańsk, Poland  Kajetan Vogtt  Center for
Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668
Warsaw, Poland  Faculty of Physics, University of Warsaw, Pasteura 5, 02-093
Warsaw, Poland  Remigiusz Augusiak  Center for Theoretical Physics, Polish
Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

###### Abstract

We introduce a new class of entangled subspaces—completely entangled subspaces
of entanglement depth $k$ ($k$-CESs). These are subspaces of multipartite
Hilbert spaces containing only pure states with the entanglement depth at
least $k$. We present an efficient construction of $k$-CESs of any achievable
dimensionality in any multipartite scenario. Further, we discuss the relation
between these subspaces and unextendible product bases (UPBs). In particular,
we establish that there is a non-trivial bound on the cardinality of a UPB
whose orthocomplement is a $k$-CES. Further, we discuss existence of such UPBs
for qubit systems.

##  I Introduction

Entanglement has long been known to be an enabling resource for a variety of
information processing protocols such as for instance quantum communication
[1] or quantum metrology [2, 3]. The ongoing rapid development of quantum
network technologies requires a thorough theoretical characterization of its
multipartite facet. It is a very challenging problem as the description of
quantum systems with multiple nodes is far more complex than that of systems
with only two subsystems [4, 5]. The topic has attracted much attention over
the recent years and many profound results touching its different aspects have
been reported in the literature (see, e.g., [6, 7, 8, 9, 10, 11]). Noteworthy,
the theoretical advancements go largely in parallel with a remarkable progress
in the experimental domain (see, e.g., [12, 13, 14, 15, 16]). Still, despite
all the efforts, many aspects of many body entanglement remain insufficiently
explored and further research towards its better understanding is highly
demanded. This work is in line with this important trend.

A particular useful tool for the characterization of entanglement in
multipartite systems is that of the entanglement depth [17], which tells one
how many particles in a given ensemble are genuinely entangled. The concept
finds applications for example in the domain of cold gases [18, 19, 20]. The
notion of entanglement can also be meaningfully considered for subspaces, not
only for individual states. The most general notion is that of completely
entangled subspaces (CESs), which are those subspaces that contain only pure
entangled states (and in consequence supporting only mixed entangled states)
[21, 22, 23, 24]. More specific classes of CESs considered in the literature
are, e.g., subspaces containing only states with bounded tensor rank
($r$-entangled subspaces) [25, 26], or genuinely entangled subspaces [22, 27],
that is subspaces containing only genuinely multipartite entangled states.

Entangled subspaces play an important role in quantum information science.
Their significance goes beyond purely theoretical as they have also proved to
be practically relevant, for example in quantum error correction [28, 29, 30],
which makes them objects of intrinsic interest (see, e.g., [31, 32, 33], for
some recent results).

Our current contribution is to marry the notions of the entanglement depth and
entangled subspaces. Precisely, we introduce completely entangled subspaces of
entanglement depth $k$ ($k$-CESs), as those containing only pure states whose
entanglement depth is at least $k$. We find the maximal dimensions of such
subspaces and present their universal construction from sets of non-orthogonal
product vectors in any multipartite setup. Further, we discuss the relation
between $k$-CESs and unextendible product bases (UPBs). This includes a
derivation of a non-trivial bound on the cardinalities of UPBs leading to
$k$-CESs and a discussion with a positive conclusion about their existence.

##  II Preliminaries

In this section we introduce the relevant notions and terminology.

We consider quantum states defined on $n$-partite ($n\geq 3$) Hilbert spaces

$\mathcal{H}_{d_{1}\dots
d_{n}}:=\bigotimes_{i=1}^{n}\mathcal{H}_{i}=\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\ldots\otimes\mathbb{C}^{d_{n}}.$
(1)

We will mainly focus on the scenario with equal local dimensions $d_{i}=d$ in
which case the space will be denoted as $\mathcal{H}_{d^{n}}$. Single-partite
subsystems (parties) are denoted $A_{1},A_{2},\dots,A_{n}:=\textbf{A}$. A
division of the set of parties into $K$ non-overlapping non-empty sets $S_{i}$
such that $\bigcup_{i}S_{i}=\textbf{A}$ is called a $K$-partition and denoted
$S_{1}|\dots|S_{K}$.

An $n$-partite state $|\Phi\rangle\in\mathcal{H}_{d_{1}\dots d_{n}}$ is called
$k$-producible if it can be written as

$|\Phi\rangle=|\phi_{1}\rangle\otimes\ldots\otimes|\phi_{M}\rangle,$ (2)

where $|\phi_{i}\rangle$’s are at most $k$-partite states. If a $k$-producible
state is not $(k-1)$-producible at the same time it is said to have the
entanglement depth $k$ [17, 34]. States with the entanglement depth $k=1$ are
named fully product as they are products of pure states on individual
subsystems $A_{i}$. All other states with $k\geq 2$ are said to be entangled
and those with the entanglement depth equal $n$, in particular, are called
genuinely multipartite entangled (GME). These are the states which cannot be
written as a product for any bipartition of the parties. Excellent examples of
GME states are the Greenberger-Horne-Zeilinger (GHZ) states [35],

$|\mathrm{GHZ}_{n}\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes
n}+|1\rangle^{\otimes n}),$ (3)

that belong to a general class of GME states called the graph states [36].

Let us now recall the concept of entangled subspaces. A subspace of a
multipartite Hilbert space $\mathcal{H}_{d_{1}\dots d_{n}}$ is called
completely entangled (CES) if all pure states belonging to it are entangled
[21, 22]. Importantly, entanglement of these states can be of any kind,
including only bipartite. In the extreme case of all states from a subspace
being GME one deals with genuinely entangled subspaces (GESs) [27]. This is
the case for instance for the antisymmetric subspace.

Entangled subspaces are closely related to so-called unextendible product
bases and we will extensively exploit this connection in further parts of the
paper. Below we remind the basic necessary facts about the discussed notions.

Consider a linearly independent set of fully product $n$-partite vectors from
$\mathcal{H}_{d_{1}\dots d_{n}}$

$\mathcal{B}:=\left\\{|\psi^{(j)}\rangle=\bigotimes_{i=1}^{n}|\varphi_{i}^{(j)}\rangle\right\\}_{j=1}^{m},$
(4)

where $|\varphi_{i}^{(j)}\rangle\in\mathbb{C}^{d_{i}}$. $\mathcal{B}$ is
called an unextendible product basis (UPB) iff it does not span the total
Hilbert space $\mathcal{H}_{d_{1}\dots d_{n}}$ and there does not exist a
fully product vector in the subspace complementary to
$\mathrm{span}\>\mathcal{B}$ [37, 38]. We will refer to the number of elements
in $\mathcal{B}$ as the size or cardinality of it.

Clearly, by the very definition of a UPB, its complementary subspace is a CES.
To illustrate this general link with a simple example let us consider the
following set of three-qubit fully product vectors introduced in Ref. [37]:

$\mathcal{S}:=\\{|000\rangle,|1+-\rangle,|-1+\rangle,|+-1\rangle\\},$ (5)

where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$. One can easily verify
that $\mathcal{S}$ is indeed a UPB and only entangled vectors belong to
$(\mathrm{span}\>\mathcal{S})^{\perp}$, which itself is thus a CES. Notably,
the latter subspace is not a GES as it contains biproduct vectors, e.g.,
$|1\rangle\otimes|\phi\rangle$, where $|\phi\rangle$ is an entangled two-qubit
vector orthogonal to $|+-\rangle$, $|1+\rangle$, and $|-1\rangle$.

Concluding this section, we give two results regarding unextendibility of a
set of product vectors that will be our main tools in further parts. The first
of them is the following observation.

######  Fact 1.

[37, 38] Consider a set of product vectors $\mathcal{B}$ defined in Eq. (4).
There exists a fully product vector orthogonal to $\mathcal{B}$ if and only if
there exists a partition of $\mathcal{B}$ into disjoint subsets:
$\mathcal{B}=\mathcal{B}_{1}\cup\dots\cup\mathcal{B}_{n}$, such that for all
subsets $\mathcal{B}_{i}$ ($i=1,2,\dots,n$), the local vectors
$\\{|\varphi_{i}^{(j)}\rangle:|\psi^{(j)}\rangle\in\mathcal{B}_{i}\\}$ do not
span the corresponding local Hilbert spaces
$\mathcal{H}_{i}=\mathbb{C}^{d_{i}}$.

This elegant result often serves as a basis for constructions of UPBs and it
will be used by us for a direct check if one of the considered bases possesses
certain property of unextendibility. Its immediate consequence is a lower
bound on the cardinalities of unextendible sets:

$m\geq\sum_{i=1}^{n}(d_{i}-1)+1.$ (6)

A stronger version of Fact 1 follows as a corollary.

######  Corollary 1.

Consider a set of product vectors $\mathcal{B}$ defined in Eq. (4) satisfying
the condition (6). If for every $i$ any $d_{i}$-tuple of local vectors
$\\{|\varphi_{i}^{(j)}\rangle\\}_{j}$ spans the corresponding local Hilbert
space $\mathcal{H}_{i}=\mathbb{C}^{d_{i}}$, then the set $\mathcal{B}$ is
unextendible with product vectors.

In fact, for sets with the minimal cardinality $\sum_{i}(d_{i}-1)+1$ the
constituent vectors must necessarily have the property of full local spanning
on subsystems given in the corollary above.

Importantly, the above results regarding unextendibility are applicable both
to sets of orthogonal and non-orthogonal vectors.

##  III Completely entangled subspaces of entanglement depth $k$ ($k$-CES)

Let us move to the main body of the paper. We introduce a class of entangled
subspaces defined through the entanglement depth of vectors belonging to them.
We propose the following.

######  Definition 2.

A subspace $S\subset\mathcal{H}_{d_{1}\dots d_{n}}$ is called a completely
entangled subspace of entanglement depth $k$ ($k$-CES) if the entanglement
depth of any vector belonging to $\mathcal{S}$ is at least $k$. Equivalently,
a $k$-CES is void of $(k-1)$-producible states.

Let us remark that for $k=2$ the above definition recovers that of completely
entangled subspaces, whereas for $k=n$ that of genuinely entangled subspaces.

Before delving into a characterization of $k$-CESs, let us give a few simple
examples of them, which will allow for a quick grasp of the notion.

As the first example let us take the discussed above CES complementary to the
UPB given in Eq. (5). Within our new terminology we can now say that it is a
$2$-CES (CES), but it is not a $3$-CES (GES) because, to recall, the
complementary subspace contains a three-qubit vector with the entanglement
depth two.

For the second example, we turn our attention to quantum error correction.
Consider the celebrated nine-qubit Shor’s code [39] described by the vectors

$\displaystyle|\psi_{0}\rangle$
$\displaystyle=|\mathrm{GHZ}_{3}\rangle\otimes|\mathrm{GHZ}_{3}\rangle\otimes|\mathrm{GHZ}_{3}\rangle,$
(7) $\displaystyle|\psi_{1}\rangle$
$\displaystyle=|\overline{\mathrm{GHZ}}_{3}\rangle\otimes|\overline{\mathrm{GHZ}}_{3}\rangle\otimes|\overline{\mathrm{GHZ}}_{3}\rangle,$
(8)

where
$|\overline{\mathrm{GHZ}}_{3}\rangle=(1/\sqrt{2})(|000\rangle-|111\rangle)$
and $|\mathrm{GHZ}_{3}\rangle$ is defined in Eq. (3). Now, the subspace
$\mathrm{span}\\{|\psi_{0}\rangle,|\psi_{1}\rangle\\}$ is a $3$-CES because
the entanglement depth of both $|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ is
three, and, at the same time, any linear combination of these vectors is GME,
i.e., it has the entanglement depth equal to nine.

As the final example we give an elementary general construction. With this aim
take a GES of a $k$-partite Hilbert space. Now, take a tensor product of all
the vectors from this subspace with an arbitrary, possibly different for each
vector, fully product $(n-k)$-partite vector. This clearly results in a
$k$-CES of an $n$-partite Hilbert space. This rather trivial method does not
lead to $k$-CESs of the maximal dimensions for $k<n$ and in Section IV we will
provide a more elaborate construction which does this job.

One of the central problems of the theory of entangled subspaces is the
problem of verification whether a given subspace is entangled and further
determining to which class it belongs to. Noteworthy, some tools have been
developed in the literature so far, which could also be applied for the case
of $k$-CESs. Precisely, the simple criterion based on the entanglement of
basis vectors from Ref. [40] is directly applicable and the hierarchical
method from Ref. [32] can be readily adapted. Another vital problem in the
area is the quantification of subspace entanglement (see [41] for recent
results). Here, also, we find certain methods which can be used in the current
case, e.g., the SDP bounds of Ref. [42].

Let us now investigate how big—in terms of their dimensions—$k$-CESs can be.
The theory of completely entangled subspaces helps us address this question
successfully. For simplicity, we concentrate on the case of equal local
dimensions in our derivation but a properly adjusted argument is applicable
for the general case too. We have the following proposition.

######  Proposition 3.

The maximal attainable dimension of a $k$-CES of $\mathcal{H}_{d^{n}}$ equals

$D_{k-CES}^{max}=d^{n}-td^{k-1}-d^{n-(k-1)t}+t,$ (9)

where

$t=\left\lfloor\frac{n}{k-1}\right\rfloor.$ (10)

###### Proof.

By definition, a $k$-CES $\mathcal{S}\subset\mathcal{H}_{d^{n}}$ does not
contain $(k-1)$-producible pure states. This implies that $\mathcal{S}$ is a
CES in any $r$-partite Hilbert space

$(\mathbb{C}^{d})^{\otimes n_{1}}\otimes(\mathbb{C}^{d})^{\otimes
n_{2}}\otimes\ldots\otimes(\mathbb{C}^{d})^{\otimes n_{r}},$ (11)

where $n_{i}\leq k-1$ and $i=1,2,\dots,r$, corresponding to an $r$-partition
$S_{1}|S_{2}|\dots|S_{r}$ of the parties, where the size of each subset
$S_{i}$ is $|S_{i}|=n_{i}$.

It was shown in Refs. [21, 22] that the maximal attainable dimension of a CES
of $\mathcal{H}_{d_{1}\dots d_{n}}$ is given by
$\prod_{i=1}^{n}d_{i}-\sum_{i=1}^{n}d_{i}+n-1$. Applying this result to (11),
we obtain

$\prod_{i=1}^{r}d^{n_{i}}-\sum_{i=1}^{r}d^{n_{i}}+r-1=d^{n}-\left[\sum_{i=1}^{r}(d^{n_{i}}-1)+1\right],$
(12)

which must further be minimized (equivalently, the term in the square brackets
must be maximized) over $n_{i}$’s to obtain an upper bound on the maximal
dimension of a $k$-CES. This is achieved for $n_{i}=k-1$, $i=1,2,\dots,t$,
$n_{t+1}=n-t(k-1)$, where $t=\left\lfloor n/(k-1)\right\rfloor$ (see Appendix
D). The minimal value of (12) is thus $d^{n}-td^{k-1}-d^{n-(k-1)t}+t$. The
choice above corresponds to the partitions of the parties as follows (we
assume that for $k-1|n$ we have $t$-partitions)

$(\mathbb{C}^{d^{k-1}})^{\otimes t}\otimes\mathbb{C}^{d^{n-(k-1)t}}.$ (13)

Further, generic subspaces are known to saturate bounds on the dimensions of
entangled subspaces (cf. [25]), meaning that a random subspace of a proper
dimension will be a $k$-CES. ∎

##  IV Universal construction of $k$-CESs

While random subspaces do have significance in quantum information theory
(see, e.g., Ref. [43]), it is the explicit constructions, which often help one
analyze these objects in more detail, for example, compute entanglement
measures of states supported on them. In this section, we provide a universal
analytical construction of $k$-CESs, that is, we introduce a method of
building $k$-CESs of any dimension for any combination of the numbers $k$,
$n$, and $d$. With this aim we utilize the approach put forward recently in
[47], which was originally designed for a construction of GESs. It turns out
that it is versatile enough to be translated directly to the current case.

The construction is based on the Vandermonde vectors:

$\displaystyle|v_{d^{n}}(x)\rangle_{\textbf{A}}:=$
$\displaystyle\left(1,x,x^{2},\dots,x^{d^{n}-1}\right)^{T}_{\textbf{A}}$
$\displaystyle=$
$\displaystyle\bigotimes_{m=1}^{n}\left(\sum_{s_{m}=0}^{d-1}x^{q_{m,s_{m}}}|s_{m}\rangle\right)_{A_{m}},$
(14)

where

$q_{m,s_{m}}=s_{m}d^{n-m-1}.$ (15)

Crucially, the Vandermonde matrices, that is matrices with rows being the
Vandermonde vectors,

$V_{K,L}=\sum_{k=0}^{K-1}|k\rangle\\!\langle v_{L}(x_{k})|,$ (16)

are totally positive (i.e., all their minors are strictly positive), whenever
the nodes $\\{x_{i}\\}_{i}$ obey the relation

$0<x_{0}<x_{1}<\dots<x_{K-1}.$ (17)

Let us now consider the following set of linearly independent vectors

$\displaystyle\left\\{|v_{d^{n}}(x_{i})\rangle_{\textbf{A}}\right\\}_{i=0}^{K-1},$
(18)

where the number $K$ satisfies the following inequality

$K\geq td^{k-1}-t+d^{n-t(k-1)},$ (19)

which follows from Eqs (6) and (9), and the nodes $x_{i}\in\mathbbm{R}$ are
ordered as in Eq. (17).

The claim is that the vectors in Eq. (18) are not extendible with vectors of
the entanglement depth less than or equal to $k-1$, i.e., the subspace
orthogonal to their span is a $k$-CES. The proof is almost immediate. We write

$|v_{d^{n}}(x_{i})\rangle_{\textbf{A}}=|\varphi_{1}^{(i)}\rangle_{S_{1}}\otimes|\varphi_{2}^{(i)}\rangle_{S_{2}}\otimes\ldots\otimes|\varphi_{r}^{(i)}\rangle_{S_{r}}$
(20)

for an $r$-partition $S_{1}|S_{2}|\dots|S_{r}$ of the parties corresponding to
(11), in particular (13). Consider matrices with rows being local vectors on
$S_{j}$’s,

$\mathfrak{S}_{j}=\sum_{k=0}^{K-1}|k\rangle\\!\langle\varphi_{j}^{(k)}|,\quad
j=1,2,\dots,r.$ (21)

From the total positivity of Vandermonde matrices with positive nodes (see
above) it follows that $\mathfrak{S}_{j}$’s are also totally positive. This
further means that for every $j$ any set of $d^{n_{j}}$ vectors
$\\{|\varphi_{j}^{(i)}\rangle\\}_{i}$ spans the Hilbert space
$(\mathbb{C}^{d})^{\otimes n_{j}}$ of $S_{j}$. This is the case for any
$r$-partition with $n_{i}\leq k-1$. By Corollary 1, we conclude that indeed
there does not exist a $(k-1)$-producible state in the orthocomplement of the
span of vectors (18), i.e., the resulting subspace is a $k$-CES. Its dimension
is $d^{n}-K$, which for the smallest $K$ corresponds to the maximal dimension
given in Eq. (9). Explicit (non-orthogonal) bases for such constructed
subspaces were given in [47].

Similarly to [47] one could also use DFT matrices in the construction. The
method presented here can be viewed as a construction of ”non-orthgonal”
$\mathrm{UPB}_{k-1}$’s (see Section V), whose elements are not mutually
orthogonal.

Observe that the construction is flexible in that for a given $k$-CES we can
always increase $k$ by adding some number of Vandermonde vectors with properly
chosen nodes to the construction set (18). For example, consider an eleven-
dimensional $2$-CES, or, simply, a CES in the standard terminology, in the
system of four qubits. Its construction requires five Vandermonde vectors. If
we now add two more vectors to (18) we will obtain a nine-dimensional 3-CES
(in Appendix A we give an example of such subspace). If we add again two
vectors we will obtain a seven-dimensional 4-CES, i.e. a GES, of the maximal
permissible dimension. Further enlargement of the set of Vandermonde vectors
leads to GESs of smaller dimensions.

##  V $k$-CES and unextendible product bases

In this section, we explore the link between $k$-CESs and UPBs. We are
interested in a certain type of UPBs, namely those bases which are
unextendible with vectors of the entanglement depth less than $k$ and as such
leading to $k$-CESs. We thus propose the following.

######  Definition 4.

A UPB which is unextendible with $(k-1)$-producible vectors, i.e., states with
the entanglement depth less than or equal to $k-1$, is called a
$(k-1)$-unextendible product basis, denoted $\mathrm{UPB}_{\mathrm{k-1}}$.

For $k=2$ we have the standard UPBs (see Section II), while for
$k=n$—genuinely unextendible product bases (GUPBs), that is UPBs unextendible
even with biproduct vectors [27, 48], whose existence in the orthogonal
version is currently unknown (see Refs. [48, 49, 50] for recent results in
this direction).

By definition, the orthocomplement of a $\mathrm{UPB}_{\mathrm{k-1}}$ is a
$k$-CES.

We observe that the theoretical minimal cardinality of a
$\mathrm{UPB}_{\mathrm{k-1}}$ is [see Eq. (10) for the definition of $t$]

$\small m^{theor.}_{min}=\begin{cases}d^{n}&d=2\;\mathrm{and\;}k=n\\\
t(d^{k-1}-1)+d^{n-(k-1)t}&(\mathrm{odd}\;d)\;\mathrm{or}\;[\mathrm{even\;}d\;\mathrm{and}\;\\{(k-1|n\;\mathrm{and\;odd}\;t)\;\mathrm{or}\;(k-1\nmid
n\;\mathrm{and\;even\;}t)\\}]\\\
t(d^{k-1}-1)+d^{n-(k-1)t}+1&\mathrm{otherwise}\end{cases}.$ (22)

These numbers correspond to the minimal cardinalities of UPBs [44, 45, 46] for
systems defined on Hilbert spaces as in Eq. (13). We will refer to these
bounds as trivial and discuss their improvement in the next section.

###  V.1 Bound on cardinalities of $\mathrm{UPB}_{\mathrm{k-1}}$

The trivial bounds from Eq. (22) can in many cases be strengthened if the
internal product structure of partitions is appropriately taken into account.
The improvement can be achieved by a generalization of the technique recently
introduced in Ref. [48] to lower-bound permissible cardinalities of UPBn-1,
i.e., GUPBs. There, the pigeonhole principle was the key resource, here, its
generalization comes naturally in handy. We have the following.

######  Proposition 5.

The number of states $m$ in a $\mathrm{UPB}_{\mathrm{k-1}}$ is bounded as
follows:

$m\geq
d^{k-1}+(n-k+1)\left(\left\lfloor\frac{d^{k-1}-2}{k-1}\right\rfloor+1\right).$
(23)

###### Proof.

Consider a set of mutually orthogonal fully product vectors

$\mathcal{K}=\\{|v_{i}\rangle_{\textbf{A}}\\}_{i=1}^{m},\quad|v_{i}\rangle_{\textbf{A}}=\bigotimes_{j=1}^{n}|u_{j}^{(i)}\rangle_{A_{j}}.$
(24)

We will show that if the number of elements in $\mathcal{K}$ is strictly
smaller than the RHS of (23) then $\mathcal{K}$ is extendible with vectors of
the entanglement depth $k-1$ or smaller, i.e., it is not a
$\mathrm{UPB}_{\mathrm{k-1}}$.

Any two vectors from $\mathcal{K}$, $|v_{p}\rangle$ and $|v_{q}\rangle$, are
orthogonal because their local vectors on at least one site $A_{k}$ are
orthogonal, i.e., $\langle u_{k}^{(p)}|u_{k}^{(q)}\rangle=0$ for at least one
value of index $k$. By the generalized pigeonhole principle [53] (see Appendix
B), we conclude that for any vector $|v_{i}\rangle$ there are $w$ sites such
that the total number of vectors orthogonal to $|v_{i}\rangle$ on these sites
is at least

$s:=w\left\lfloor\frac{m-1}{n}\right\rfloor+\min\left(w,m-1-n\left\lfloor\frac{m-1}{n}\right\rfloor\right).\displaystyle$
(25)

Let us focus on one of the vectors from $\mathcal{K}$, e.g., $|v_{1}\rangle$.
Assume that the said $s$ vectors which are orthogonal to $|v_{1}\rangle$ are
$\mathcal{K}_{1}=\\{|v_{2}\rangle,|v_{3}\rangle,\ldots,|v_{s+1}\rangle\\}$
with orthogonality holding on sites from
$\textbf{A}_{w}:=A_{1},A_{2},\cdots,A_{w}$. Since there exist a vector
orthogonal to $\mathcal{K}_{1}$ on one of the sites from $\textbf{A}_{w}$, one
immediately infers that the elements of $\mathcal{K}_{1}$ do not span the
whole Hilbert space corresponding to $\textbf{A}_{w}$, that is

$\dim\mathrm{span}\left\\{\otimes_{j=1}^{w}|u_{j}^{(i)}\rangle_{A_{j}}\right\\}_{i=2}^{s+1}<\dim\mathcal{H}_{\textbf{A}_{w}}.$
(26)

Now, if the remaining vectors
$\mathcal{K}_{2}:=\mathcal{K}\setminus\mathcal{K}_{1}=\\{|v_{1}\rangle,|v_{s+2}\rangle,\ldots,|v_{m}\rangle\\}$
did not span on $\mathbf{A}\setminus\textbf{A}_{w}$ the corresponding Hilbert
space, i.e.,

$\displaystyle\dim\mathrm{span}\>\mathcal{K}_{2}^{\mathbf{A}\setminus\textbf{A}_{w}}<\dim\mathcal{H}_{\textbf{A}\setminus\textbf{A}_{w}},$
(27)

where

$\displaystyle\mathcal{K}_{2}^{\mathbf{A}\setminus\textbf{A}_{w}}=\left\\{\otimes_{j={w+1}}^{n}|u_{j}^{(i)}\rangle_{A_{j}}\right\\}_{i=1,s+2,s+3,\dots,m},$
(28)

then one would find a vector orthogonal to $\mathcal{K}$ in the form

$\left(\otimes_{j=1}^{w}|u_{j}^{(1)}\rangle_{A_{j}}\right)\otimes|\xi\rangle_{\textbf{A}\setminus\textbf{A}_{w}}$
(29)

with an arbitrary $(n-w)$-partite vector $|\xi\rangle$ orthogonal to
$\mathcal{K}_{2}^{\mathbf{A}\setminus\textbf{A}_{w}}$. Clearly, the vector
$|\xi\rangle$ would have in this case the entanglement depth at most $(n-w)$
and in consequence $\mathcal{K}$ would not be a $\mathrm{UPB}_{\mathrm{n-w}}$.

We are interested in the permissible cardinalities of
$\mathrm{UPB}_{\mathrm{k-1}}$’s in which case we need to consider

$w=n-k+1$ (30)

and verify when the condition (27) holds true. This will provide us with the
forbidden cardinalities of the bases and by negating the obtained bound we
will arrive at (23). A sufficient condition for (27) is simply that the number
of states in $\mathcal{K}_{2}^{\mathbf{A}\setminus\textbf{A}_{w}}$ is smaller
than the dimension of $\mathcal{H}_{\textbf{A}\setminus\textbf{A}_{w}}$, that
is

$m-s\leq d^{n-w}-1:=r,$ (31)

which needs to be solved for $m$.

Let us define the following function corresponding to the LHS of Eq. (31)

$\displaystyle f_{w}(m)=m-s$
$\displaystyle=m-w\left\lfloor\frac{m-1}{n}\right\rfloor-\min\left(w,m-1-n\left\lfloor\frac{m-1}{n}\right\rfloor\right).$
(32)

Using the identities $\min(x,y)=x+y-\max(x,y)$ and
$\max(x,y)=\max(x-z,y-z)+z$, we can write this function as

$\displaystyle f_{w}(m)=$
$\displaystyle(n-w)\left\lfloor\displaystyle\frac{m-1}{n}\right\rfloor$
$\displaystyle+\max\left(m-w-n\left\lfloor\displaystyle\frac{m-1}{n}\right\rfloor,1\right).$
(33)

It is straightforward to show that this function has the following property
(see Appendix C)

$f_{w}(m+1)=\left\\{\begin{array}[]{ccc}f_{w}(m)+1&\mathrm{if}&m-n\displaystyle\left\lfloor\frac{m}{n}\right\rfloor\geq
w+1\;\mathrm{or}\;n\mid m,\\\\[8.61108pt]
f_{w}(m)&\mathrm{if}&m-n\displaystyle\left\lfloor\frac{m}{n}\right\rfloor\leq
w\;\mathrm{and}\;n\nmid m.\end{array}\right.$ (34)

It then follows that for a fixed $w$ function $f_{w}(m)$ is non-decreasing in
$m$ and takes all integer values. This allows us to look for the largest
solution of $f_{w}(m)=r$, that is

$(n-w)\left\lfloor\frac{m-1}{n}\right\rfloor+\max{\left(m-w-n\left\lfloor\frac{m-1}{n}\right\rfloor,1\right)}=r.$
(35)

We now observe that we can replace the maximum with the function inside of it,
as for any $m$ for which this function is less than one, we can find another
argument $\tilde{m}>m$ attaining one without changing the value of the whole
function under scrutiny; precisely, we can take
$\tilde{m}=n\lfloor\frac{m-1}{n}\rfloor+w+1$ for which
$f_{w}(m)=f_{w}(\tilde{m})$. In turn, we can consider

$(n-w)\left\lfloor\frac{m-1}{n}\right\rfloor+m-w-n\left\lfloor\frac{m-1}{n}\right\rfloor=r$
(36)

with the assumption that

$\left\lfloor\frac{m-1}{n}\right\rfloor\leq\frac{m-w-1}{n}.$ (37)

Eq. (36) simplifies to

$m-w\>q=r+w,\quad\quad q:=\left\lfloor\frac{m-1}{n}\right\rfloor.$ (38)

From equations (37) and (38) we have

$q\leq\frac{r+qw-1}{n},$ (39)

which after solving for $q$ gives

$q\leq\frac{r-1}{n-w},$ (40)

implying further that the largest solution to Eq. (38) is

$m=r+w+w\left\lfloor\frac{r-1}{n-w}\right\rfloor.$ (41)

Plugging back $r=d^{n-w}-1$ and substituting $w=n-k+1$ [(30)], we finally
obtain the largest excluded cardinality

$m=d^{k-1}+n-k+(n-k+1)\left\lfloor\frac{d^{k-1}-2}{k-1}\right\rfloor,$ (42)

meaning that the minimal permissible size of a $\mathrm{UPB}_{\mathrm{k-1}}$
is

$d^{k-1}+(n-k+1)\left(\left\lfloor\frac{d^{k-1}-2}{k-1}\right\rfloor+1\right),$
(43)

as claimed in Eq. (23). ∎

We have proved the statement for equal local dimension but a similar reasoning
can be applied to the general case. We omit the derivation here.

We now show when the obtained bound is non-trivial, i.e., it gives a lower
bound strictly larger than the one provided by Eq. (22). We have the following
chain of inequalities:

$\displaystyle
d^{k-1}+(n-k+1)\left(\left\lfloor\frac{d^{k-1}-2}{k-1}\right\rfloor+1\right)$
$\displaystyle\geq d^{k-1}+(n-k+1)\frac{d^{k-1}-1}{k-1}$ (44)
$\displaystyle=t(d^{k-1}-1)+1+(d^{k-1}-1)\left(\frac{n}{k-1}-t\right)$
$\displaystyle\geq t(d^{k-1}-1)+d^{(k-1)(\frac{n}{k-1}-t)}$ (45)
$\displaystyle=t(d^{k-1}-1)+d^{n-(k-1)t},$ (46)

where $t=\lfloor\frac{n}{k-1}\rfloor$. The first inequality follows from the
fact that for any pair of integers $x$ and $y>0$ the following inequality
holds true:

$\left\lfloor\frac{x}{y}\right\rfloor=\left\lceil\frac{x+1}{y}\right\rceil-1\geq\frac{x+1}{y}-1$
(47)

and the equality in (44) holds iff $k-1$ divides $d^{k-1}-1$. The second
inequality, Eq. (45), follows from Bernoullie’s inequality, stating that
$(1+x)^{r}\leq 1+xr$ with $x>-1$ and $0\leq r\leq 1$, here applied with
$x=d^{k-1}-1$ and $r=n/(k-1)-t$. The equality in this case holds iff $k-1$
divides $n$ (then $r=0$; $r=1$ is never the case).

Confronting the above conditions for equalities with Eq. (22) provides us with
the cases when Eq. (23) is certainly non-trivial. For example, this is the
case whenever $d$ is odd and $k-1\nmid d^{k-1}-1$ or $k-1\nmid n$. On the
other hand, there are clearly cases when it is trivial. This happens for
example in the case of $3$-CESs in systems of four qubits ($n=4$, $d=2$,
$k=3$). Then, our bound gives $8$ as the minimal cardinality and the same
number is obtained from the theory of bipartite UPBs applied to a $4\otimes 4$
system.

Note that for $k=n$ (GUPBs) the bound reproduces, as it should, the one from
[48], in which case it is always non-trivial. This bound has been recently
improved in [50]. At this point, it is not clear whether our current bound on
the permissible cardinalities of $\mathrm{UPB}_{\mathrm{k-1}}$’s can be
improved using the graph theory techniques of [50].

###  V.2 Construction of $k$-CESs from UPBs

While the bound of Proposition 5 puts limitations on sizes of
$\mathrm{UPB_{k-1}}$’s, it does not say anything about the possibility of the
actual constructions of bases with permissible cardinalities. This is
particularly important in view of the unknown status of this problem for GUPBs
($k=n$). It turns out that $\mathrm{UPB_{k-1}}$’s do exist for $k<n$ and in
fact there are already examples in the literature for systems of four qubits
[51].

We will now reconstruct one of the four-qubit basis from Ref. [51] and show
existence of a $\mathrm{UPB_{3}}$ in the case of five qubits. In both
situations, we are interested in the case of $k=n-1$. With this aim we use the
results of Ref. [52], where a recursive procedure was given for a construction
of UPBs of size $2^{n-1}$ for any $n$.

Let us begin with $n=4$. The UPB from Ref. [52] is given by

$\displaystyle\mathcal{K}^{(4)}=\\{$
$\displaystyle|0000\rangle,\quad|01fe\rangle,\quad|1e1e\rangle,\quad|1fe0\rangle,$
$\displaystyle|e001\rangle,\quad|e1ff\rangle,\quad|fe1f\rangle,\quad|ffe1\rangle\\},$
(48)

where $\\{|e\rangle,|f\rangle\\}$ is any orthonormal basis different from
$\\{|0\rangle,|1\rangle\\}$, e.g., $\\{|+\rangle,|-\rangle\\}$. It is
immediate to realize that there exists a vector with the entanglement depth
equal to $2$ orthogonal to $\mathcal{K}^{(4)}$, i.e., this set is not a
$\mathrm{UPB_{2}}$ . For example, such a vector is given by
$|\varphi\rangle_{A_{1}A_{4}}\otimes|\psi\rangle_{A_{2}A_{3}}$, where
$|\varphi\rangle\perp\\{|10\rangle,|f1\rangle\\}$ and
$|\psi\rangle\perp\\{|00\rangle,|1f\rangle,|e1\rangle\\}$. One realizes that
this stems from the fact that there are pairs of vectors in
$\mathcal{K}^{(4)}$, which are identical on two subsystems. Here, there are
three such pairs, but for sets with cardinality eight this would already be an
obstacle if there were only two of them. However, the following amendment can
be made to $\mathcal{K}^{(4)}$ to avoid this problem: in the pairs of vectors
with $|1\rangle$ and $|e\rangle$ on the first site ($A_{1}$), we swap vectors
on the third site ($A_{3}$). This leads to the following set

$\displaystyle\overline{\mathcal{K}^{(4)}}=\\{$
$\displaystyle|0000\rangle,\quad|01fe\rangle,\quad|1eee\rangle,\quad|1f10\rangle,$
$\displaystyle|e0f1\rangle,\quad|e10f\rangle,\quad|fe1f\rangle,\quad|ffe1\rangle\\},$
(49)

which recovers one of the UPBs given in Ref. [51], where it was shown that it
is unextendible for any of the two vs two parties bipartitions, i.e.,
$\overline{\mathcal{K}^{(4)}}$ is a $\mathrm{UPB_{2}}$. As noted in Section
V.1, its cardinality is the minimal possible.

In the $n=5$ case the UPB from Ref. [52] is as follows:

$\displaystyle\mathcal{K}^{(5)}=\\{$ $\displaystyle|00000\rangle,$
$\displaystyle|001fe\rangle,\;|e001f\rangle,\;|fe001\rangle,\;|1fe00\rangle,\;|01fe0\rangle,$
$\displaystyle|01e1e\rangle,\;\;|e01e1\rangle,\;|1e01e\rangle,\;\;|e1e01\rangle,\;|1e1e0\rangle,$
$\displaystyle|1fffe\rangle,\;|e1fff\rangle,\>|fe1ff\rangle,\;|ffe1f\rangle,|fffe1\rangle\\}.$

The set has a visibly cyclic structure and when proving (non)existence of a
vector with the entanglement depth three orthogonal to all its elements one
needs to consider only two bipartitions as the rest are equivalent to one of
them. These bipartitions are $A_{1}A_{2}|A_{3}A_{4}A_{5}$ and
$A_{1}A_{3}|A_{2}A_{4}A_{5}$. With an exhaustive search based on Lemma 1 we
have verified that $\mathcal{K}^{(5)}$ is already a $\mathrm{UPB}_{3}$ and no
modifications of it are necessary.

We conjecture that the bases from Ref. [52] can be used, directly or after
some modifications similar to those shown above in the four qubit case, to
obtain $\mathrm{UPB_{n-2}}$’s for any $n$.

Let us conclude with an observation that in general the case $k-1|n$ is
somewhat special since then we deal with systems with $t$ subsystems of equal
local dimensions $d^{k-1}$ (recall our assumptions of dimension $d$ of each
subsystem $A_{i}$) in the ”coarse-grained” Hilbert space [Eq. (13)].
Constructions of UPBs for homogeneous systems (i.e., with equal local
dimensions) are far better explored that those for heterogeneous systems
(i.e., different local dimensions). A possible route to find
$\mathrm{UPB}_{\mathrm{k-1}}$’s could then be to build UPBs first for (13) and
then look for their proper ”fine-grained” versions in the original space.

##  VI Conclusions and outlook

We have proposed to consider the notion of the entanglement depth in the
context of entangled subspaces by introducing the notion of completely
entangled subspaces of entanglement depth $k$ ($k$-CESs), that is subspaces
composed solely of pure states with the entanglement depth at least $k$. We
have presented a universal construction of such subspaces working for any
multipartite systems. We have also considered the relationship between
unextendible product bases (UPBs) and $k$-CESs. In particular, we have
provided a non-trivial bound on the cardinalities of UPBs whose
orthocomplements are $k$-CESs. Further, we have discussed the problem of
constructing such UPBs and provided their examples in systems of several
qubits.

From the general point of view, it is natural to expect that the newly
introduced notion may turn out relevant for the analyses of many-body systems,
where the entanglement depth is a figure of merit. Furthermore, the already
established connection between entangled subspaces and quantum error
correction may suggest that $k$-CESs could be of some use also in this area
and their analysis could help understand this relationship deeper. There are
also several concrete open problems naturally emerging from our study. For
example, it would be desirable to see if the obtained bound on the
cardinalities of UPBs leading to $k$-CESs can be further improved, or, to
research the possibility of a general practical construction of such UPBs, as
this would automatically entail a construction of PPT states. Finally, there
is constant demand for practical methods of subspace entanglement
certification and quantification tailored to specific types of entangled
subspaces and this line or research is also worth exploration.

##  VII Acknowledgments

K.V. and R. A. acknowledge the support of the National Science Center (Poland)
through the SONATA BIS project (grant no. 2019/34/E/ST2/00369).

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## Appendices

##  Appendix A Maximal $3$-CES: four qubit case

Below we give a non-orthogonal basis for a nine-dimensional, i.e., maximal,
$3$-CES in the case of four qubits ($n=4$, $d=2$). It was obtained using the
construction of Section IV by setting the nodes $x_{i}=i$ in (18). We present
it as a matrix whose rows are the basis vectors:

$\displaystyle\footnotesize\left(\begin{array}[]{cccccccccccccccc}-5040&13068&-13132&6769&-1960&322&-28&1&0&0&0&0&0&0&0&0\\\
-141120&360864&-354628&176400&-48111&7056&-462&0&1&0&0&0&0&0&0&0\\\
-2328480&5896296&-5706120&2772650&-729120&100653&-5880&0&0&1&0&0&0&0&0&0\\\
-29635200&74511360&-71319864&34095600&-8752150&1164240&-63987&0&0&0&1&0&0&0&0&0\\\
-322494480&806546916&-765765924&361808139&-91318920&11851664&-627396&0&0&0&0&1&0&0&0&0\\\
-3162075840&7876316448&-7432417356&3481077600&-867888021&110702592&-5715424&0&0&0&0&0&1&0&0&0\\\
-28805736960&71527084992&-67178631520&31255287700&-7721153440&972478507&-49329280&0&0&0&0&0&0&1&0&0\\\
-248619571200&615829294080&-576265019968&266731264800&-65430101100&8162874720&-408741333&0&0&0&0&0&0&0&1&0\\\
-2060056318320&5092812168444&-4751761890876&2190505063109&-534401747880&66184608126&-3281882604&0&0&0&0&0&0&0&0&1\\\
\end{array}\right).$ (59)

##  Appendix B Generalized pigeonhole principle

For convenience, we recall here the statement of the generalized pigeonhole
principle [53].

######  Fact 2.

If $pq+r$ objects are put into $q$ boxes, then for each $0\leq s\leq q$ there
exist $s$ boxes with the total number of objects in them at least
$ps+\min(r,s)$.

##  Appendix C Proof of Eq. (34)

Let us start with recalling the definition of the function in question:

$\displaystyle f_{w}(m)=$
$\displaystyle(n-w)\left\lfloor\displaystyle\frac{m-1}{n}\right\rfloor$
$\displaystyle+\max\left(m-w-n\left\lfloor\displaystyle\frac{m-1}{n}\right\rfloor,1\right)$
(60)

with $w=n-k+1$.

Case (i): $n|m.$ We begin with the case of $n$ being a divisor of $m$. We then
have

$\displaystyle\left\lfloor\displaystyle\frac{m}{n}\right\rfloor=\frac{m}{n}\quad\mathrm{and}\quad\left\lfloor\displaystyle\frac{m-1}{n}\right\rfloor=\frac{m}{n}-1.$
(61)

It follows that

$\displaystyle f_{w}(m)$
$\displaystyle=(n-w)\left(\frac{m}{n}-1\right)+\max\left(n-w,1\right)$
$\displaystyle=(n-w)\frac{m}{n}$ (62)

since $n-w=k-1\geq 1$. Further

$\displaystyle f_{w}(m+1)=$
$\displaystyle(n-w)\left\lfloor\displaystyle\frac{m}{n}\right\rfloor$
$\displaystyle+\max\left(m+1-w-n\left\lfloor\displaystyle\frac{m}{n}\right\rfloor,1\right)$
(63) $\displaystyle=$ $\displaystyle(n-w)\frac{m}{n}+\max\left(1-w,1\right)$
(64) $\displaystyle=$ $\displaystyle(n-w)\frac{m}{n}+1=f_{w}(m)+1$ (65)

because $w=n-k+1\geq 1$.

Case (ii): $n\nmid m$. We now consider the opposite case of $n$ not being a
divisor of $m$. It now holds

$\displaystyle\left\lfloor\frac{m-1}{n}\right\rfloor=\left\lfloor\frac{m}{n}\right\rfloor.$
(66)

Within the current case we analyze two subcases related to the value of the
function under the maximum in $f_{w}(m)$.

Subcase (ii,a): $m-n\displaystyle\left\lfloor\frac{m}{n}\right\rfloor-w\geq
1$. We just note that this condition can never be satisfied simultaneously
with case (i) and is only valid for $n\nmid m$. We have

$\displaystyle
m-n\displaystyle\left\lfloor\frac{m-1}{n}\right\rfloor-w=m-n\displaystyle\left\lfloor\frac{m}{n}\right\rfloor-w\geq
1$ (67)

in view of Eq. (66). In consequence, in both $f_{w}(m)$ and $f_{w}(m+1)$ we
can take the function under the maximum as its value. We then have after
trivial simplifications

$\displaystyle f_{w}(m)=$ $\displaystyle
m-w-w\left\lfloor\displaystyle\frac{m-1}{n}\right\rfloor$ (68)

and

$\displaystyle f_{w}(m+1)=$ $\displaystyle
m-w+1-w\left\lfloor\displaystyle\frac{m}{n}\right\rfloor=f_{w}(m)+1.$ (69)

Subcase (ii,b): $m-n\displaystyle\left\lfloor\frac{m}{n}\right\rfloor-w\leq
0$. Now, the maximum is easily seen to be equal to one for both $f_{w}(m)$ and
$f_{w}(m+1)$, which, taking into account Eq. (66), results in this subcase in
$f_{w}(m)=f_{w}(m+1)$.

This concludes the proof of Eq. (34).

##  Appendix D Minimal value of (12)

Let us start with two auxiliary results.

######  Lemma 1.

Let $n$ and $k\leq n$ be positive integers. Let further

$\boldsymbol{x}=(x_{1},x_{2},\dots,x_{n})$ (70)

be a non-increasing sequence of non-negative integers, such that

$x_{i+1}\leq x_{i}\leq k-1,\quad\quad\sum_{i}x_{i}=n$ (71)

and

$\boldsymbol{y}=\Big{(}\underbrace{k-1,\dots,k-1}_{\text{t}\;\mathrm{times}},n-t(k-1),0,\dots,0\Big{)},$
(72)

where

$t=\left\lfloor\frac{n}{k-1}\right\rfloor.$ (73)

Then, for any $p\leq n$ it holds

$\sum_{i=1}^{p}x_{i}\leq\sum_{i=1}^{p}y_{i},$ (74)

i.e., $\boldsymbol{x}$ is majorized by $\boldsymbol{y}$, written as
$\boldsymbol{x}\prec\boldsymbol{y}$.

###### Proof.

The result is obvious. ∎

######  Lemma 2.

(see, e.g., [54]) Let $g(\cdot)$ be a convex function and
$\boldsymbol{x}\prec\boldsymbol{y}$. It then holds

$\sum_{i}g(x_{i})\leq\sum_{i}g(y_{i}).$ (75)

We can now prove the following.

######  Fact 3.

The maximal value of

$\sum_{i=1}^{n}(d^{n_{i}}-1)+1$ (76)

over all $\boldsymbol{n}=(n_{1},n_{2},\dots,n_{n})$ such that
$\sum_{i}n_{i}=n$ and $0\leq n_{i}\leq k-1$ is

$td^{k-1}-t+d^{n-t(k-1)},$ (77)

where $t=\left\lfloor\frac{n}{k-1}\right\rfloor$.

###### Proof.

From Lemma 1 and Lemma 2 with $g(x)=d^{x}$ it follows immediately that the
maximal value of $\sum_{i}d^{n_{i}}$ over all permissible $\boldsymbol{n}$’s
corresponds to the choice $\boldsymbol{n}=\boldsymbol{y}$ with
$\boldsymbol{y}$ defined as in Lemma 1. Further, since for any other
$\boldsymbol{n}$ its number of non-zero elements is at least as large as in
$\boldsymbol{y}$, also the maximal value of $\sum_{i}(d^{n_{i}}-1)$ is
achieved for the same choice. Eq. (77) then follows from the direct
substitution of the optimal $n_{i}$’s. ∎

The minimal value of the expression in Eq. (12) is obtained by substracting
the obtained value (77) from $d^{n}$.


# Lee-Yang zeros at $O(3)$ and deconfined quantum critical points

Jonathan D’Emidio<EMAIL_ADDRESS>Donostia International Physics
Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain

###### Abstract

Lee-Yang theory, based on the study of zeros of the partition function, is
widely regarded as a powerful and complimentary approach to the study of
critical phenomena and forms a foundational part of the theory of phase
transitions. Its widespread use, however, is complicated by the fact that it
requires introducing complex-valued fields that create an obstacle for many
numerical methods, especially in the quantum case where very limited studies
exist beyond one dimension. Here we present a simple and statistically exact
method to compute partition function zeros with general complex-valued
external fields in the context of large-scale quantum Monte Carlo simulations.
We demonstrate the power of this approach by extracting critical exponents
from the leading Lee-Yang zeros of 2D quantum antiferromagnets with a complex
staggered field, focusing on the Heisenberg bilayer and square-lattice $J$-$Q$
models. The method also allows us to introduce a complex field that couples to
valence bond solid order, where we observe extended rings of zeros in the
$J$-$Q$ model with purely imaginary staggered and valence bond solid fields.

Introduced over a half century ago, the Lee-Yang theory of phase transitions
[Yang and Lee, 1952,Lee and Yang, 1952] provides deep insights into how non-
analyticities of the free energy arise in the thermodynamic limit of finite-
size systems. The theory rests on the fact that while finite systems are
manifestly analytic in the domain of real-valued physical control parameters,
they can display singularities—zeros of the partition function—when the
control parameters are extended to the complex plane. As the thermodynamic
limit is approached, these zeros can accumulate near and eventually pinch the
real axis, producing a genuine phase transition. In fact, even when zeros
accumulate away from the real-axis they provide an interesting example of non-
unitary critical points [Fisher, 1978, Cardy, 1985].

While the study of Lee-Yang zeros may seem relegated to the realm of pure
theory, they have in fact been demonstrated in several experimental systems
[Binek, 1998; Peng  _et al._ , 2015; Brandner  _et al._ , 2017] and quantum
computers [Francis  _et al._ , 2021]. But perhaps the greatest impact of the
theory lies in computational studies of many-body systems where the themes
range from protein folding [Lee, 2013] and DNA zippers [Deger  _et al._ ,
2018,Majumdar and Bhattacharjee, 2020] to quantum chromodynamics [Nagata  _et
al._ , 2015], to name but a few. See Ref. [Bena  _et al._ , 2005] for a
review. However, despite the broad range of studies, there remains a glaring
disparity between the strength of numerical methods in the classical versus
the quantum case, an issue that we seek to address in this work.

While classical systems with complex fields can be efficiently treated with a
variety of methods, from histogram-based approaches [Ferrenberg and Swendsen,
1988; Kenna and Lang, 1993, 1994] or high-order cumulants [Flindt and
Garrahan, 2013,Deger  _et al._ , 2020] to tensor network methods [Shimizu and
Kuramashi, 2014a, b; García-Saez and Wei, 2015; Hong and Kim, 2022], the
quantum case is more restrictive, mainly being limited to 1D [Heyl  _et al._ ,
2013; Ananikian  _et al._ , 2014; Peotta  _et al._ , 2021] or small 2D
lattices [Kist  _et al._ , 2021; Vecsei  _et al._ , 2022; Brange  _et al._ ,
2022; Vecsei  _et al._ , 2023]. Here we develop a statistically exact method
for extracting Lee-Yang zeros in large-scale quantum Monte Carlo (QMC)
simulations of spin systems. The method builds on a simple, yet so far unused,
formula for computing free energy differences in QMC, and allows for the
computation of partition function ratios for a whole range of complex external
fields based a single simulation in zero field.

Free energy differences in QMC: We work in the context of the stochastic
series expansion (SSE) quantum Monte Carlo algorithm [Sandvik, 2010a], which
is a statistically exact method for sampling the partition function as a
Taylor expansion: $Z(\beta)=\text{Tr}\left(e^{-\beta
H}\right)=\textstyle\sum^{\infty}_{n=0}\frac{\beta^{n}}{n!}\text{Tr}\left((-H)^{n}\right).$
The trace is then further decomposed into a sum of powers of Hamiltonian
matrix elements by inserting complete sets of states, giving a sum over SSE
configurations [Sandvik, 2010a].

In this formulation it is simple to show that partition function ratios at
different inverse temperatures can be computed as follows:

$R_{\beta}(\tilde{\beta})\equiv\frac{Z(\tilde{\beta})}{Z(\beta)}=\left\langle\left(\frac{\tilde{\beta}}{\beta}\right)^{n}\right\rangle_{\beta},$
(1)

where $n$ is the expansion order of the SSE configuration and the average is
taken in the ensemble with inverse temperature $\beta$. To the best of our
knowledge, Eqn. (1) has not yet been used nor has appeared in the literature.

The ratio formula can be extended to the case where the partition functions
differ only in the value of a Hamiltonian coupling $J$. In this case we simply
replace $\beta\to J$, $\tilde{\beta}\to\tilde{J}$, and $n\to n_{J}$ in Eq. (1)
where $n_{J}$ refers to the number of $J$-type matrix elements in the SSE
configuration.

Note that in these computations, only the denominator partition function needs
to be simulated, and the ratio with any “nearby” partition function can be
computed by recording a histogram of the number of operators ($n$) during an
SSE simulation. This formula then allows us to extend the ratio estimator to
complex values of its argument, which is the focus of this work. Before
demonstrating this, we first will describe how to introduce a complex-valued
staggered field that couples to the Néel order parameter, which will allow for
the meaningful extraction of Lee-Yang zeros in the case of quantum anti-
ferromagnets.

Figure 1: Main panel: A histogram of the number operators from Eq. (2) for the
Heisenberg bilayer with $L=80$, $J_{\perp}=2.5222$ and $\beta=40$ with $J=1$.
Inset: Eq. (2) applied to the histogram with complex-valued external Néel
field $h$, where we plot $-\ln(|R_{0}(h)|)$ and the Lee-Yang zeros appear as
the dark points.

Lee-Yang fields for quantum antiferromagnets: The classic case for Lee-Yang
zeros is that of the Ising model in an external magnetic field, which is taken
to be complex. The Lee-Yang circle theorem [Lee and Yang, 1952] then states
that all of the Ising partition function zeros with a complex magnetic field
lie on the imaginary axis. In order to extend this picture to quantum
antiferromagnets we introduce a field that couples to the Néel order
parameter. Here, remarkably, we will show that it is possible to compute the
ratio at finite complex field while simulating the partition function in zero
field.

The idea is to imagine embedding the $z-$component of the Néel field operator
inside nearest-neighbor operators that are already present in the Hamiltonian
$h_{i,j}=J(\vec{S}_{i}\cdot\vec{S}_{j}-\tfrac{1}{4})\to
J(\vec{S}_{i}\cdot\vec{S}_{j}-\tfrac{1}{4})+\tfrac{h}{N_{c}}(S^{z}_{i}-S^{z}_{j})$.
Here $i\in$ sublattice $A$ and $j\in$ sublattice $B$ and $N_{c}$ is the number
of $J$-operators that touch each site (usually the coordination number of the
lattice). The diagonal matrix elements then change as
$-\langle\uparrow_{i}\downarrow_{j}|h_{i,j}|\uparrow_{i}\downarrow_{j}\rangle=\tfrac{J}{2}\to\tfrac{J}{2}-\tfrac{h}{N_{c}}$
and
$-\langle\downarrow_{i}\uparrow_{j}|h_{i,j}|\downarrow_{i}\uparrow_{j}\rangle=\tfrac{J}{2}\to\tfrac{J}{2}+\tfrac{h}{N_{c}}$.
The ratio of these matrix elements before and after the change gives
$1\mp\frac{2h}{N_{c}J}$. If we denote the number of such matrix elements in an
SSE configuration by $n_{\uparrow\downarrow}$ and $n_{\downarrow\uparrow}$
then the ratio formula with a Néel field reads

$R_{0}(h)=\left\langle\left(1-\frac{2h}{N_{c}J}\right)^{n_{\uparrow\downarrow}}\left(1+\frac{2h}{N_{c}J}\right)^{n_{\downarrow\uparrow}}\right\rangle_{h=0}.$
(2)

We emphasize that this formula only requires a histogram of the values
$(n_{\uparrow\downarrow},n_{\downarrow\uparrow})$ obtained by QMC simulation
(see Fig. (1)) of typical Heisenberg models in the absence of external fields,
and allows for the statistically exact extraction of partition function zeros
with a complex-valued Néel field. We now demonstrate this with a concrete
example, the $O(3)$ quantum critical Heisenberg bilayer.

Heisenberg bilayer: As a first test case for this technique we select a well
studied 2D quantum spin model for the $O(3)$ transition in $d=2+1$ dimensions,
the Heisenberg bilayer given by the Hamiltonian

$H_{\text{bl}}=J\sum_{\langle
i,j\rangle,a=1,2}(\vec{S}_{i,a}\cdot\vec{S}_{j,a}-\tfrac{1}{4})+J_{\perp}\sum_{i}(\vec{S}_{i,1}\cdot\vec{S}_{i,2}-\tfrac{1}{4}),$
(3)

where $\langle i,j\rangle$ are nearest neighbor pairs of a square lattice and
$a$ is the layer index. When $J_{\perp}=0$ the model reduces to two
independent Heisenberg models, which exhibits long-range Néel order at zero
temperature, while for strong $J_{\perp}$ singlets are formed on the
interlayer bonds and Néel order is destroyed. A continuous transition in the
$O(3)$ universality class occurs at $J_{\perp}/J=2.5220(1)$ [Wang  _et al._ ,
2006].

Here we will focus on extracting the Lee-Yang zeros, zeros of Eq. (2) at the
critical point. First we show that the zeros of the partition function with a
complex-valued Néel field lie exactly on the imaginary axis, akin to the Lee-
Yang circle theorem for the Ising model [Lee and Yang, 1952] and its
generalization to the classical Heisenberg model [Harris, 1970]. This is shown
if Fig. (1), where in the main panel we display the histogram of the number
operators from Eq. (2) that are used to compute the partition function ratio.
The data is collected for a bilayer system with side length $L=80$ at the
critical point, fixing $J=1,J_{\perp}=2.5222$ and $\beta=L/2$. The inset shows
$-\ln(|R_{0}(h)|)$ computed with this histogram, where the dark spikes
indicate the first five zeros near $|h|=0$, which lie directly on the
imaginary axis.

Figure 2: Main panel: The location on the imaginary axis of the first three
Lee-Yang zeros of the Heisenberg bilayer in a complex Néel field at the
critical point $J_{\perp}/J=2.5222$ and $\beta=L/2$. Panel (a): the exponent
extracted from a fit of the data to the form $h_{\text{LY}}(L)\sim
L^{-(d+2-\eta)/2}(1+bL^{-\omega})$, where the fit is performed separately for
each of the first three zeros. The grey line is the best estimate of the
exponent from the literature [Campostrini  _et al._ , 2002]. Panel (b): the
raw data minus the fit.

Now we investigate the critical scaling of the leading zeros of the Heisenberg
bilayer as we approach the thermodynamic limit, using finite-size scaling of
the location of the zeros to extract critical exponents. At a critical point,
Lee-Yang zeros are expected to scale as $h_{\text{LY}}(L)\sim
L^{-\beta\delta/\nu}$ [Itzykson  _et al._ , 1983], where, through scaling
relations, the exponent can be expressed as $\beta\delta/\nu=(d+2-\eta)/2$.
For the Heisenberg bilayer, we find it necessary to include corrections to
scaling of the form $h_{\text{LY}}(L)\sim L^{-(d+2-\eta)/2}(1+bL^{-\omega})$,
as has been previously observed with Lee-Yang zeros for the 3D classical
Heisenberg model [Gordillo-Guerrero  _et al._ , 2013]. Fig. (2) shows the
first three zeros extracted from bilayer systems collected at
$J=1,J_{\perp}=2.5222$ and $\beta=L/2$. The extracted exponent is given in
inset $(a)$, which agrees perfectly with the best estimate from classical
Monte Carlo simulations [Campostrini  _et al._ , 2002]. Interestingly, we also
find the correction to scaling $\omega\approx 1$ in agreement with the
classical case [Gordillo-Guerrero  _et al._ , 2013].

Deconfined criticality: Now that we have demonstrated the ability to extract
critical exponents from the finite-size scaling of Lee-Yang zeros for the
first time at a quantum critical point, we move on to a more exotic quantum
spin model that shows criticality and emergent symmetry between two seemingly
unrelated order parameters.

The $J$-$Q$ model [Sandvik, 2007] is given by the following Hamiltonian:

$H_{JQ}=J\sum_{\langle
i,j\rangle}(\vec{S}_{i}\cdot\vec{S}_{j}-\tfrac{1}{4})-Q\sum_{\langle
i,j,k,l\rangle}(\vec{S}_{i}\cdot\vec{S}_{j}-\tfrac{1}{4})(\vec{S}_{k}\cdot\vec{S}_{l}-\tfrac{1}{4}),$
(4)

where $J$ is the nearest-neighbor coupling on a square lattice and $Q$ is a
four-spin interaction that acts on elementary plaquettes of the square
lattice. $Q$ can be thought of as a product of two $J$’s and the sum over
$\langle i,j,k,l\rangle$ includes both $\hat{x}$ and $\hat{y}$ orientations to
preserve lattice symmetries. When $Q=0$ the $J$-$Q$ model reduces to the
Heisenberg antiferromagnet, which exhibits long-range Néel order at $T=0$.
However, for large $Q$, magnetic order is destroyed by locally forming
singlets that stack along columns of the square lattice, referred to as
valence-bond solid (VBS) order, which breaks lattice translation symmetry. At
$J/Q\approx 0.045$ the system undergoes a seemingly continuous transition
between these two distinct ordered phases, which belongs to a class of
transitions known as deconfined quantum critical points (DQCP) [Senthil  _et
al._ , 2004a, Senthil  _et al._ , 2004b].

The true nature of the phase transition in the $J$-$Q$ model has been the
topic of continued debate. While most studies observe a direct continuous
transition between Néel and VBS order [Sandvik, 2007; Lou  _et al._ , 2009;
Sandvik, 2010b; Harada  _et al._ , 2013; Block  _et al._ , 2013; Shao  _et
al._ , 2016; Sandvik and Zhao, 2020; Zhao  _et al._ , 2020], other studies
show evidence of a weak first-order transition [Kuklov  _et al._ , 2008; Jiang
_et al._ , 2008; Chen  _et al._ , 2013; D’Emidio  _et al._ , 2021; Zhao  _et
al._ , 2022; Song  _et al._ , 2023]. The $J$-$Q$ model therefore provides us
with an important case for the study of Lee-Yang zeros.

Having already developed the methodology to extract Lee-Yang zeros associated
with fluctuations of the Néel order parameter, we would now like to develop an
analogous treatment for the VBS order parameter. Here we must introduce a
complex-valued field that couples to the VBS order parameter, which is a field
that preserves spin rotation symmetry but breaks lattice translation symmetry.
In order to do so, we make use of the $Q$ plaquette terms that are already
present in the Hamiltonian, and we introduce the field $d\sum_{\langle
i,j,k,l\rangle\in\hat{x}_{e}}P_{i,j,k,l}-d\sum_{\langle
i,j,k,l\rangle\in\hat{x}_{o}}P_{i,j,k,l}$ where
$P_{i,j,k,l}\equiv(\vec{S}_{i}\cdot\vec{S}_{j}-\tfrac{1}{4})(\vec{S}_{k}\cdot\vec{S}_{l}-\tfrac{1}{4})$
and $\hat{x}_{e}$ are all $\hat{x}$-oriented plaquettes on even columns of the
square lattice and $\hat{x}_{o}$ are all $\hat{x}$-oriented plaquettes on odd
columns. $d$ is the coupling strength of the VBS field, which we will take to
be complex-valued.

Figure 3: Main panel: the locations of the first Lee-Yang zeros in the $J$-$Q$
model at the critical point in two separate cases: with a complex Néel field
and with a complex VBS field. The thin black line shows the power law behavior
expected at a first order transition with $\eta=0$. Inset: The fitted exponent
from a simple power law fit of the zeros to the form $\sim L^{-(d+2-\eta)/2}$
as a function of the smallest system size used in the fit.

Following the same prescription as the ratio formula with a complex Néel
field, we have the ratio with a complex VBS field as follows:

$R_{0}(d)=\left\langle\left(1-\frac{d}{Q}\right)^{n_{Q_{xe}}}\left(1+\frac{d}{Q}\right)^{n_{Q_{xo}}}\right\rangle_{d=0}.$
(5)

Here $n_{Q_{xe}}$ is the number $\hat{x}$-oriented $Q$ matrix elements on even
columns of the square lattice and $n_{Q_{xo}}$ is the same but for odd
columns. Again, we only require a histogram of the values
$(n_{Q_{xe}},n_{Q_{xo}})$ obtained from QMC simulation of the pure $J$-$Q$
model without any fields, and Eqn. (5) allows for the statistically exact
extraction of Lee-Yang zeros of the partition function with a complex VBS
field. For the $J$-$Q$ model we can simultaneously gather histograms for both
the Néel zeros and VBS zeros in a single simulation.

In Fig. (3) we plot the location of the Lee-Yang zeros
$h_{\text{LY}},d_{\text{LY}}$ as a function of system size at the critical
point $J=0.04502$, $Q=1$ with $\beta=L/2$. Indeed, as with the Néel zeros, we
find the nearest VBS zeros lie exactly on the imaginary axis:
re$(h_{\text{LY}})=$ re$(d_{\text{LY}})=0$. For simplicity we plot only the
location of the leading zeros. In the $J$-$Q$ model, contrary to the
Heisenberg bilayer, we achieve much better fits using only a simple power law
with no corrections to scaling. The extracted exponents are far below the
values of the bilayer, signaling a much larger value of $\eta$ here, with our
values in the range of previous studies. We do, however, observe drifting of
the critical exponents, as shown in the inset, which is also similar to what
was observed in other studies [Harada  _et al._ , 2013, Nahum  _et al._ ,
2015].

Combined Néel and VBS fields: The transition in the $J$-$Q$ model is
frequently described in terms of an emergent SO(5) symmetry that arises
between the Néel and VBS order parameters exactly at the critical point. From
this point of view, it is interesting to ask the how the zeros of the
partition function are distributed in the presence of combined Néel and VBS
fields. Fortunately, our formulation allows us to simply compute the ratio
when both fields are nonzero, again, while the actual simulation is carried
out in zero field. We can straightforwardly define the combined ratio as:

$\begin{split}R_{0}(h,d)&=\left\langle\left(1-\frac{2h}{N_{c}J}\right)^{n_{\uparrow\downarrow}}\left(1+\frac{2h}{N_{c}J}\right)^{n_{\downarrow\uparrow}}\right.\\\
&\left.\quad\quad\left(1-\frac{d}{Q}\right)^{n_{Q_{xe}}}\left(1+\frac{d}{Q}\right)^{n_{Q_{xo}}}\right\rangle_{h=0,d=0}.\end{split}$
(6)

In Fig. (4) we plot $-\ln(|R_{0}(h,d)|)$ for purely imaginary values of $h$
and $d$ at the critical point. Interestingly, we see that the partition
function zeros form extended rings in the plane (im($h$),im($d$)). On either
side of the transition the ovals become elongated in either the vertical or
horizontal direction (see supplemental materials). Our conclusion is that at
the critical point these rings shrink down to the origin in the thermodynamic
limit, whereas on either side of the transition they become squashed in either
the horizontal or vertical direction. It is interesting to note that although
the values of $J$ and $Q$ differ by an order of magnitude at the transition,
the lines of zeros are nearly circular in terms of $h$ and $d$.

Conclusions: We have presented the first large-scale QMC computations of Lee-
Yang zeros in quantum spin systems. Our formulation is simple to implement, in
that it only requires gathering a histogram of the number of certain types of
matrix elements that appear during standard QMC simulations in the absence of
any external fields. Furthermore, we find that the location of Lee-Yang zeros
computed in this way is extremely precise, easily allowing for the extraction
of the the first three zeros for the critical Heisenberg bilayer on system
sizes up to $L=128$ and excellent agreement with known critical exponents. We
also have demonstrated the generality of this approach by using it to extract
Lee-Yang zeros associated with critical Néel, VBS and combined Néel-VBS
fluctuations in the $J$-$Q$ model, the emblematic model of deconfined
criticality.

Moving forward, we envision that this technique could be applied to extract
properties of non-unitary critical points that occur at finite imaginary field
values. Specifically, with the partition function ratio in hand, it allows one
to compute expectation values in the ensemble with a complex field. This seems
promising given that the ratio is computed to high precision in this
framework. On a speculative note, these types of studies could aid in
understanding lattice models with pseudo-critical behavior, where the true
critical point lies in the complex plane but with a small imaginary component,
as has been proposed to explain anomalous behavior in models of deconfined
criticality [Nahum  _et al._ , 2015; Gorbenko  _et al._ , 2018; Ma and Wang,
2020; Nahum, 2020; Song  _et al._ , 2023].

Figure 4: A density plot of $-\ln(|R_{0}(h,d)|)$ for the $J$-$Q$ model at the
critical coupling with $L=32$. The dark lines indicate zeros of the partition
function in the $J$-$Q$ model subjected to combined Néel and VBS fields.

Acknowledgements: We thank Román Orús for his support for this project.
Computing resources were used from the XSEDE allocation NSF DMR-130040 using
the Expanse cluster at the San Diego Supercomputer Center.

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Figure 5: A comparison of data obtained with the QMC method versus exact
diagonalization. The system is an $L=2$ Heisenberg bilayer with
$J=1,J_{\perp}=2.52205$ with three different values of $\beta$. The ratio
$|R_{0}(h)|$ is plotted as a function of the Néel field $h$ along the
imaginary axis.

Figure 6: A density plot of $-\ln(|R_{0}(h,d)|)$ for the $J$-$Q$ model at the
critical coupling with $L=40$ at three different values of the coupling:
$J/Q=0.015,0.045,0.085$ from left to right.

##  I Supplemental material

###  I.1 QMC versus exact diagonalization

Here we demonstrate the validity of Eqn. (2) by comparing against exact
results for a small system size. Fig. (5) shows $|R_{0}(h)|$ computed with a
purely imaginary Néel field on a $2\times 2$ Heisenberg bilayer system
compared against exact diagonalization of the whole spectrum. We make the
comparison for three different values of $\beta$, where in each case the first
three Lee-Yang zeros are visible. The QMC values match the exact results
perfectly within the statistical uncertainty.

###  I.2 Data away from deconfined critical point

Here we would like to show what happens to the nearly circular distribution of
zeros, with combined imaginary Néel and VBS fields, away from the critical
point in the $J$-$Q$ model. In Fig. (6) we show density plots (similar to Fig.
(4)) for three different values of the coupling: $J/Q=0.015,0.045,0.085$ from
left to right for system size $L=40$. Here we see on the left, which is on the
VBS side of the transition, the Néel zeros get pushed away from the origin,
relative to their values at the transition given, and in fact are not clearly
detectable on this plot. A similar story goes for the right panel. While we
believe our data to be statistically converged for the field values shown,
which would indicate a potential rearrangement of the lines of zeros away from
the critical point, we cannot rule out the possibility of under sampling at
the larger field values.