Two-dimensional and absolutely entanglement-breaking subspaces

Jian Yan; Lin Chen

doi:10.1088/1572-9494/ad43d2

Communications in Theoretical Physics >

2024 , Vol. 76 >Issue 9: 95102

DOI: https://doi.org/10.1088/1572-9494/ad43d2

Quantum Physics and Quantum Information

Two-dimensional and absolutely entanglement-breaking subspaces

Jian Yan ^,¹ ,
Lin Chen ^,¹^,²

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¹LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing 100191, China
²International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China

Received date: 2024-02-06

Revised date: 2024-04-25

Accepted date: 2024-04-26

Online published: 2024-07-24

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Abstract

Entanglement-breaking (EB) subspaces determine the additivity of entanglement of formation (EOF), which is a long-standing issue in quantum information. We explicitly construct the two-dimensional EB subspaces of any bipartite system, when system dimensions are equal, and we apply the subspaces to construct EB spaces of arbitrary dimensions. We also present partial construction when system dimensions are different. Then, we present the notion and properties of EB subspaces for some systems, and in particular the absolute EB subspaces. We construct some examples of absolute EB subspaces, as well as EB subspaces for some systems by using multiqubit Dicke states.

Key words： entanglement-breaking subspaces; entanglement of formation; absolutely EB subspaces; entanglement cost; additivity

Cite this article

Jian Yan , Lin Chen . Two-dimensional and absolutely entanglement-breaking subspaces[J]. Communications in Theoretical Physics, 2024 , 76(9) : 095102 . DOI: 10.1088/1572-9494/ad43d2

1. Introduction

Entanglement has played a key role in various quantum information tasks. Of such tasks as quantum teleportation, the maximally entangled states, namely the Bell states are considered to be more useful than other forms of entanglement. As a result, it is a vital problem to quantify how much entanglement a state has. Several entanglement measures for both bipartite and multipartite states have been proposed to tackle the problem over the past decades, such as the distillable entanglement, entanglement costs, geometric measurement of entanglement, relative entropy of entanglement, and robustness of entanglement [1]. In particular, the entanglement cost evaluates the minimum number of Bell states required to asymptotically generate a given state. Such physical interpretation makes entanglement cost one of the most fundamental entanglement measures [2], which has received plenty of attention. Nevertheless, the quantification of entanglement cost turns out to be a challenging problem. A key simplification for this problem was proposed by Hayden et al [3], namely that the entanglement cost is the regularized form of entanglement of formation (EOF). The latter calculates the minimal entanglement to generate exactly one copy of the target bipartite state [4–7]. Researchers have found the analytical expression of EOF for two-qubit states [8], isotropic states [9] and some special families of states [10]. To explore the entanglement cost, one has to study whether the EOF is additive, i.e., whether the EOF of the tensor product of two bipartite states is the sum of their EOF. While the answer has been shown to be negative [11], the constructive state with non-additive EOF remains unknown. As a result, understanding the states with additive EOF is far from satisfying. A systematic method for determining the additivity is to study the entanglement breaking (EB) space [12, 13], and we shall explain the details in the section. The additivity is also related to the study of quantum channels, minimum entropy, as well as the additivity of tensor product states [14–16], and coherence measures from resource theory [17, 18]. Apart from the role of serving entanglement cost, EOF helps to investigate many-body quantum correlations, monogamy of coherent states, symmetric Gaussian states [19–21], as well as robust entanglement preparation against noise [22], propagating optical modes and entanglement certification [23, 24].

The idea of [12, 13] is related to the generation of separable states through the function of entanglement breaking channels, by which some EB subspaces of low dimensions were proposed. As the first contribution of this paper, we extend the previous study to two-dimensional bipartite EB subspaces of arbitrary dimensions. We explicitly give the expressions of such spaces when the system dimensions are equal in theorem 3. This is assisted by the existing facts on matrix ranks and EB spaces concluded in lemmas 1 and 2. Then we construct EB spaces of large dimensions in corollary 4. When the system dimensions are different, we investigate a few cases in lemma 5 and theorems 6 and 7. We construct explicit expressions for certain two-dimensional EB subspaces in terms of the parameters of basis vectors.

As the second contribution of this paper, we will present the notion of EB subspaces for some systems. In particular, the absolute EB subspace means an EB subspace for any subsystems. We establish some properties to construct novel absolute EB subspaces by using known ones in theorem 8. Such properties include the fact that a subspace of an absolutely EB subspace is still an absolutely EB subspace. In addition, the tensor product of two absolutely EB subspaces is also an absolutely EB subspace. We present examples of absolutely EB subspaces, as well as EB subspaces for some systems by using multiqubit Dicke states. Our results explore the further connection between EB spaces and multipartite systems.

The remainder of this paper is organized as follows. In section 2, we introduce the preliminary facts used in this paper. In section 3, we introduce our main results and applications. Then, we present the notions and examples of EB subspaces for some systems, as well as absolute EB subspaces in section 4. Then, we propose some physical application in section 5. We present our summary in section 6.

2. Preliminaries

In this section, we introduce the fundamental techniques used throughout the remaining sections. Given two n-partite states α and β, we say that β is convertible to α under stochastic local operations and classical communications (SLOCC), when there is a product square matrix $X={\otimes }_{j=1}^{n}{X}_{j}$ such that β = XαX^†. If the matrix X is invertible then we say that α and β are SLOCC equivalent. The above matrices X can be physically implemented using the positive operator-valued measurement (POVM) with a nonzero probability. The implementation occurs with certainty when X is a unitary matrix. In this case, we say that α and β are locally unitarily (LU) equivalent. One can directly extend the above convertibility and equivalence to subspaces. For example, if there exists a product square matrix X such that X∣a_j⟩ ∝ ∣b_j⟩ for every j, then the subspace spanned by ∣a_j⟩'s is SLOCC convertible to the subspace spanned by ∣b_j⟩'s. One can similarly define equivalence subspaces.

While the SLOCC equivalence has the advantage in certain quantum information tasks such as state classification [25], we shall show that the LU equivalence is essential in the construction of entanglement breaking (EB) subspaces. For this purpose, we recall that an EB subspace ${ \mathcal V }\subseteq {{ \mathcal H }}_{{AB}}$ is defined as a bipartite subspace such that that the partial trace over system B of every pure state in the space ${{ \mathcal H }}_{{AB}:{ab}}$ results in a bipartite separable state of the system A and ab. From the definition, one can straightforwardly verify that every subspace of an EB space is still an EB space, and every 1-dimensional subspace is also an EB space [12]. Further, every space convertible to an EB subspace via the product matrix X = V ⨂ W with a unitary W is also an EB subspace. We refer to this fact as the EB convertibility. Evidently, if V is invertible then the two subspaces are EB subspaces at the same time. On the other hand, it has been shown that if the range of a bipartite state is an EB space, then the EOF of this state is additive [12]. As a result, the entanglement cost E_c of this state is the same as its EOF E_f, due to the regularized form ${E}_{c}(\rho )={\mathrm{lim}}_{n\to \infty }\displaystyle \frac{1}{n}{E}_{f}({\rho }^{\otimes n})$ [3].

To characterize EB subspaces, we recall that an m × n state ρ is a positive semidefinite matrix satisfying that rank ρ_A = m and rank ρ_B = n. The non-entangled (i.e., separable) state is the convex sum of the product state, so its partial transpose is still a positive semidefinite matrix [26, 27]. We say that the separable state is positive partial transpose (PPT), while a PPT state may be not separable [28]. The separable and PPT states have been widely studied in the past decades [29–31]. Here, we present some facts useful for subsequent sections.

Lemma 1.

1.	1. (i) If a bipartite quantum state has a rank at most three, then it is PPT if and only if it is separable.
2.	2. (ii) If $m\leqslant n$, then the m × n state of rank n is PPT if and only if it is separable. In this case, the state is the convex sum of n pure product states.
3.	3. (iii) The m × n state of rank r is entangled if $\max \{m,n\}\gt r$.

In the following, we briefly review the findings on EB subspaces from [13]. They are EB subspaces in ${{\mathbb{C}}}^{m}\otimes {{\mathbb{C}}}^{3}$, and the two-dimensional EB spaces in ${{\mathbb{C}}}^{2}\otimes {{\mathbb{C}}}^{n}$.

Lemma 2.

1.	1. (i) Every bipartite EB subspace in ${{\mathbb{C}}}^{m}\otimes {{\mathbb{C}}}^{n}$ has dimension at most n. The upper bound is saturated when the subspace is spanned by $\| {a}_{1},1\rangle ,\ldots ,\| {a}_{n},n\rangle $ up to EB convertibility.
2.	2. (ii) Every 2-dimensional subspace in ${{\mathbb{C}}}^{m}\otimes {{\mathbb{C}}}^{2}$ is an EB space if and only if it is spanned by $\{\| x,0\rangle ,\| y,1\rangle \}$ up to EB convertibility.
3.	3. (iii) The 2-dimensional subspace in ${{\mathbb{C}}}^{m}\otimes {{\mathbb{C}}}^{3}$ is an EB space if and only if one of the following three spaces is EB convertible to the subspace. The spaces are respectively spanned by $\{\| 0,0\rangle ,\| 1,1\rangle +\| 2,2\rangle \}$, $\{\| 0,0\rangle +\| 1,1\rangle ,\| 1,1\rangle +\| 2,2\rangle \}$, and $\{\| 0,0\rangle +\| 1,1\rangle ,\quad \| 0,1\rangle +\| 1\rangle (d\| 0\rangle +{{f}{\rm{e}}}^{i\theta }\| 1\rangle +g\| 2\rangle )\}$, where $g\gt 0$, $d,f,\theta \geqslant 0$, such that $-1-{f}^{2}+{g}^{2}-{d}^{2}(-2+{d}^{2}\,+{f}^{2}+{g}^{2})+2{{df}}^{2}\cos 2\theta \geqslant 0.$
4.	4. (iv) The 2-dimensional EB subspace V in ${{\mathbb{C}}}^{2}\otimes {{\mathbb{C}}}^{n}$ is LU equivalent to those in ${{\mathbb{C}}}^{2}\otimes {{\mathbb{C}}}^{k}$ with k = 2,3 or 4. The space V with k = 2,3 has been characterized in (ii) and (iii). Up to EB convertibility, V with k = 4 is spanned by $\| 0,0\rangle +{b}_{0}({a}_{0}\| 0\rangle +{a}_{1}\| 1\rangle )\| 2\rangle +{d}_{0}({c}_{0}\| 0\rangle +{c}_{1}\| 1\rangle )\| 3\rangle ,\| 1,1\rangle \,+{b}_{1}({a}_{0}\| 0\rangle +{a}_{1}\| 1\rangle )\| 2\rangle +{d}_{1}({c}_{0}\| 0\rangle +{c}_{1}\| 1\rangle )\| 3\rangle ,$ for any complex numbers ${a}_{j},{b}_{j},{c}_{j},{d}_{j}$ for j=0,1 and $\det \left[\begin{array}{cc}{a}_{0}{b}_{1} & {a}_{1}{b}_{0}\\ {c}_{0}{d}_{1} & {c}_{1}{d}_{0}\end{array}\right]\ne 0.$ □

We shall apply these facts to derive more EB subspaces and their properties in the next sections.

3. Construction of two-dimensional EB subspaces and applications

Characterizing two-dimensional EB spaces ${ \mathcal S }\subseteq {{ \mathcal H }}_{{AB}}\,={{\mathbb{C}}}^{m}\otimes {{\mathbb{C}}}^{n}$ with $\min \{m,n\}\gt 1$ is an open problem. So far, only a few examples with small m or n have been constructed in lemma 2. Let ${ \mathcal S }$ be spanned by the linearly independent bipartite states ∣α⟩ and ∣β⟩. Using the Schmidt decomposition and EB convertibility, we can assume that

(1)$\begin{eqnarray}\begin{array}{l}| \alpha \rangle =\displaystyle \sum _{j=1}^{r}| j,j\rangle ,\quad | \beta \rangle =\displaystyle \sum _{i=1}^{m}\displaystyle \sum _{j=1}^{n}{c}_{{ij}}| i,j\rangle ,\end{array}\end{eqnarray}$

with r ∈ [1, m] and coefficients ${c}_{{ij}}\in {\mathbb{C}}$. By definition of EB spaces, we need to show that the state ${\rho }_{{Aab}}={\mathrm{Tr}}_{B}| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}{| }_{{ABab}}$ is a bipartite separable state of the system A and ab, where

(2)$\begin{eqnarray}\begin{array}{l}| {\rm{\Psi }}{\rangle }_{{ABab}}=| \alpha {\rangle }_{{AB}}| 0{\rangle }_{{ab}}+| \beta {\rangle }_{{AB}}| 1{\rangle }_{{ab}}.\end{array}\end{eqnarray}$

We consider the case that $| {\rm{\Psi }}{\rangle }_{{ABab}}\subseteq {{ \mathcal S }}_{{AB}}\otimes {({{\mathbb{C}}}^{2})}_{{ab}}$ is an m × n × 2 state of system A, B and ab. If m > n then lemma 1 (iii) implies that ρ_Aab is not separable, and the subspace ${ \mathcal S }$ is not EB. We have m ≤ n. We present the following observation.

Theorem 3.

1.	1. (i) The two-dimensional EB subspace ${ \mathcal S }\subseteq {{ \mathcal H }}_{{AB}}\,={{\mathbb{C}}}^{m}\otimes {{\mathbb{C}}}^{n}$ satisfies $m\leqslant n$.
2.	2. (ii) If m = n, then ${ \mathcal S }$ is EB if and only if ${ \mathcal S }$ is spanned by the two linearly independent states ${\sum }_{i=1}^{m}{p}_{i}\| {a}_{i},i\rangle $ and ${\sum }_{i=1}^{m}{q}_{i}\| {a}_{i},i\rangle $ up to EB convertibility, where ${p}_{i},{q}_{i}$ are complex numbers.

Proof.

1. (i) The claim has been proven above this lemma.

2. (ii) We prove the ‘if’ part. Suppose $\mathcal{S}$ is spanned by the two linearly independent states $\sum_{i=1}^{m} p_{i}\left|a_{i}, i\right\rangle$ and $\sum_{i=1}^{m} q_{i}\left|a_{i}, \ i\right\rangle$ up to EB convertibility. Let $\ |\ \Psi\rangle_{A B a b}=(V \otimes W)_{A B}$ $\sum_{i=1}^{m} p_{i}\left|a_{i}, i \ \right\rangle_{A B}|0\rangle_{a b}$ $\ $+ $\ $ $(V \otimes W)_{A B}$ $\ $ $\sum_{i=1}^{m} q_{i}\left|a_{i}, i\right\rangle_{A B}|1\rangle_{a b}$ $\ $ with a unitary matrix W. One can show that the state $\ $ $\rho_{A a b}=\operatorname{Tr}_{B}\ | \ \Psi\rangle\left\langle\left.\Psi\right|_{A B a b}\right.$ is a bipartite separable state of system A and ab. Hence, ${\mathcal S}$ is an EB subspace.□

Next, we prove the ‘only if’ part. Suppose the subspace ${ \mathcal S }\subset {{\mathbb{C}}}^{m}\otimes {{\mathbb{C}}}^{m}$ is EB. Let $| {\rm{\Psi }}{\rangle }_{{AB}:{ab}}\in {{ \mathcal S }}_{{AB}}\otimes {({{\mathbb{C}}}^{2})}_{{ab}}$ be an m × m × 2 state. So, the state ${\rho }_{{Aab}}={\mathrm{Tr}}_{B}| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}{| }_{{ABab}}$ is a separable state of system A and ab. Using lemma 1 (ii), we have ${\rho }_{{Aab}}={\sum }_{i=1}^{m}| {a}_{j},{b}_{j}\rangle \langle {a}_{j},{b}_{j}| $. Its purification state, namely ∣$\Psi$⟩_ABab can be written as $| {\rm{\Psi }}{\rangle }_{{ABab}}={\sum }_{i=1}^{m}| {a}_{j},{u}_{j},{b}_{j}\rangle $ with an orthonormal basis ∣u_j⟩'s in ${\mathbb{C}}$^m. Setting ∣b_j⟩ = p_j∣0⟩ + q_j∣1⟩ implies the claim.

Using the definition of EB subspaces, we know that the rank two bipartite state whose range is spanned by ${\sum }_{i=1}^{m}{p}_{i}| {a}_{i},i\rangle $ and ${\sum }_{i=1}^{m}{q}_{i}| {a}_{i},i\rangle $ has additive EOF. Hence such a state has equal EOF and entanglement cost. Because the tensor product of two EB subspaces and every subspace of an EB space are both an EB subspace [13], we obtain the following fact.

Corollary 4. Let $| a\rangle ={\sum }_{i=1}^{m}{p}_{i}| {a}_{i},i\rangle $ and $| b\rangle ={\sum }_{i=1}^{m}{q}_{i}| {a}_{i},i\rangle $. Let $f({x}_{1},\ldots ,{x}_{n})=| {x}_{1}\rangle \otimes ...\otimes | {x}_{n}\rangle $ with ${x}_{j}=a$ or b. The subspace spanned by the ${2}^{n}$ basis vectors $f(a,\ldots a,a),f(a,\ldots ,a,b),\ldots ,f(b,\ldots ,b,b)$ is an EB subspace.

In the remainder of this section, we shall study (2) with m < n. We say that two bipartite states ∣x⟩ and ∣y⟩ on ${{ \mathcal H }}_{{AB}}$ have the same local subspaces in the sense that the range spaces of the reduced density operators x_A (resp. x_B) and y_A (resp. y_B) are the same, i.e., ${ \mathcal R }({x}_{A})={ \mathcal R }({y}_{A})$ and ${ \mathcal R }({x}_{B})\,={ \mathcal R }({y}_{B})$. Using (1), we write

(3)$\begin{eqnarray}\begin{array}{rcl}| \beta \rangle & = & | {\beta }_{1}\rangle +| {\beta }_{2}\rangle +| {\beta }_{3}\rangle +| {\beta }_{4}\rangle \\ & & := \displaystyle \sum _{i=1}^{r}\displaystyle \sum _{j=1}^{r}{c}_{{ij}}| i,j\rangle \\ & & +\displaystyle \sum _{i=r+1}^{m}\displaystyle \sum _{j=1}^{r}{c}_{{ij}}| i,j\rangle \\ & & +\displaystyle \sum _{i=1}^{r}\displaystyle \sum _{j=r+1}^{n}{c}_{{ij}}| i,j\rangle \\ & & +\displaystyle \sum _{i=r+1}^{m}\displaystyle \sum _{j=r+1}^{n}{c}_{{ij}}| i,j\rangle ,\end{array}\end{eqnarray}$

where the coefficients c_ij's are complex numbers. Using the Schmidt decomposition of ∣β₄⟩ and EB convertibility, we can find an invertible V₁ and a unitary W₁ such that

(4)$\begin{eqnarray}\begin{array}{l}({V}_{1}\otimes {W}_{1})| {\beta }_{j}\rangle =| {\beta }_{j}^{\prime} \rangle ,\end{array}\end{eqnarray}$

for j = 1, 2, 3, 4 where ∣β_j⟩ and $| {\beta }_{j}^{\prime} \rangle $ have the same local subspaces, and $| {\beta }_{4}^{\prime} \rangle ={\sum }_{i=r+1}^{s}| {ii}\rangle $ with s ∈ [r, m]. We begin by presenting the following observation.

Lemma 5. If $| {\beta }_{1}\rangle =| {\beta }_{2}\rangle =0$ in (3), then the space ${ \mathcal S }=\mathrm{span}\{| \alpha \rangle ,| \beta \rangle \}$ is EB.

Proof. Because $| {\beta }_{1}\rangle =| {\beta }_{2}\rangle =0$, we have ${\mathrm{Tr}}_{B}| \alpha \rangle \langle \beta {| }_{{AB}}=0$ by using (2). We have

(5)$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{Aab}} & = & {\mathrm{Tr}}_{B}| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}{| }_{{ABab}}\\ & = & {\mathrm{Tr}}_{B}| \alpha \rangle \langle \alpha {| }_{{AB}}\otimes | 0\rangle \langle 0{| }_{{ab}}+{\mathrm{Tr}}_{B}| \beta \rangle \langle \beta {| }_{{AB}}\otimes | 1\rangle \langle 1{| }_{{ab}}.\end{array}\end{eqnarray}$

So it is a bipartite separable state of the system A and ab. By definition, the subspace ${ \mathcal S }$ is EB breaking.□

Next, we present a more complex case.

Theorem 6.

1.	1. (i) If $\| {\beta }_{3}\rangle =\| {\beta }_{4}\rangle =0$ in (3), then the space ${ \mathcal S }\,=\mathrm{span}\{\| \alpha \rangle ,\| \beta \rangle \}$ is EB when up to LU equivalence, $\| \alpha \rangle ={\sum }_{j=1}^{r}\| j,j\rangle $ and $\| \beta \rangle ={\sum }_{j=1}^{r}{f}_{j}\| j,j\rangle $ with ${f}_{j}\in {\mathbb{C}}$.
2.	2. (ii) If $\| {\beta }_{3}\rangle =0$ in (3), then the space ${ \mathcal S }=\mathrm{span}\{\| \alpha \rangle ,\| \beta \rangle \}$ is EB when up to LU equivalence, $\| \alpha \rangle ={\sum }_{j=1}^{r}\| j,j\rangle $ and $\| \beta \rangle ={\sum }_{j=1}^{r}{f}_{j}\| j,j\rangle +{\sum }_{j\gt r}{g}_{j}\| j,j\rangle $ with ${f}_{j},{g}_{j}\in {\mathbb{C}}$.

Proof.

1. (i) Using (3), we know that the state $| {\rm{\Psi }}{\rangle }_{{ABab}}$ in (2) is a $d\times r\times 2$ state, where $m\geqslant d\geqslant r$. Because ${ \mathcal S }$ is EB, it follows from theorem 3 (i) that $d\leqslant r$. Hence d = r. Theorem 3 (ii) implies that $| {\rm{\Psi }}{\rangle }_{{ABab}}={\sum }_{i=1}^{r}| {a}_{j},{u}_{j},{b}_{j}\rangle $ with orthonormal states $| {u}_{j}\rangle $'s in ${\mathbb{C}}$^m. Using (1) and (2), we obtain that

$\begin{eqnarray*}| {\rm{\Psi }}{\rangle }_{{ABab}}=\displaystyle \sum _{j=1}^{r}| {u}_{j}^{* },{u}_{j}{\rangle }_{{AB}}{(| 0\rangle +{f}_{j}| 1\rangle )}_{{ab}},\end{eqnarray*}$

where ${f}_{j}\in {\mathbb{C}}$. Up to LU equivalence, we may assume that $| {u}_{j}\rangle =| j\rangle $ for every j, and the claim holds.

2. (ii) The proof is similar to that of (i).

□

The above facts heavily rely on the relation between the ranks of global and local density matrices. We note that such relation has been used to detect the entanglement of non-PPT states when the rank of global density matrix is smaller, as shown in lemma 1.

The characterization of EB spaces turns out to be a larger challenge when the relation does not help, as we show below.

Theorem 7. Suppose $| {\beta }_{4}\rangle =0$, and r and c_ij’s are given in (3). If the space ${ \mathcal S }=\mathrm{span}\{| \alpha \rangle ,| \beta \rangle \}$ is EB, then $| {\beta }_{2}\rangle =0$ in (3) and the r × r matrix X is positive semidefinite, where the matrix element ${X}_{{iy}}={\sum }_{j=1}^{n}{c}_{{ij}}{c}_{{yj}}^{* }-{\sum }_{j=1}^{r}{c}_{{ji}}^{* }{c}_{{jy}}$.

It is possible to simplify the matrix in (A.6). We can find some r × r unitary matrices U and (n − r) × (n − r) invertible matrix V, and apply the matrices to (A.1) to obtain

(6)$\begin{eqnarray}\left(U\otimes \left[\begin{array}{cc}{U}^{* } & 0\\ 0 & V\end{array}\right]\right)\displaystyle \sum _{j=1}^{r}| j,j\rangle =\displaystyle \sum _{j=1}^{r}| j,j\rangle =| \alpha \rangle ,\end{eqnarray}$

and

(7)$\begin{eqnarray}\left(U\otimes \left[\begin{array}{cc}{U}^{* } & 0\\ 0 & V\end{array}\right]\right)\left|{\beta }_{3}\right\rangle =\displaystyle \sum _{j=1}^{k}| j,j+r\rangle ,\end{eqnarray}$

where $k\leqslant \min \{r,n-r\}$. Nevertheless, such a reduction is not enough to determine whether the PPT state in (A.4) is separable. A further investigation towards the coefficients c_ij's may work in the future, in spite of the absence of an operational criterion on determining the separability of qubit-qudit states yet.

4. Entanglement breaking subspaces for some systems

So far, we have discussed EB subspaces of exactly two systems. In this section, we extend the EB subspaces to that for some systems. Let $| {a}_{j}\rangle \in {{ \mathcal H }}_{{A}_{1}...{A}_{n}}$ be n-partite states for j=1,…,k. Let

(8)$\begin{eqnarray}\begin{array}{l}| {\rm{\Psi }}\rangle =\displaystyle \sum _{j=1}^{k}{c}_{j}| {a}_{j}\rangle | j\rangle \in {{ \mathcal H }}_{{A}_{1}...{A}_{n}}\otimes {{ \mathcal H }}_{{B}_{1}...{B}_{n}},\end{array}\end{eqnarray}$

with complex numbers c_j's. We say that the states ∣a_j⟩ span an EB subspace for systems ${A}_{{j}_{1}},\ldots ,{A}_{{j}_{m}}$ when tracing out these systems of the state ∣$\Psi$⟩ generates a bipartite separable state of systems ${A}_{{j}_{m+1}},\ldots ,{A}_{{j}_{n}}$ and B₁,…,B_n. For example, if the states ∣a_j⟩ = ∣j⟩^⨂n, then one can straightforwardly show that the states span a k-dimensional EB subspace for the systems ${A}_{{j}_{1}},\ldots ,{A}_{{j}_{m}}$ for any j₁,…,j_m ∈ [1, n] with 0 < m < n. In this case, we say that the subspace is a k-dimensional absolutely EB subspace. As another example, if k = 1 then any ∣a₁⟩ spans a one-dimensional absolutely EB subspace by (8). We present the following observation on absolute EB subspaces.

Theorem 8.

1.	1. (i) Every subspace of an absolutely EB space is also an absolutely EB subspace.
2.	2. (ii) If ${ \mathcal H }$ and ${ \mathcal K }$ are two absolutely EB spaces, then so is the tensor product ${ \mathcal H }\otimes { \mathcal K }$.

In claim (ii), we shall refer to ${ \mathcal H }={{ \mathcal H }}_{{A}_{1}....{A}_{n}}$ and ${ \mathcal K }={{ \mathcal K }}_{{C}_{1}....{C}_{n}}$. The tensor product ${ \mathcal H }\otimes { \mathcal K }$ means the n-partite Hilbert space ${\otimes }_{j=1}^{n}({{ \mathcal H }}_{{A}_{j}}\otimes {{ \mathcal K }}_{{C}_{j}})$. Note that the claim extends some conclusions on bipartite EB subspaces [13].

Proof.

1. (i) The claim follows from the definition of absolute EB subspaces.

2. (ii) Let ${ \mathcal H }$ be spanned by the basis $| {a}_{1}\rangle ,\ldots ,| {a}_{p}\rangle $, ${ \mathcal K }$ spanned by the basis $| {b}_{1}\rangle ,\ldots ,| {a}_{q}\rangle $, and $| \alpha \rangle \in { \mathcal H }\otimes { \mathcal K }\otimes {{ \mathcal H }}_{{ab}}$. We have

(9)$\begin{eqnarray}\begin{array}{l}| \alpha \rangle =\displaystyle \sum _{i=1}^{p}\displaystyle \sum _{j=1}^{q}| {a}_{i},{b}_{j}\rangle | {x}_{i,j}\rangle .\end{array}\end{eqnarray}$

Because ${ \mathcal H }$ is an absolute subspace, tracing over the systems ${A}_{{j}_{1}},\ldots ,{A}_{{j}_{m}}$ from the state $| \alpha \rangle $ generates a bipartite separable state of system ${A}_{{j}_{m+1}},\ldots ,{A}_{{j}_{n}}$ and the remaining systems. We have

(10)$\begin{eqnarray}\begin{array}{l}{\mathrm{Tr}}_{{A}_{{j}_{1}},\ldots ,{A}_{{j}_{m}}}| \alpha \rangle \langle \alpha | =\displaystyle \sum _{i}{r}_{i}| {\beta }_{i}\rangle \langle {\beta }_{i}{| }_{{A}_{{j}_{m+1}},\ldots ,{A}_{{j}_{n}}}\otimes | {\gamma }_{i}\rangle \langle {\gamma }_{i}{| }_{{C}_{1}...{C}_{n}{ab}},\end{array}\end{eqnarray}$

where the reduced density operators of $| {\gamma }_{i}\rangle $'s on systems ${C}_{1}...{C}_{n}$ have the range spaces spanned by $| {b}_{j}\rangle $'s. Because ${ \mathcal K }$ is an absolute subspace, tracing over the systems ${B}_{{j}_{1}},\ldots ,{B}_{{j}_{m}}$ from the state $| {\gamma }_{i}\rangle $ generates a bipartite separable state of system ${B}_{{j}_{m+1}},\ldots ,{B}_{{j}_{n}}$ and the remaining systems. Hence, tracing over the systems ${A}_{{j}_{1}}{B}_{{j}_{1}},\ldots ,{A}_{{j}_{m}}{B}_{{j}_{m}}$ from the state $| \alpha \rangle $ generates a bipartite separable state of system ${A}_{{j}_{m+1}}{B}_{{j}_{m+1}},\ldots ,{A}_{{j}_{n}}{B}_{{j}_{n}}$ and the remaining systems. By definition, the tensor product ${ \mathcal H }\otimes { \mathcal K }$ is an absolutely EB subspace.

□

In the rest of this section, we construct some non-absolutely EB subspaces. One can employ the n-qubit Dicke states $| (n,k)\rangle ={\displaystyle \left(\genfrac{}{}{0em}{}{n}{k}\right)}^{-1/2}{\sum }_{j\,=\,1}^{\displaystyle \left(\genfrac{}{}{0em}{}{n}{k}\right)}{P}_{j}\left(| 0{\rangle }^{\otimes k}\otimes | 1{\rangle }^{\otimes (n-k)}\right)$, where P_j is an operator of the permutation group over n distinct elements. For example, when n = 3 then we can use ∣(3, 0)⟩ and ∣(3, 2)⟩. One can show that they span the two-dimensional EB subspace ${{ \mathcal H }}_{0}$ for systems ${A}_{{j}_{1}},{A}_{j,2}$ where j₁, j₂ ∈ {1, 2, 3}, because tracing out systems ${A}_{{j}_{1}},{A}_{j,2}$ of the state $| {x}_{0}\rangle ={c}_{0}| (3,0)\rangle | 0\rangle +{c}_{1}| (3,2)\rangle | 1\rangle )\in {{ \mathcal H }}_{{A}_{1}{A}_{2}{A}_{3}}\otimes {{ \mathcal H }}_{{ab}}$ results in a bipartite separable state of system ${A}_{{j}_{3}}$ and ab. On the other hand, one can verify that tracing out the system ${A}_{{j}_{1}}$ of the state ∣x₀⟩ may result in a bipartite entangled state of the system ${A}_{{j}_{2}},{A}_{{j}_{3}}$ and ab. So ${{ \mathcal H }}_{0}$ is not an absolutely EB subspace. A similar idea using multipartite Dicke states can be used to construct EB subspaces for any subsystems.

5. Physical application

In this section, we apply the results obtained in the preceding section for some physical scenarios. We consider the atoms A₁,…,A_n of the same site. They have correlation with the quantized single mode of the field C as the Tavis–Cummings (TC) model [32, 33], as well as the calculation process in [34]. We consider the model in the space ${({{\mathbb{C}}}^{2})}^{\otimes n}\otimes {{\mathbb{C}}}^{n}$. Suppose the initial state of the atoms is ∣0⟩^⨂n. The cavity mode will evolve within a Hilbert space spanned by {∣0⟩, ∣1⟩,…,∣k − 1⟩}, which can be practically constructed by ∣j⟩ → ∣j⟩^⨂n with the n atoms A₁,…,A_n. As a result, the atom system is involved into state $| {\rm{\Psi }}\rangle ={\sum }_{j=1}^{k}| j{\rangle }^{\otimes n}| {a}_{j}\rangle $, and thus the range space of A₁,…,A_n is an absolutely EB subspace, as we have shown in the last section. One can show that the state ∣$\Psi$⟩ is genuinely entangled. It is the strongest form of entanglement widely used in quantum computing and cryptography, and hence we have found an application of EB subspaces. Further to this, one can see that many copies of states ∣$\Psi$⟩ still leads to absolute subspaces of higher dimensions.

6. Conclusions

We have studied the two-dimensional EB subspace of any bipartite system, so as to study the additivity of EOF. We also presented the notions of EB subspaces of many systems and EB subspaces, as well as some examples to demonstrate the notions. An open problem is to extend our results to the high-dimensional EB subspace of any bipartite system, so that more entangled states of rank two with additive EOF can be constructed. Another question is how to construct all absolute EB subspaces.

Appendix

In this appendix section, we explain the proof of theorem 7.

Proof. By definition, we require that the state ${\rho }_{{Aab}}\,={\mathrm{Tr}}_{B}(| {\rm{\Psi }}{\rangle }_{{ABab}}\langle {\rm{\Psi }}{| }_{{ABab}})$ is a bipartite separable state of system A and ab, where

(A.1)$\begin{eqnarray}\begin{array}{rcl}| {\rm{\Psi }}{\rangle }_{{ABab}} & = & | \alpha {\rangle }_{{AB}}| 0{\rangle }_{{ab}}+{\left|\beta \right\rangle }_{{AB}}| 1{\rangle }_{{ab}}\\ & = & \displaystyle \sum _{j=1}^{r}| j,j,0\rangle +\displaystyle \sum _{i=1}^{r}\displaystyle \sum _{j=1}^{r}{c}_{{ij}}| i,j,1\rangle \\ & & +\,\displaystyle \sum _{i=r+1}^{m}\displaystyle \sum _{j=1}^{r}{c}_{{ij}}| i,j,1\rangle \\ & & +\,\displaystyle \sum _{i=1}^{r}\displaystyle \sum _{j=r+1}^{n}{c}_{{ij}}| i,j,1\rangle .\end{array}\end{eqnarray}$

By calculation, we have

(A.2)$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{Aab}} & = & \displaystyle \sum _{j=1}^{r}| j,0\rangle \langle j,0| +\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i=1}^{m}{c}_{{ij}}^{* }| j,0\rangle \langle i,1| \\ & & +\,\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i=1}^{m}{c}_{{ij}}| i,1\rangle \langle j,0| \\ & & +\,\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i,y=1}^{m}{c}_{{ij}}{c}_{{yj}}^{* }| i,1\rangle \langle y,1| \\ & & +\,\displaystyle \sum _{j=r+1}^{n}\displaystyle \sum _{i,y=1}^{r}{c}_{{ij}}{c}_{{yj}}^{* }| i,1\rangle \langle y,1| .\end{array}\end{eqnarray}$

Because $r\leqslant m\lt n$, the Peres-Horodecki criterion says that ${\rho }_{{Aab}}$ is PPT in terms of the system $A:{ab}$. It implies that ${c}_{{ij}}=0$ for $i=r+1,\ldots ,m$. Furthermore, if $m\leqslant 3$ then the PPT ${\rho }_{{Aab}}$ is separable. In the following, we investigate the conditions by which ${\rho }_{{Aab}}$ is separable for $m\gt 3$. Using the above implication, we have

(A.3)$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{Aab}} & = & \displaystyle \sum _{j=1}^{r}| j,0\rangle \langle j,0| +\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i=1}^{r}{c}_{{ij}}^{* }| j,0\rangle \langle i,1| \\ & & +\,\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i=1}^{r}{c}_{{ij}}| i,1\rangle \langle j,0| \\ & & +\,\displaystyle \sum _{j=1}^{n}\displaystyle \sum _{i,y=1}^{r}{c}_{{ij}}{c}_{{yj}}^{* }| i,1\rangle \langle y,1| .\end{array}\end{eqnarray}$

Because ${\rho }_{{Aab}}$ is separable, we obtain that ${\sigma }_{{abA}}:= S{\rho }_{{Aab}}S$ is also separable, where S is the swap operator for system A and ab. We write ${\sigma }_{{abA}}$ in the matrix form, i.e.,

(A.4)$\begin{eqnarray}\begin{array}{l}{\sigma }_{{abA}}=\left[\begin{array}{cc}{I}_{r} & {F}^{\dagger }\\ F & G\end{array}\right],\end{array}\end{eqnarray}$

where the order-r matrices $F={\sum }_{j=1}^{r}{\sum }_{i=1}^{r}{c}_{{ij}}| i\rangle \langle j| $ and $G={\sum }_{j=1}^{n}{\sum }_{i,y=1}^{r}{c}_{{ij}}{c}_{{yj}}^{* }| i\rangle \langle y| $. Because ${\sigma }_{{abA}}$ is separable, it is also PPT. Hence the matrix $G-{F}^{\dagger }F\geqslant 0$. We write up the matrix using c_ij's as follows.

(A.5)$\begin{eqnarray}\begin{array}{rcl}G-{F}^{\dagger }F & = & \displaystyle \sum _{j=1}^{n}\displaystyle \sum _{i,y=1}^{r}{c}_{{ij}}{c}_{{yj}}^{* }| i\rangle \langle y| \\ & & -{\left(\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i=1}^{r}{c}_{{ij}}| i\rangle \langle j| \right)}^{\dagger }\left(\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i=1}^{r}{c}_{{ij}}| i\rangle \langle j| \right)\\ & = & \displaystyle \sum _{j=1}^{n}\displaystyle \sum _{i,y=1}^{r}{c}_{{ij}}{c}_{{yj}}^{* }| i\rangle \langle y| \\ & & -\,\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i=1}^{r}{c}_{{ij}}^{* }| j\rangle \langle i| \displaystyle \sum _{y=1}^{r}\displaystyle \sum _{x=1}^{r}{c}_{{xy}}| x\rangle \langle y| \\ & = & \displaystyle \sum _{j=1}^{n}\displaystyle \sum _{i,y=1}^{r}{c}_{{ij}}{c}_{{yj}}^{* }| i\rangle \langle y| \\ & & -\,\displaystyle \sum _{j=1}^{r}\displaystyle \sum _{i,y=1}^{r}{c}_{{ij}}^{* }{c}_{{iy}}| j\rangle \langle y| \\ & = & \displaystyle \sum _{j=1}^{n}\displaystyle \sum _{i,y=1}^{r}{c}_{{ij}}{c}_{{yj}}^{* }| i\rangle \langle y| \\ & & -\,\displaystyle \sum _{i=1}^{r}\displaystyle \sum _{j,y=1}^{r}{c}_{{ji}}^{* }{c}_{{jy}}| i\rangle \langle y| .\end{array}\end{eqnarray}$

Therefore, the matrix $G-{F}^{\dagger }F$ is a r × r matrix with elements as

(A.6)$\begin{eqnarray}{(G-{F}^{\dagger }F)}_{{iy}}=\displaystyle \sum _{j=1}^{n}{c}_{{ij}}{c}_{{yj}}^{* }-\displaystyle \sum _{j=1}^{r}{c}_{{ji}}^{* }{c}_{{jy}}.\end{eqnarray}$

We have proven the assertion.□

This work was supported by the NNSF of China (Grant No. 11871089).

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模态框（Modal）标题

Abstract

Cite this article

1. Introduction

2. Preliminaries

3. Construction of two-dimensional EB subspaces and applications

4. Entanglement breaking subspaces for some systems

5. Physical application

6. Conclusions

Appendix

References