Open AccessArticle

New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States

Key Laboratory of Optoelectronic Devices and Systems, College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China

Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Authors to whom correspondence should be addressed.

Mathematics 2022, 10(15), 2624; https://doi.org/10.3390/math10152624

Submission received: 30 May 2022 / Revised: 6 July 2022 / Accepted: 20 July 2022 / Published: 27 July 2022

(This article belongs to the Section Computational and Applied Mathematics)

Download Versions Notes

Abstract

This study focuses on tensor Z-eigenvalue localization and its application in the geometric measure of entanglement for multipartite quantum states. A new Z-eigenvalue localization theorem and the bounds for the Z-spectral radius are derived, which are more precise than some of the existing results. On the other hand, we present theoretical bounds of the geometric measure of entanglement for a weakly symmetric multipartite quantum state with non-negative amplitudes by virtue of different distance measures. Numerical examples show that these conclusions are superior to the existing results in quantum physics in some cases.

Keywords:

Z-eigenvalue; non-negative tensors; spectral radius; geometric measure of entanglement

MSC:

15A18; 15A69; 15A21

1. Introduction

Tensors, namely multidimensional arrays, have become more and more important in many different fields of applied mathematics and computational mathematics, and they have promoted the development of numerical multilinear algebra. They have a very rich diversity in practical applications, especially in the positive definiteness of even-order multivariate forms in automatical control [1], higher-order statistics [2,3] and multiple dimensional data analysis [4,5].

Tensor eigenvalues are widely used in a large amount of scientific and engineering problems. However, the calculation of the Z-eigenvalue of a higher-order tensor is usually NP hard, which is different from the case of matrices. Nevertheless, there are some algorithms for calculating one or more eigenvalues of tensors, such as [6,7,8,9,10,11,12]. Unfortunately, these methods do not work well in larger-sized tensors, even on a medium scale. In this situation, the eigenvalue localization methods can capture all eigenvalues of a high-order tensor in a certain interval. For example, Ger

\overset{ˇ}{s}

gorin and Brauer-type tensor eigenvalue inclusion sets are introduced in [13]. Therefore, eigenvalue localization is one of the important methods to investigate the spectral radius of higher-order tensors.

Entanglements in composite systems are a basic and important feature of quantum physics and the core resource of the field of quantum information science [14], but it has been proved difficult to quantify. It makes all the difference that we know whether a quantum state is entangled or not in many practical applications [15]. There are many elegant entanglement criteria, such as the Bell inequality [16], entanglement witness [17] and the positive partial transposition (PPT) criterion [18,19]. However, in the case of multipartite systems, the situation is substantially more complicated. The geometric measure of entanglement has become the most basic method for measuring the entanglement of multipartite systems, which is proposed to Shimony [20] for bipartite systems and extended to multipartite systems by Wei and Goldbart [21]. Despite its significance, the explicit value of the geometric measures of entanglement can be derived for only a few entangled states, such as generalized W states [22], Dicke states and m-qubit GHZ states [21]. Since the definition involves the optimization process of all separable state sets, it is still impossible to obtain geometric measures for most of the multipartite states, which represents a formidable task in the general case even with numerical approaches.

The geometric measure of entanglement can be attributed to the spectral radius of the normalized tensors [11] from a mathematical perspective. Recently, the authors in [23] indicate that the maximal overlap of the state

Ψ

with a pure separable state is equivalent to the Z-spectral radius of symmetric nonnegative tensor

A_{Ψ}

; that is, the geometric measure of entanglement is derived. Moreover, these conclusions are generalized to the weakly symmetric non-negative tensor case [24]. On the other hand, based on Bures distance, the authors in [25] propose an upper bound for a maximally geometric measure of entanglement for an m-partite system composed of subsystems of dimensions

d_{1}, \dots, d_{m}

Highly entangled multipartite states are very important in the fields of quantum information processing, quantum error correction and quantum communication, especially in the exponential acceleration of quantum algorithms; for details, see [26,27,28]. However, the authors in [29] argue that the entanglement in symmetric case is much smaller than in the general case, and most symmetric quantum states are close to being maximally entangled. They also present the upper bound of the maximal possible geometric measure of entanglement for Boson quantum states. On this basis, an upper bound for the entanglement is derived in [27].

In this literature, we focus on the Z-eigenvalue localization set for a tensor and its application in the geometric measure of entanglement for multipartite quantum states, which is beneficial to the cross development of tensor theory and quantum information. A new Z-eigenvalue localization theorem and the bounds for the Z-spectral radius are derived, which is more precise than some of the existing results. As applications, we are devoted to the geometric measure of entanglement on the ground of tensor Z-eigenvalue theory. Based on different distance measures, we present theoretical bounds of the geometric measure of entanglement for a weakly symmetric pure state with non-negative amplitudes. Numerical examples show that our bounds are more precise than some existing conclusions in quantum physics.

2. Preliminaries

2.1. Preliminaries for Tensors

For a positive integer

n \in N

, we denote

[n]

by the set of positive integers

{1, \dots, n}

. An mth-order n-dimensional real tensor denoted by

A : = (a_{i_{1} \dots i_{m}}) \in R^{n_{1} \times \dots \times n_{m}}

is a multidimensional array consisting of

n^{m}

numbers

a_{i_{1} \dots i_{m}} \in R

for all

i_{j} \in [n_{j}]

and

j \in [m] .

A symmetric tensor is a square tensor, that is

i_{k} = n, k = 1, \dots, m

, if its entries

a_{i_{1} \dots i_{m}}

are invariant under any permutation of m indices

(i_{1}, i_{2}, \dots, i_{m})

, which are denoted as

S^{m} (R^{n})

. We use

R^{[m \times n]} (C^{[m \times n]})

to represent the set of all m-order n-dimensional real (complex) tensors. For a tensor

A \in R^{[m \times n]}

A

is non-negative (positive) if every entry

a_{i_{1} i_{2} \dots i_{m}} \geq (>) 0

A = (t_{i_{1} i_{2} \dots i_{m}}) \in R^{[m \times n]}

is weakly symmetric [30] if the associated homogeneous polynomial

A x^{m} = \sum_{i_{1}, i_{2}, \dots, i_{m} = 1}^{n} a_{i_{1} i_{2} \dots i_{m}} x_{i_{1}} x_{i_{2}} \dots x_{i_{m}}

satisfies

\nabla A x^{m} = m A x^{m - 1}

, where ∇ denotes the gradient of the associated multivariable function and

x = {(x_{1}, x_{2}, \dots, x_{n})}^{⊤} \in R^{n}

, where

A x^{m - 1}

is an n dimension vector in

C^{n}

, whose ith component is

{(A x^{m - 1})}_{i} = \sum_{i_{2}, i_{3}, \dots, i_{m} = 1}^{n} a_{i i_{2} \dots i_{m}} x_{i_{2}} \dots x_{i_{m}} .

It is worth noting that a symmetric tensor must be a weakly symmetric tensor but not vice versa. In a word, some conclusions for weakly symmetric tensors are applicable for symmetric tensors.

Definition 1

([31,32]). Let

A = (a_{i_{1} i_{2} \dots i_{m}})

be an m-order n-dimensional real tensor. If there is a real number λ and a nonzero real vector x such that

A x^{m - 1} = λ x, x^{⊤} x = 1,

where

A x^{m - 1}

is an n-dimension vector in

R^{n}

, whose ith component is

{(A x^{m - 1})}_{i} : = \sum_{i_{2}, \dots, i_{m} \in [n]} a_{i i_{2} \dots i_{m}} x_{i_{2}} \dots x_{i_{m}}, \forall i \in [n] .

Then, we say that λ is an Z-eigenvalue of

A

and x is an Z-eigenvector of

A

associated with λ.

We denote the Z-spectrum of tensor

A

σ (A)

: that is the set of all Z-eigenvalues of

A

. The Z-spectrum radius of

A

is defined as

ρ (A) = max {| λ |; λ \in σ (A)} .

For non-negative tensor

A

, the authors in [30] imply that the Z-spectrum radius

ρ (A)

is a Z-eigenvalue of

A

A

is weakly symmetric.

Ger

\overset{ˇ}{s}

gorin and Brauer-type tensor eigenvalues inclusion sets are introduced in [13].

Theorem 1

([13]). Let

A = (a_{i_{1} i_{2} \dots i_{m}}) \in C^{[m \times n]}

be anm-order n-dimension tensor. It follows that

σ (A) \subseteq K (A) = ⋃_{i \in N} K_{i} (A),

where

K_{i} (A) = {z \in C : | z | \leq R_{i} (A)}, R_{i} (A) = \sum_{i_{2}, \dots, i_{m}} | a_{i i_{2} \dots i_{m}} |

Theorem 2

([13]). Let

A = (a_{i_{1} i_{2} \dots i_{m}}) \in C^{[m \times n]}

be an m-order n-dimension tensor. It follows that

σ (A) \subseteq N (A) = ⋃_{i, j \in N, i \neq j} N_{i, j} (A),

where

N_{i, j} (A) = {z \in C : [| z | - (R_{i} (A) - P_{i}^{i} (A))] | z | \leq P_{i}^{i} (A) R_{j} (A), P_{i}^{i} (A) = \sum_{\begin{matrix} i_{2} \dots i_{m} \in N, \\ i \notin {i_{2}, \dots, i_{m}} \end{matrix}} | a_{i i_{2} \dots i_{m}} | .

2.2. Tensor Representation of Quantum States

For a composite m-partite quantum system, an m-partite pure state

| Ψ 〉

can be interpreted as a normalized element in a tensor product Hilbert space

H = \otimes_{k = 1}^{m} H_{k} = \otimes_{k = 1}^{m} R^{n_{k}}

, where the dimension of

H_{k}

n_{k}

. We suppose that

\{|e_{i_{k}}^{(k)}〉 : i_{k} \in [n_{k}]\}

is an orthogonal basis of

R^{n_{k}}

, which yields

\{|e_{i_{1}}^{(1)} e_{i_{2}}^{(2)} \dots e_{i_{m}}^{(m)}〉 : i_{k} \in [n_{k}]; k \in [m]\},

that is also an orthogonal basis of

\otimes_{k = 1}^{m} R^{n_{k}}

. In this expression,

| Ψ 〉

can be regarded as

| Ψ 〉 : = \sum_{i_{1}, \dots, i_{m} = 1}^{n_{1}, \dots, n_{m}} a_{i_{1} \dots i_{m}} |e_{i_{1}}^{(1)} e_{i_{2}}^{(2)} \dots e_{i_{m}}^{(m)}〉,

where

a_{i_{1} \dots i_{m}} \in R

. Under the orthogonal basis, the quantum state

| Ψ 〉

has a corresponding tensor representation denoted by

A_{| Ψ 〉} : = (a_{i_{1} \dots i_{m}}) .

In this sense, a weakly symmetric pure state always has a corresponding weakly symmetric tensor. A separable m-partite pure state can be considered as a product state

| ϕ 〉 : = \otimes_{k = 1}^{m} |ϕ^{(k)}〉, | ϕ^{(k)} 〉 = \sum u_{i_{k}}^{(k)} | e_{i_{k}}^{(k)} 〉 \in H_{k}, | | | ϕ^{(k)} 〉 | | = 1, k = 1, \dots, m .

We denote the set of all separable pure states in

H

S e p a r (H)

. We call the state an entangled state if it is inseparable.

The Hilbert–Schmidt distance is the Hilbert–Schmidt norm, such as trace operators and Hilbert–Schmidt operators (

| | \cdot {| |}_{2}

). Based on Hilbert–Schmidt distance, the geometric measure of entanglement for multipartite pure states

| Ψ 〉

is defined as

{GME}_{Ψ} ≜ min {| | | Ψ 〉 - | Φ 〉 | | : | Φ 〉 = \otimes_{k = 1}^{m} | ϕ^{(k)} 〉 \in S e p a r (H)} .

(1)

The minimization of

{GME}_{Ψ}

always has a solution because the minimization in (1) is taken with a continuous function on a compact set

S e p a r (H)

in a finite dimensional space

H

. It is evident that the nearest separable state

| Φ_{Ψ} 〉

can be chosen as a symmetric one.

Based on Bures distance, the authors in [25] propose an upper bound for a maximally geometric measure of entanglement for an m-partite system composed of subsystems of dimensions

d_{1}, \dots, d_{m}

Theorem 3

([25]). For any normalized pure state

| Ψ 〉 \in H

, we have

{GME}_{Ψ} \leq \sqrt{2 - 2 / \sqrt{d_{1} \dots d_{m - 1}}} .

On the basis of von Neumann entropy, a commonly used entropy form of geometric measure [27] can be defined as:

E_{G} (| Ψ 〉) = - {log}_{2} max_{| Φ 〉 = \otimes_{k = 1}^{m} | ϕ^{(k)} 〉 \in H} {| 〈 Ψ | Φ 〉 |}^{2} = - {log}_{2} G_{Ψ}^{2} .

(2)

The following upper bound for the entanglement is derived in [27].

Theorem 4

([27]). For all unit length tensors

A_{Ψ} \in R^{[m \times n]}

, one has

E_{G} (| Ψ 〉) \leq (m - 1) {log}_{2} (n) .

3. New $Z$ -Eigenvalue Localization Set and the Bounds for $Z$ -Spectral Radius

In this section, we present a new Z-eigenvalue inclusion theorem of tensors and show that our localization set is tighter than some existing localization sets. On this basis, lower and upper bounds for the Z-spectral radius of weakly symmetric non-negative tensors are available.

For a tensor

A = (a_{i_{1} i_{2} \dots i_{m}}) \in C^{[m \times n]}

, we denote

P_{i}^{i} (A) = \sum_{\begin{matrix} i_{2} \dots i_{m} \in N, \\ i \notin {i_{2}, \dots, i_{m}} \end{matrix}} | a_{i i_{2} \dots i_{m}} |, r_{i} (A) = \sum_{i_{2}, \dots, i_{m} \in N} | a_{i i_{2} \dots i_{m}} | .

Theorem 5.

Let

A = (a_{i_{1} i_{2} \dots i_{m}}) \in C^{[m \times n]}, n \geq 2

, there is the following Z-eigenvalue localization sets.

σ (A) \subseteq Υ (A) = (⋃_{i, j \in N, i \neq j} {\hat{Υ}}_{i, j} (A)) ⋃ (⋃_{i, j \in N, i \neq j} {\tilde{Υ}}_{i, j} (A) ⋂ K_{i} (A)),

where

{\hat{Υ}}_{i, j} (A) = {z \in C : | z | \leq r_{i} (A) - P_{i}^{i} (A), | z | \leq r_{j} (A) - | a_{j i \dots i} |},

\begin{matrix} {\tilde{Υ}}_{i, j} (A) = {z \in C : (| z | - (r_{i} (A) - P_{i}^{i} (A))) (| z | - (r_{j} (A) - | a_{j i \dots i} |)) \leq P_{i}^{i} (A) | a_{j i \dots i} |} . \end{matrix}

Proof.

Let

λ

be a Z-eigenvalue of

A

with the corresponding eigenvector x, then

A x^{m - 1} = λ x, x^{⊤} x = 1 .

(3)

We assume that

| x_{p} | \geq | x_{q} | \geq max_{i \in N, i \neq p, q} | x_{i} |

, then

0 < | x_{p} |^{m} \leq | x_{p} | \leq 1

. Based on (3), it follows that

\begin{matrix} λ x_{p} = \sum_{\begin{matrix} i_{2} \dots i_{m} \in N, \\ p \in {i_{2}, \dots, i_{m}} \end{matrix}} a_{p i_{2} \dots i_{m}} x_{i_{2}} \dots x_{i_{m}} + \sum_{\begin{matrix} i_{2} \dots i_{m} \in N, \\ p \notin {i_{2}, \dots, i_{m}} \end{matrix}} a_{p i_{2} \dots i_{m}} x_{i_{2}} \dots x_{i_{m}} . \end{matrix}

There are the following inequalities by virtue of the absolute value and the triangle inequality:

\begin{matrix} | λ | \leq \sum_{\begin{matrix} i_{2} \dots i_{m} \in N, \\ p \in {i_{2}, \dots, i_{m}} \end{matrix}} | a_{p i_{2} \dots i_{m}} | + \sum_{\begin{matrix} i_{2} \dots i_{m} \in N, \\ p \notin {i_{2}, \dots, i_{m}} \end{matrix}} | a_{p i_{2} \dots i_{m}} | \frac{| x_{q} |}{| x_{p} |}, \end{matrix}

which is equivalent to

| λ | \leq (r_{p} (A) - P_{p}^{p} (A)) + P_{p}^{p} (A) \frac{| x_{q} |}{| x_{p} |} .

(4)

From inequalities (4), it is obvious that

λ \in K_{p} (A)

| x_{q} | = 0

, it yields

| λ | - (r_{p} (A) - P_{p}^{p} (A)) \leq 0

on the ground of

| x_{p} | > 0

. If

| λ | \geq r_{q} (A) - | a_{q p \dots p} |

, there is

\begin{matrix} (| λ | - (r_{p} (A) - P_{p}^{p} (A))) (| λ | - (r_{q} (A) - | a_{q p \dots p} |)) \leq 0 \leq P_{p}^{p} (A) | a_{q p \dots p} |, \end{matrix}

which indicates

λ \in {\tilde{Υ}}_{p, q} (A)

. Otherwise,

| λ | < r_{q} (A) - | a_{q p \dots p} |

, we have

λ \in {\hat{Υ}}_{p, q} (A) \subseteq Υ (A)

| x_{q} | > 0

, we can derive the following inequalities in a similar way

\begin{matrix} | λ | \leq (r_{q} (A) - | a_{q p \dots p} |) + | a_{q p \dots p} | \frac{| x_{p} |}{| x_{q} |} . \end{matrix}

(5)

Multiplying (4) and (5) yields

\begin{matrix} (| λ | - (r_{p} (A) - P_{p}^{p} (A))) (| λ | - (r_{q} (A) - | a_{q p \dots p} |)) \leq P_{p}^{p} (A) | a_{q p \dots p} |, \end{matrix}

which indicates that

λ \in {\tilde{Υ}}_{p, q} (A) \subseteq Υ (A)

| λ | \leq r_{p} (A) - P_{p}^{p} (A)

and

| z | \leq r_{q} (A) - | a_{q p \dots p} |

, then

λ \in {\hat{Υ}}_{p, q} (A) \subseteq Υ (A)

. Therefore, the conclusion is proved. □

In order to further compare Theorems 2 and 5, we introduce the following Lemma.

Lemma 1

([33]). Let

a, b, c \geq 0

and

d > 0

. If

\frac{a}{b + c + d} \geq 1

, then

\frac{a - (b + c)}{d} \geq \frac{a - b}{c + d} \geq \frac{a}{b + c + d} .

Theorem 6.

Let

A = (a_{i_{1} i_{2} \dots i_{m}}) \in C^{[m \times n]}

, it follows that

σ (A) \subseteq Υ (A) \subseteq N (A) \subseteq K (A) .

Proof.

The authors in reference [13] have shown that

N (A) \subseteq K (A)

. Therefore, the proof of

Υ (A) \subseteq N (A)

is only needed. For any

z \in Υ (A)

, there are

i, j \in N, j \neq i

such that

z \in {\hat{Υ}}_{i, j} (A)

z \in ({\tilde{Υ}}_{i, j} (A) ⋂ K_{i} (A))

. In this situation, we prove our result from two cases.

In the case of

z \in {\hat{Υ}}_{i, j} (A)

, there are

| z | \leq R_{i} (A) - P_{i}^{i} (A), | z | \leq r_{j} (A) - | a_{j i \dots i} |,

These indicate that

z \in N_{i, j} (A)

In the case of

z \in ({\tilde{Υ}}_{i, j} (A) ⋂ K_{i} (A))

, there are

| z | \leq R_{i} (A)

[| z | - (R_{i} (A) - P_{i}^{i} (A))] [| z | - (R_{j} (A) - | a_{j i \dots i} |)] \leq P_{i}^{i} (A) | a_{j i \dots i} | .

(6)

P_{i}^{i} (A) | a_{j i \dots i} | = 0

, it yields

R_{i} (A) - P_{i}^{i} (A) \leq | z | \leq R_{j} (A) - | a_{j i \dots i} |,

(7)

R_{j} (A) - | a_{j i \dots i} | \leq | z | \leq R_{i} (A) - P_{i}^{i} (A) .

(8)

When inequalities (7) hold, there are

R_{i} (A) - P_{i}^{i} (A) \leq | z | \leq R_{i} (A), | z | \leq R_{j} (A) .

This implies

z \in N_{i, j} (A)

When inequalities (8) hold, it follows that

[| z | - (R_{i} (A) - P_{i}^{i} (A))] | z | \leq 0 \leq P_{i}^{i} (A) R_{j} (A),

which also implies

z \in N_{i, j} (A)

P_{i}^{i} (A) | a_{j i \dots i} | > 0

, then inequalities (6) show

\frac{| z | - (R_{i} (A) - P_{i}^{i} (A))}{P_{i}^{i} (A)} \frac{| z | - (R_{j} (A) - | a_{j i \dots i} |)}{| a_{j i \dots i} |} \leq 1 .

When

\frac{| z | - (R_{j} (A) - | a_{j i \dots i} |)}{| a_{j i \dots i} |} \leq 1

, there is

| z | \leq R_{j} (A)

, that is to say,

z \in N_{i, j} (A)

When

\frac{| z | - (R_{j} (A) - | a_{j i \dots i} |)}{| a_{j i \dots i} |} \geq 1

, according to Lemma 1, one has

\frac{| z | - (R_{i} (A) - P_{i}^{i} (A))}{P_{i}^{i} (A)} \frac{| z |}{R_{j} (A)} \leq \frac{| z | - (R_{i} (A) - P_{i}^{i} (A))}{P_{i}^{i} (A)} \frac{| z | - (R_{j} (A) - | a_{j i \dots i} |)}{| a_{j i \dots i} |} \leq 1 .

This indicates that

z \in N_{i, j} (A)

. In summary, we have completed the proof that

Υ (A) \subseteq N (A) \subseteq K (A)

. □

The following simple numerical example can verify the superiority of our conclusion in the bounds of the tensor spectrum.

Example 1.

Let

A = (a_{i j k}) \in R^{[3, 3]}

with 10 nonzero elements defined as follows;

A (:, :, 1) = (\begin{matrix} 1 & 0 & 1 \\ - 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}), A (:, :, 2) = (\begin{matrix} - 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 0 \end{matrix}), A (:, :, 3) = (\begin{matrix} 0 & 0 & - 1 \\ 0 & 0 & 0 \\ 0 & 1 & 3 \end{matrix}) .

It follows from Theorem 1 that

K = (A) = ⋃_{i \in N} K_{i} (A) = {z \in C : | z | \leq 5} .

It follows from Theorem 2 that

N (A) = ⋃_{i, j \in N, i \neq j} N_{i, j} (A) = {z \in C : | z | \leq 2 + 2 \sqrt{2} ≐ 4.8184} .

However, it follows from Theorem 5 that

σ (A) \subseteq Υ (A) = ⋃_{i, j \in N, j \neq i} ({\hat{Υ}}_{i, j} (A) ⋃ ({\tilde{Υ}}_{i, j} (A) ⋂ K_{i} (A))) = {z \in C : | z | \leq \frac{7 + \sqrt{5}}{2} ≐ 4.618} .

Lemma 2

([30]). For a weakly symmetric non-negative tensor

A

, there is

ρ (A) = λ^{*},

where

λ^{*}

denotes the largest Z-eigenvalue of tensor

A

Theorem 7

([13]). Let

A = (a_{i_{1} i_{2} \dots i_{m}}) \in C^{[m \times n]}

be a weakly symmetric non-negative tensor, it follows that

ρ (A) \leq \bar{ω} = max_{i, j \in N, i \neq j} \frac{1}{2} \{R_{i} (A) - P_{i}^{i} (A) + \sqrt{{(R_{i} (A) - P_{i}^{i} (A))}^{2} + 4 P_{i}^{i} (A) R_{j} (A)}\} .

In a similar manner, based on Theorem 5 and Lemma 2, we derive the following low and upper bounds for a weakly symmetric non-negative tensor, which is tighter than bound in Theorem 7.

Theorem 8.

Let

A = (a_{i_{1} i_{2} \dots i_{m}}) \in C^{[m \times n]}

be a weakly symmetric non-negative tensor; it follows that

min_{i, j \in N, j \neq i} {\underset{̲}{φ}}_{i, j} (A) = \underset{̲}{φ} \leq ρ (A) \leq \bar{φ} = max_{i, j \in N, j \neq i} {\bar{φ}}_{i, j} (A),

(9)

where

{\underset{̲}{φ}}_{i, j} (A) = max \{\frac{1}{2} [R_{i} (A) - P_{i}^{i} (A) + R_{j} (A) - | a_{j i \dots i} | - {\hat{φ}}_{i, j} (A)], 0\},

\begin{matrix} {\bar{φ}}_{i, j} (A) & = max {max \{R_{i} (A) - P_{i}^{i} (A), R_{j} (A) - | a_{j i \dots i} |\}, \\ min \{R_{i} (A), \frac{1}{2} [R_{i} (A) - P_{i}^{i} (A) + R_{j} (A) - | a_{j i \dots i} | + {\hat{φ}}_{i, j} (A)]\}}, \end{matrix}

and

{\hat{φ}}_{i, j} (A) = \sqrt{{(R_{i} (A) - P_{i}^{i} (A) - R_{j} (A) + | a_{j i \dots i} |)}^{2} + 4 P_{i}^{i} (A) | a_{j i \dots i} |} .

Moreover, it follows that

0 \leq \underset{̲}{φ} \leq ρ (A) \leq \bar{φ} \leq \bar{ω} \leq max_{i \in N} R_{i} (A) .

4. The Geometric Measure of Entanglement of Multipartite Pure States

This section is devoted to the geometric measure of entanglement on the ground of tensor Z-eigenvalue localization theory. Theoretical bounds of the geometric measure of entanglement for a weakly symmetric pure state with non-negative amplitudes are proposed. It is worth noting that the geometric measures derived based on different distance measures will be slightly different, such as Hilbert–Schmidt distance, Bures distance and trace distance.

As we know, a multipartite quantum state

| Ψ 〉

has a corresponding tensor representation

A_{| Ψ 〉} : = (a_{i_{1} \dots i_{m}})

under the orthogonal basis. We define the product of the tensor and vector as follows:

A_{Ψ} u^{(1)} \dots u^{(m)} ≜ \sum a_{i_{1} i_{2} \dots i_{m}} u_{i_{1}}^{(1)} \dots u_{i_{m}}^{(m)} .

In other words, the inner product between the entangled state

| Ψ 〉

and separable states

| Φ 〉

can be regarded as

〈 Ψ | Φ 〉 = A_{Ψ} u^{(1)} \dots u^{(m)} .

In this situation, the spectral radius of the tensor

A_{Ψ}

is denoted as

ρ (A_{Ψ}) ≜ max_{| | | ϕ^{(k)} 〉 | | = 1, k = 1, \dots, m .} | A_{Ψ} u^{(1)} \dots u^{(m)} | .

In general, we consider as follows instead of solving (1) directly:

{GME}_{Ψ}^{2} ≜ min {| | | Ψ 〉 - {| Φ 〉 | |}^{2} : | Φ 〉 \in S e p a r (H)},

which yields

| | | Ψ 〉 - {| Φ 〉 | |}^{2} = 2 - 〈 Ψ | Φ 〉 - 〈 Φ | Ψ 〉 .

In other words, the minimization problem in (1) transforms into the maximization problem as follows:

max_{| | | ϕ^{(k)} 〉 | | = 1, k = 1, \dots, m} {〈 Ψ | \otimes_{j = 1}^{m} | ϕ^{(j)} 〉 + \otimes_{j = 1}^{m} 〈 ϕ^{(j)} | Ψ 〉} .

(10)

By introducing Lagrange multipliers

λ

and applying complex differentiation, we show that the maximization problem in (10) is regarded as the largest entanglement eigenvalue

λ

, satisfying

\{\begin{matrix} 〈 Ψ | (\otimes_{j \neq k} | ϕ^{(j)} 〉) = λ 〈 ϕ^{(k)} |, \\ (\otimes_{j \neq k} 〈 ϕ^{(j)} |) Ψ 〉 = λ | ϕ^{(k)} 〉, \\ | | | ϕ^{(k)} 〉 | | = 1, k = 1, \dots, m . \end{matrix}

(11)

According to (11), it follows that

λ = 〈 Ψ | Φ 〉 = 〈 Φ | Ψ 〉,

is a real number in the interval

[- 1, 1]

. For an m-partite pure state

| Ψ 〉 \in H

, we denote the maximal overlap by

G_{Ψ} ≜ max_{| Φ 〉 = \otimes_{k = 1}^{m} | ϕ^{(k)} 〉 \in H} | 〈 Ψ | Φ 〉 | .

(12)

where

| Φ 〉 = \otimes_{k = 1}^{m} | ϕ^{(k)} 〉

is the closest separable state to

| Ψ 〉

Based on Bures distance, the geometric measure of entanglement for a multipartite pure state

| Ψ 〉

is defined as

{GME}_{Ψ} = \sqrt{2 - 2 G_{Ψ}} .

(13)

In other words, the geometric measure of entanglement for a multipartite pure state

| Ψ 〉

, expressed in (13), becomes

{GME}_{Ψ} = \sqrt{2 - 2 ρ (A_{Ψ})} .

(14)

In a similar way, (2) can be regarded as:

E_{G} (| Ψ 〉) = - 2 {log}_{2} ρ (A_{Ψ}) .

(15)

When

H_{1} = \dots = H_{m}

A_{Ψ}

is symmetric if and only if

| Ψ 〉

is permutation symmetric. The geometric measure of symmetric states attracted much attention recently [23,34,35,36,37]. When

| Ψ 〉

is symmetric, (12) becomes

G_{Ψ} = max_{| Φ 〉 = {| ϕ 〉}^{\otimes m} \in H} | 〈 Ψ | Φ 〉 | .

Therefore, the nearest separable state can be chosen as a symmetric one; for details, see [34,35].

In [24], the authors show that the maximal overlap for the geometric measure of entanglement for

| Ψ 〉 \in H

is equivalent to the Z-spectral radius of the corresponding tensor

A_{Ψ}

in a weakly symmetric non-negative case.

We know that a weakly symmetric m-partite pure state

| Ψ 〉 \in H

with non-negative amplitude corresponding always has a corresponding m-order weakly symmetric non-negative tensor

A_{| Ψ 〉} : = (a_{i_{1} \dots i_{m}}) .

Thus, we consider the lower and upper bounds for the geometric measure of entanglement for

| Ψ 〉

by virtue of Bures distance. It follows from Theorem 8 and (14) that the desired lower and upper bounds can be obtained.

Theorem 9.

For a weakly symmetric pure state with non-negative amplitudes

| Ψ 〉 \in H

, there are the following lower and upper bounds for the geometric measure of entanglement for

| Ψ 〉

\sqrt{2 - 2 \bar{φ}} \leq {GME}_{Ψ} \leq \sqrt{2 - 2 \underset{̲}{φ}},

where

\bar{φ}

and

\underset{̲}{φ}

are as in Equation (9).

On the other side, we consider the bounds for the geometric measure of entanglement for

| Ψ 〉

on the grounds of von Neumann entropy; there are the following conclusions on the basis of Theorem 8 and (15).

Theorem 10.

For a weakly symmetric pure state with non-negative amplitudes

| Ψ 〉 \in H

, according to von Neumann entropy, there are the following bounds for the geometric measure of entanglement for

| Ψ 〉

- 2 {log}_{2} \bar{φ} \leq E_{G} (| Ψ 〉) \leq - 2 {log}_{2} \underset{̲}{φ},

where

\bar{φ}

and

\underset{̲}{φ}

are as in Equation (9).

On the one hand, an m-order n-dimensional real tensor

A

has

n^{m}

independent entries, and a symmetric m-order n-dimensional real tensor

A

has

C_{m + n - 1}^{m} = (\binom{m + n - 1}{m})

independent entries [38]. However, a weakly symmetric tensor has at least

n^{m} - n (\binom{m + n - 2}{m}) + n

independent entries [39], and the number of independent elements still increases exponentially.

It is worth noting that our conclusion in Theorems 9 and 10 depends on the characteristics of elements of the tensor

A_{Ψ}

corresponding to the m-partite quantum state

| Ψ 〉 \in H

, while Theorems 3 and 4 only depending on the dimension and order of the tensor. Therefore, we have numerical advantages in most cases, regardless of the size of the tensor.

Example 2.

We consider a simple

t h r e e

-partite state with a three-level system, such as 3-qutrit GHZ state (3-qutrit system), as follows:

| Ψ 〉 = \frac{1}{\sqrt{3}} (| 000 〉 + | 111 〉 + | 222 〉) .

From Theorem 3, the upper bound is

{GME}_{Ψ} \leq \sqrt{2 - 2 / \sqrt{d_{1} \dots d_{m - 1}}} = 1.1547 .

However, by virtue of Theorem 9, the lower and upper bounds are

0.9194 = \sqrt{2 - 2 \bar{φ}} \leq {GME}_{Ψ} \leq \sqrt{2 - 2 \underset{̲}{φ}} = 0.9194 .

In fact, the

{GME}_{Ψ}

of the 3-qutrit GHZ state is 0.9194 with the closest product state

| ϕ_{Ψ} 〉 = | 000 〉

In addition, it follows from Theorem 4 that

E_{G} (| Ψ 〉) \leq (m - 1) {log}_{2} (n) = 3.1699 .

However, based on Theorem 10, the lower and upper bounds are

1.5850 = - 2 {log}_{2} \bar{φ} \leq E_{G} (| Ψ 〉) \leq - 2 {log}_{2} \underset{̲}{φ} = 1.5850 .

which is less than the upper bound in Theorem 4.

Example 3.

We consider the following more general 3-qutrit weakly symmetric state in a three-level with non-negative amplitudes

\begin{matrix} | Ψ 〉 = & 0.6805 | 000 〉 + 0.4990 | 111 〉 + 0.4768 | 222 〉 \\ + 0.0500 (| 001 〉 + | 010 〉 + | 100 〉 + | 200 〉 + | 020 〉 + | 002 〉 \\ + | 022 〉 + | 202 〉 + | 220 〉 + | 011 〉 + | 101 〉 + | 110 〉) \\ + 0.0150 (| 122 〉 + | 212 〉 + | 221 〉) \\ + 0.1000 (| 112 〉 + | 121 〉 + | 211 〉) . \end{matrix}

It is easy to verify

| | A_{Ψ} {| |}_{F} = 1.0000, ρ (A_{Ψ}) = 0.6940 {GME}_{Ψ} = \sqrt{2 - 2 ρ (A_{Ψ})} = 0.7823 .

According to Theorem 3 in [25], there is

{GME}_{Ψ} \leq \sqrt{2 - 2 / 3} = 1.1547 .

However, it follows from Theorem 9 that

0.3728 = \sqrt{2 - 2 \bar{φ}} \leq {GME}_{Ψ} \leq \sqrt{2 - 2 \underset{̲}{φ}} = 0.8713 .

On the other side, the upper bound from Theorem 4 is

E_{G} (| Ψ 〉) \leq (m - 1) {log}_{2} (n) = 3.1699 .

It follows from Theorem 10 that

0.4156 = - 2 {log}_{2} \bar{φ} \leq E_{G} (| Ψ 〉) \leq - 2 {log}_{2} \underset{̲}{φ} = 2.7550 .

In fact, we can verify that

E_{G} (| Ψ 〉) = - {log}_{2} ρ (A_{Ψ}) = 1.4168 .

Therefore, it is evident that Theorem 10 not only obtains a smaller upper bound compared with Theorem 4 but also a lower bound. This lower bound plays a significant role in the geometric measure of entanglement and other quantum information topic.

5. Conclusions

In this paper, we concentrate on the tensor Z-eigenvalue inclusion theorem and its application in the geometric measure of entanglement for multipartite quantum states. Firstly, we propose a new Z-eigenvalue localization theorem and bounds for the Z-spectral radius of non-negative tensors, which prove to be tighter than existing results. As applications, on the basis of the connection between the geometric measure of entanglement and the Z-spectral radius for a weakly symmetric non-negative tensor, we present theoretical bounds of the geometric measure of entanglement for a weakly symmetric multipartite quantum state with non-negative amplitudes by virtue of different distance measures. Numerical examples show that our bounds are tighter than the existing results in quantum physics in some cases. We believe that our results may be beneficial to the development of the intersection between tensor theory and quantum information.

Author Contributions

Conceptualization, L.X.; Data curation, L.X., Z.J. and J.L.; Formal analysis, L.X.; Funding acquisition, Q.Q.; Investigation, L.X.; Methodology, L.X. and J.L.; Project administration, Z.J. and Q.Q.; Software, Z.J.; Supervision, Z.J. and Q.Q.; Validation, L.X.; Visualization, J.L.; Writing—original draft, L.X.; Writing—review & editing, L.X. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No.11971413) and Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515011995).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

Ni, Q.; Qi, L.; Wang, F. An eigenvalue method for the positive definiteness identification problem. IEEE Trans. Automat. Control. 2008, 53, 1096–1107. [Google Scholar] [CrossRef] [Green Version]
Ng, M.; Qi, L.; Zhou, G. Finding the largest eigenvalue of a nonnegative tensor. SIAM J. Matrix Anal. Appl. 2009, 31, 1090–1099. [Google Scholar] [CrossRef] [Green Version]
Wen, L.; Michael, K.N. On the limiting probability distribution of a transition probability tensor. Linear Multilinear Algebra 2014, 62, 362–385. [Google Scholar]
Lathauwer, L.; Moor, B.; Vandewalle, J. On the best rank-1 and rank-(R1,R2, …,RN) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 2000, 21, 1324–1342. [Google Scholar] [CrossRef]
Panagakis, Y.; Kossaifi, J.; Chrysos, G.G.; Oldfield, J.; Nicolaou, M.A.; Anandkumar, A.; Zafeiriou, S. Tensor methods in computer vision and deep learning. Proc. IEEE 2021, 109, 863–890. [Google Scholar] [CrossRef]
Kolda, T.G.; Mayo, J.R. Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 2011, 32, 1095–1124. [Google Scholar] [CrossRef] [Green Version]
Kolda, T.G.; Mayo, J.R. An adaptive shifted power method for computing generalized tensor eigenpairs. SIAM J. Matrix Anal. Appl. 2014, 35, 1563–1581. [Google Scholar] [CrossRef] [Green Version]
Cui, C.-F.; Dai, Y.-H.; Nie, J. All real eigenvalues of symmetric tensors. SIAM J. Matrix Anal. Appl. 2014, 35, 1582–1601. [Google Scholar] [CrossRef]
Chen, L.; Han, L.; Zhou, L. Computing tensor eigenvalues via homotopy methods. SIAM J. Matrix Anal. Appl. 2016, 37, 290–319. [Google Scholar] [CrossRef] [Green Version]
Chen, L.; Han, L.; Yin, H.; Zhou, L. A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor. J. Comput. Appl. Math. 2019, 355, 174–181. [Google Scholar] [CrossRef] [Green Version]
Qi, L.; Chen, H.; Chen, Y. Tensor Eigenvalues and Their Applications; Springer: Singapore, 2018. [Google Scholar]
Wei, Y.; Ding, W. Theory and Computation of Tensors; Elsevier: Amsterdam, The Netherlands; Academic Press: Cambridge, MA, USA, 2016. [Google Scholar]
Wang, G.; Zhou, G.; Caccetta, L. Z-eigenvalue inclusion theorems for tensors. Discrete Contin. Dyn. Syst. Ser. B 2017, 22, 187–198. [Google Scholar] [CrossRef] [Green Version]
Nielsen, M.A.; Chuang, I.L. Quantum Computing and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
Horodecki, R.; Horodecki, P.; Horodecki, M.; Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 2009, 81, 865–942. [Google Scholar] [CrossRef] [Green Version]
Bell, J.S. On the einstein podolsky rosen paradox. Phys. Phys. Fiz. 1964, 1, 195–200. [Google Scholar] [CrossRef] [Green Version]
Filip, R. Overlap and entanglement-witness measurements. Phys. Rev. A 2002, 65, 062320. [Google Scholar] [CrossRef] [Green Version]
Peres, A. Separability criterion for density matrices. Phys. Rev. Lett. 1996, 77, 1413–1415. [Google Scholar] [CrossRef] [Green Version]
Horodecki, M.; Horodecki, P.; Horodecki, R. Separability of mixed states: Necessary and sufficient conditions. Phys. Lett. A 1996, 223, 1–8. [Google Scholar] [CrossRef] [Green Version]
Shimony, A. Degree of entanglement. Ann. N. Y. Acad. Sci. 1995, 755, 675. [Google Scholar] [CrossRef]
Wei, T.-C.; Goldbart, P.M. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 2003, 68, 042307. [Google Scholar] [CrossRef] [Green Version]
Tamaryan, S.; Sudbery, A.; Tamaryan, L. Duality and the geometric measure of entanglement of general multiqubit w states. Phys. Rev. A 2010, 81, 052319. [Google Scholar] [CrossRef] [Green Version]
Hu, S.; Qi, L.; Zhang, G. Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys. Rev. A 2016, 93, 012304. [Google Scholar] [CrossRef] [Green Version]
Xiong, L.; Liu, J.; Qin, Q. The geometric measure of entanglement of multipartite states and the Z-eigenvalue of tensors. Quntum. Inf. Process. 2022, 21, 102. [Google Scholar] [CrossRef]
Qi, L.; Zhang, G.; Ni, G. How entangled can a multi-party system possibly be? Phys. Lett. A 2018, 382, 1465–1471. [Google Scholar] [CrossRef] [Green Version]
Gross, D.; Flammia, S.T.; Eisert, J. Most quantum states are too entangled to be useful as computational resources. Phys. Rev. Lett. 2009, 102, 190501. [Google Scholar] [CrossRef] [PubMed] [Green Version]
Derksen, H.; Makam, V. Highly entangled tensors. Linear Multilinear A 2022, 70, 380–393. [Google Scholar] [CrossRef] [Green Version]
Teng, P. Accurate calculation of the geometric measure of entanglement for multipartite quantum states. Quntum. Inf. Process. 2017, 16, 181. [Google Scholar] [CrossRef] [Green Version]
Friedland, S.; Kemp, T. Most boson quantum states are almost maximally entangled. Proc. Amer. Math. Soc. 2018, 146, 5035–5049. [Google Scholar] [CrossRef] [Green Version]
Chang, K.; Pearson, K.; Zhang, T. Some variational principles for Z-eigenvalues of nonnegative tensors. Linear Algebra Appl. 2013, 438, 4166–4182. [Google Scholar] [CrossRef]
Qi, L. Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 2005, 40, 1302–1324. [Google Scholar] [CrossRef] [Green Version]
Lim, L. Singular values and eigenvalues of tensors: A variational approach. In Proceedings of the 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Puerto Vallarta, Mexico, 13–15 December 2005; pp. 129–132. [Google Scholar]
Li, C.; Li, Y. An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors. Linear Multilinear A 2016, 64, 587–601. [Google Scholar] [CrossRef]
Hbener, R.; Kleinmann, M.; Wei, T.-C.; González-Guillén, C.; Gühne, O. Geometric measure of entanglement for symmetric states. Phys. Rev. A 2009, 80, 032324. [Google Scholar] [CrossRef]
Wei, T.-C.; Severini, S. Matrix permanent and quantum entanglement of permutation invariant states. J. Math. Phys. 2010, 51, 092203. [Google Scholar] [CrossRef] [Green Version]
Ors, R.; Dusuel, S.; Vidal, J. Equivalence of critical scaling laws for many-body entanglement in the lipkin-meshkov-glick model. Phys. Rev. Lett. 2008, 101, 025701. [Google Scholar] [CrossRef] [PubMed] [Green Version]
Hayashi, M.; Markham, D.; Murao, M.; Owari, M.; Virmani, S. The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes. J. Math. Phys. 2009, 50, 122104. [Google Scholar] [CrossRef] [Green Version]
Comon, P.; Golub, G.; Lim, L.-H.; Mourrain, B. Symmetric tensors and symmetric tensor rank. SIAM. J. Matrix. Anal. A 2008, 30, 1254–1279. [Google Scholar] [CrossRef] [Green Version]
Chang, K.; Pearson, K.; Zhang, T. On eigenvalue problems of real symmetric tensors. J. Math. Anal. Appl. 2009, 350, 416–422. [Google Scholar] [CrossRef] [Green Version]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Xiong, L.; Jiang, Z.; Liu, J.; Qin, Q. New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States. Mathematics 2022, 10, 2624. https://doi.org/10.3390/math10152624

AMA Style

Xiong L, Jiang Z, Liu J, Qin Q. New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States. Mathematics. 2022; 10(15):2624. https://doi.org/10.3390/math10152624

Chicago/Turabian Style

Xiong, Liang, Zhanfeng Jiang, Jianzhou Liu, and Qi Qin. 2022. "New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States" Mathematics 10, no. 15: 2624. https://doi.org/10.3390/math10152624

APA Style

Xiong, L., Jiang, Z., Liu, J., & Qin, Q. (2022). New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States. Mathematics, 10(15), 2624. https://doi.org/10.3390/math10152624

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States

Abstract

1. Introduction

2. Preliminaries

2.1. Preliminaries for Tensors

2.2. Tensor Representation of Quantum States

3. New $Z$ -Eigenvalue Localization Set and the Bounds for $Z$ -Spectral Radius

4. The Geometric Measure of Entanglement of Multipartite Pure States

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Article Menu

New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States

Abstract

1. Introduction

2. Preliminaries

2.1. Preliminaries for Tensors

2.2. Tensor Representation of Quantum States

3. New Z -Eigenvalue Localization Set and the Bounds for Z -Spectral Radius

4. The Geometric Measure of Entanglement of Multipartite Pure States

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

3. New $Z$ -Eigenvalue Localization Set and the Bounds for $Z$ -Spectral Radius