Basis-Dependent Quantum Nonlocality

Quantum nonlocality represents correlation properties between subsystems of a composite quantum system, usually including the four types: Bell nonlocality, steerability, entanglement, and quantum correlation (quantum discord). Given a basis $e_{AB}=\{|e_{ij}\>\}_{i\in[d_A],j\in[d_B]}$ for the Hilbert space $\H_A\otimes \H_B$ of a bipartite system $AB$, a density operator $\rho^{AB}$ (quantum state) of $AB$ can be represented as a $d_Ad_B\times d_Ad_B$ matrix $\hat{\rho}_{e_{AB}}=[\]$, called the density matrix of a density operator $\rho^{AB}$. A natural question is what is the relationship between the quantum nonlocality of a density operator $\rho^{AB}$ and its corresponding density matrix $\hat{\rho}_{e_{AB}}$? In this work, we discuss the relationships between quantum locality and basis, and prove that one type of quantum locality of density operators and that of their density matrices under a basis are the same if and only if the chosen basis is the tensor product of the bases of subsystems. Consequently, different bases define different quantum nonlocality density operators.

Keywords:

Subject: Physical Sciences - Quantum Science and Technology

1. Introduction

Quantum nonlocality, also called quantum correlation, represents correlation properties between subsystems of a composite quantum system, including Bell nonlocality, steerability, entanglement, and quantum correlation.

Bell nonlocality of bipartite states is demonstrated by some local quantum measurements whose statistics of the measurement outcomes cannot be explained by a local hidden variable (LHV) model [1,2]. Such a nonclassical feature of quantum mechanics can be used in device-independent quantum information processing [2]. For more discussions on Bell nonlocality, please refer to Clauser and Shimony [3], Home and Selleri [4], Khalfin and Tsirelson [5], Tsirelson [6], Zeilinger [7], Werner and Wolf [8], Genovese [9], and Buhrman et al. [10], and [11,12,13,14].

Einstein-Podolsky-Rosen (EPR) steering, as a form of quantum correlation, was first observed by Schr

\ddot{o}

dinger [15] in the context of the well-known EPR paradox [16,17,18,19]. EPR steering arises in scenarios wherein some local quantum measurements on one part of a bipartite system are used to steer the other part. This scenario demonstrates EPR steering if the obtained ensembles cannot be explained by a local hidden state (LHS) model [20]. Following a close analogy of criteria for other forms of quantum nonlocality, Cavalcanti et al. [21] developed a general theory of experimental EPR steering criteria and derived several criteria applicable to discrete and continuous-variable observables. Saunders et al. [22] contributed experimental EPR steering by using Bell local states. Bennet et al. [23] derived arbitrarily loss-tolerant tests, thereby enabling us to perform a detection loophole free demonstration of EPR steering with parties separated by a coiled 1-km-long optical fiber. Händchen et al. [24] presented an experimental realization of two entangled Gaussian modes of light that shows a steering effect in one direction but not in the other. The generated one-way steering provides new insight into quantum physics and may open a new field of applications in quantum information. EPR steering, as an intermediate form of quantum correlation between entanglement and Bell nonlocality, allows for entanglement certification when the measurements performed by one of the parties are not characterized (or are untrusted); it has many applications, such as quantum key distribution [25,26,27], quantum benchmark for qubit teleportation[28], subchannel discrimination in a quantum evolution [29], and so on. Therefore, it has been widely researched, see [30,31,32,33,34,35,36,37,38,39].

Just like quantum steering, quantum entanglement was also recognized by Einstein, Podolsky, Rosen [16], and Schr

\ddot{o}

dinger [15] in 1935. It is a property of a composite quantum system involving nonclassical correlations between subsystems and then having potential for many quantum processes, including canonical ones: quantum cryptography, quantum teleportation, and dense coding.

A bipartite entanglement state reveals correlations between the two subsystems. However, some separable states may reveal some correlations. Such states are just the states that have nonzero quantum discord. Quantum discord induced by H. Ollivier and W.H. Zurek in [40] is a measure of the quantumness of correlations but not a quantum property of states. Considered measurement-induced disturbance, Luo [41] classified correlations between subsystems into classical correlations and quantum correlations by introducing classical correlated (CC) states and quantum correlated states. It was found that a CC state is a separable state [40,41,42,43] and so any entangled state must be a QC state. Thus, quantum correlations have been applied in some quantum computing tasks without entanglement [44], the study of quantum key distribution [45], three-spin XXZ chain with three-spin interaction [46] and so on.

It is well-known that the four types of quantum nonlocality have the following relations:

Bell nonlocality \Rightarrow steerability \Rightarrow entanglement \Rightarrow quantum correlation,

equivalently,

Bell locality \Leftarrow unsteerability \Leftarrow separability \Leftarrow classical correlation .

From the mathematical definitions of quantum locality (Bell locality, separability and classical correlation), we see that these properties depend only on the algebraic structures of the density operators

ρ^{A B}

(quantum states) and should be independent of the choice of basis for the Hilbert space

H_{A} \otimes H_{B}

of a system

A B

. However, in applications and experiments, density operators are usually written as matrices under a chosen basis

e_{A B}

for

H_{A} \otimes H_{B}

. As we known, different bases lead to different matrix representations of operators. Thus, different choice of bases may induce different quantum locality of density matrices, which we called basis-dependent quantum nonlocality.

In the sequence sections, we discuss the relationships between these quantum locality and basis, and find that quantum locality of density operators and their density matrices under a basis are the same only when the basis is a tensor product of the bases of subsystems.

2. Basis Dependent Separability

According to quantum mechanics, a quantum system S is described by a

d_{S}

-dimensional complex Hilbert space

H_{S}

(called the state space of S) with a right-linear inner product

〈 \cdot | \cdot 〉

and the states of the system are denoted by positive operators of trace 1 on

H_{S}

. The set of all states of S is denoted by

D (H_{S})

. Thus,

D (H_{S}) = {ρ \in B (H_{S}) : ρ \geq 0, tr ρ = 1},

where

B (H_{S})

is the

C^{*}

-algebra of all bounded linear operators on

H_{S}

. The elements of

D (H_{S})

are called the mixed states of S. A unit vector

| ψ 〉

H_{S}

is said to be a pure state of S and the set of all pure states of S is denoted by

P S (H_{S}) .

We also use

[d]

to denote the set

{1, 2, \dots, d} .

By the postulates of quantum mechanics, the state space of the composite system

A B

of A and B is given by the tensor product space

H_{A} \otimes H_{B}

of the state spaces

H_{A}

and

H_{B}

of A and B, respectively.

2.1. Concepts and Notations

Recall that a state

ρ \in D (H_{A} \otimes H_{B})

is said to be separable if it can be written as the form of

ρ = \sum_{i = 1}^{k} c_{i} ρ_{i}^{A} \otimes ρ_{i}^{B}

(1)

for some states

ρ_{i}^{A} \in D (H_{A}), ρ_{i}^{B} \in D (H_{B})

and

c_{i} \geq 0 (i \in [k])

with

\sum_{i = 1}^{k} c_{i} = 1

. Otherwise, it is said to be entangled. Moreover, a pure state

| ψ 〉 \in P S (H_{A} \otimes H_{B})

is said to be separable if it can be written as the form

| ψ 〉 = | ψ_{A} 〉 \otimes | ψ_{B} 〉

for some

| ψ_{A} 〉 \in P S (H_{A})

and

| ψ_{B} 〉 \in P S (H_{B})

. Otherwise, it is called to be entangled.

2.2. Separability Depending on Basis

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

be an orthonormal basis (ONB) for

H_{A} \otimes H_{B}

and

U : H_{A} \otimes H_{B} \to C^{d_{A} d_{B}} = C^{d_{A}} \otimes C^{d_{B}}

be the coordinate mapping:

U | ψ^{A B} 〉 = {(c_{11}, c_{12}, \dots, c_{1 d_{B}}, c_{21}, c_{22}, \dots, c_{2 d_{B}}, \dots, c_{d_{A} 1}, c_{d_{A} 2}, \dots, c_{d_{A} d_{B}})}^{T}

(2)

for all

| ψ^{A B} 〉 = \sum_{i, j} c_{i j} | e_{i j} 〉

H_{A} \otimes H_{B}

. We call the vector

U | ψ^{A B} 〉

the coordinate state of a state

| ψ^{A B} 〉

H_{A} \otimes H_{B}

We want to discuss the relationship between the separability of the coordinate state

U | ψ^{A B} 〉

and that of the original (abstract) state

| ψ^{A B} 〉

. For convenience, we write

U | ψ^{A B} 〉 = | ψ^{A B} 〉_{e_{A B}}

. Let

{| i_{A} 〉}_{i = 1}^{d_{A}}

and

{| j_{B} 〉}_{j = 1}^{d_{B}}

be the canonical

{0, 1}

-bases for

C^{d_{A}}

and

C^{d_{B}}

, respectively. Then we get the canonical

{0, 1}

-basis (with

m = d_{A}, n = d_{B}

)

{| i_{A} j_{B} 〉}_{i, j} = {| 1_{A} 1_{B} 〉, | 1_{A} 2_{B} 〉, \dots, | 1_{A} n_{B} 〉, \dots, | m_{A} 1_{B} 〉, | m_{A} 2_{B} 〉, \dots, | m_{A} n_{B} 〉}

for

C^{d_{A}} \otimes C^{d_{B}}

. Thus, the definition (2) of U becomes

U | ψ^{A B} 〉 = \sum_{i, j} c_{i j} | i_{A} j_{B} 〉

(3)

for all

| ψ^{A B} 〉 = \sum_{i, j} c_{i j} | e_{i j} 〉

H_{A} \otimes H_{B}

. For example,

U | e_{i j} 〉 = | i_{A} 〉 \otimes | j_{B} 〉 \equiv | i_{A} 〉 | j_{B} 〉 \equiv | i_{A} j_{B} 〉, \forall i \in [d_{A}], j \in [d_{B}],

(4)

leading to

〈 i_{A} j_{B} | U ρ^{A B} U^{†} | k_{A} ℓ_{B} 〉 = 〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉 = 〈 i_{A} j_{B} | ρ_{e_{A B}}^{A B} | k_{A} ℓ_{B} 〉

(5)

for all

i, k \in [d_{A}], j, ℓ \in [d_{B}]

, where

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is the

d_{A} d_{B} \times d_{A} d_{B}

matrix with

(i j, k ℓ)

-entry

〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉

, called the density matrix of

ρ^{A B}

under basis

e_{A B}

and written as

ρ_{e_{A B}}^{A B} = U ρ^{A B} U^{†}

due to the relation (5).

Equation (4) shows that the coordinate mapping U transforms the basis states

| e_{i j} 〉

as separable states

| i_{A} 〉 \otimes | j_{B} 〉

. Generally,

U | ψ^{A B} 〉

is not necessarily separable even if

| ψ^{A B} 〉

is separable. For example, we let

d_{A} = d_{B} = 2

and

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{2}

and

e_{B} = {| e_{i}^{B} 〉}_{i = 1}^{2}

for

H_{A}

and

H_{B}

, respectively, and define an ONB

e_{A B} = {| e_{11} 〉, | e_{12} 〉, | e_{21} 〉, | e_{22} 〉}

for

H_{A} \otimes H_{B}

\{\begin{matrix} | e_{11} 〉 = \frac{\sqrt{2}}{2} (| e_{1}^{A} e_{1}^{B} 〉 + | e_{2}^{A} e_{2}^{B} 〉); \\ | e_{12} 〉 = \frac{\sqrt{2}}{2} (| e_{1}^{A} e_{2}^{B} 〉 + | e_{2}^{A} e_{1}^{B} 〉); \\ | e_{21} 〉 = \frac{\sqrt{2}}{2} (| e_{1}^{A} e_{2}^{B} 〉 - | e_{2}^{A} e_{1}^{B} 〉); \\ | e_{22} 〉 = \frac{\sqrt{2}}{2} (| e_{1}^{A} e_{1}^{B} 〉 - | e_{2}^{A} e_{2}^{B} 〉) . \end{matrix}

Clearly,

| ψ^{A B} 〉 : = | e_{1}^{A} e_{1}^{B} 〉 = \frac{\sqrt{2}}{2} (| e_{11} 〉 + | e_{22} 〉)

is a separable state of

H_{A} \otimes H_{B}

, but

U | ψ^{A B} 〉 = | ψ^{A B} 〉_{e_{A B}} = {(\frac{\sqrt{2}}{2}, 0, 0, \frac{\sqrt{2}}{2})}^{T} \in C^{2} \otimes C^{2},

which can not be written as a tensor product of two qubits and then not separable. If, however, we choose a “separable basis":

e_{A} \otimes e_{B} = {| e_{i}^{A} 〉 \otimes | e_{j}^{B} {〉}}_{i, j = 1}^{2} = {| e_{1}^{A} 〉 \otimes | e_{1}^{B} 〉, | e_{1}^{A} 〉 \otimes | e_{2}^{B} 〉, | e_{2}^{A} 〉 \otimes | e_{1}^{B} 〉, | e_{2}^{A} 〉 \otimes | e_{2}^{B} 〉},

then the coordinate state of

| ψ^{A B} 〉 = | e_{1}^{A} e_{1}^{B} 〉 = \frac{\sqrt{2}}{2} (| e_{11} 〉 + | e_{22} 〉)

reads

| ψ^{A B} 〉_{e_{A} \otimes e_{B}} = {(1, 0, 0, 0)}^{T} \in C^{2} \otimes C^{2},

which is separable.

This observation leads to the following conclusion.

Theorem 1.

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is a separable state of

H_{A} \otimes H_{B}

if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is a separable state of

C^{d_{A}} \otimes C^{d_{B}}

(b) A density operator

ρ^{A B}

is a separable state of

H_{A} \otimes H_{B}

if and only if its density matrix

ρ_{e_{A B}}^{A B}

is a separable state of

C^{d_{A}} \otimes C^{d_{B}}

Proof.

(a)

For every separable state

| ψ^{A B} 〉 = | ψ^{A} 〉 | ψ^{B} 〉

, it holds that

c_{i j} : = 〈 e_{i j} | ψ^{A B} 〉 = 〈 e_{i}^{A} | ψ^{A} 〉 \cdot 〈 e_{j}^{B} | ψ^{B} 〉

for all

i, j

. Writing

a_{i} = 〈 e_{i}^{A} | ψ^{A} 〉

and

b_{j} = 〈 e_{j}^{B} | ψ^{B} 〉

implies that

| ψ^{A B} 〉_{e_{A B}} = {(a_{1}, a_{2}, \dots, a_{d_{A}})}^{T} \otimes {(b_{1}, b_{2}, \dots, b_{d_{B}})}^{T},

(6)

and so

| ψ^{A B} 〉_{e_{A B}}

is a separable state in

C^{d_{A}} \otimes C^{d_{B}}

Conversely, let

| ψ^{A B} 〉_{e_{A B}}

be a separable state. Then Equation (6) holds for some states

| a 〉 = {(a_{1}, a_{2}, \dots, a_{d_{A}})}^{T}

and

| b 〉 = {(b_{1}, b_{2}, \dots, b_{d_{B}})}^{T}

C^{d_{A}}

and

C^{d_{B}}

, respectively. Put

| ψ^{A} 〉 = \sum_{i = 1}^{d_{A}} a_{i} | e_{i}^{A} 〉

and

| ψ^{B} 〉 = \sum_{j = 1}^{d_{B}} b_{j} | e_{j}^{B} 〉

, then

U (| ψ^{A} 〉 | ψ^{B} 〉) = | a 〉 | b 〉 = U | ψ^{A B} 〉

and so

| ψ^{A B} 〉 = | ψ^{A} 〉 | ψ^{B} 〉

since U is injective. Hence,

| ψ^{A B} 〉

is separable. This shows that

U | ψ^{A B} 〉 = | ψ^{A B} 〉_{e_{A B}}

is separable if and only if

ψ^{A B} 〉

is separable.

(b)

First, we let

ρ^{A B} = \sum_{t} c_{t} ρ_{t}^{A} \otimes ρ_{t}^{B}

be a separable state of

A B

. Then

[〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉] = \sum_{t} c_{t} [〈 e_{i}^{A} | ρ_{t}^{A} | e_{k}^{A} 〉 \cdot 〈 e_{j}^{B} | ρ_{t}^{B} | e_{ℓ}^{B} 〉] = \sum_{t} c_{t} [〈 e_{i}^{A} | ρ_{t}^{A} | e_{k}^{A} 〉] \otimes [〈 e_{j}^{B} | ρ_{t}^{B} | e_{ℓ}^{B} 〉],

which is the tensor product of two density matrices. Thus,

ρ_{e_{A B}}^{A B}

is a separable density matrix. □

Conversely, we let

ρ_{e_{A B}}^{A B}

be a separable density matrix. Then

ρ_{e_{A B}}^{A B} = \sum_{n = 1}^{m} t_{n} [a_{i k}^{(n)}] \otimes [b_{j ℓ}^{(n)}]

for some density matrices

[a_{i k}^{(n)}]

and

[b_{j ℓ}^{(n)}]

where

t_{n} \geq 0

and

\sum_{n} t_{n} = 1

. Define density operators

ρ_{n}^{A}

H_{A}

and

ρ_{n}^{B}

H_{B}

〈 e_{i}^{A} | ρ_{n}^{A} | e_{k}^{A} 〉 = a_{i k}^{(n)}, 〈 e_{j}^{B} | ρ_{n}^{B} | e_{ℓ}^{B} 〉 = b_{j ℓ}^{(n)},

for all

i, j, k, ℓ

, then we compute that

\begin{matrix} [〈 e_{i j} | \sum_{n = 1}^{m} t_{n} (ρ_{n}^{A} \otimes ρ_{n}^{B}) | e_{k ℓ} 〉] & = & \sum_{n = 1}^{m} t_{n} [〈 e_{i}^{A} | ρ_{n}^{A} | e_{k}^{A} 〉 \cdot 〈 e_{j}^{B} | ρ_{n}^{B} | e_{ℓ}^{B} 〉] \\ = & \sum_{n = 1}^{m} t_{n} [〈 e_{i}^{A} | ρ_{n}^{A} | e_{k}^{A} 〉] \otimes [〈 e_{j}^{B} | ρ_{n}^{B} | e_{ℓ}^{B} 〉] \\ = & \sum_{n = 1}^{m} t_{n} {[a_{i k}^{(n)}]}_{i, k} \otimes [b_{j ℓ}^{(n)}] \\ = & ρ_{e_{A B}}^{A B} \\ = & [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉], \end{matrix}

and so

ρ^{A B} = \sum_{n = 1}^{m} t_{n} (ρ_{n}^{A} \otimes ρ_{n}^{B}),

which is separable. The proof is completed.

Similarly, one can check the following.

Theorem 2.

Let

d_{A} = d_{B} = d

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is separable if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is separable.

(b) A density operator

ρ^{A B}

is separable if and only if its density matrix

ρ_{e_{A B}}^{A B}

is separable.

The following theorem shows that the condition

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

(resp.

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

) is necessary for the conditions

(a)

and

(b)

in Theorem 1 (resp. Theorem 2) to be satisfied.

Theorem 3.

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

be an orthonormal basis for

H_{A} \otimes H_{B}

and and U be fined by Equation (2). Then the following statements are equivalent (TFSAE).

(a) The coordinate mapping U preserves separability of pure states in both directions, i.e.,

| ψ^{A B} 〉

is separable if and only if

U | ψ^{A B} 〉

is separable.

(b) The coordinate mapping U preserves separability of density operators in both directions, i.e., a density operator

ρ^{A B}

is separable if and only if its density matrix

ρ_{e_{A B}}^{A B}

is separable.

W_{X} : H_{X} \to C^{d_{X}} (X = A, B)

such that when

d_{A} \neq d_{B}

U = W_{A} \otimes W_{B}

; when

d_{A} = d_{B} = d

, either

U = W_{A} \otimes W_{B}

, or

U = S (W_{A} \otimes W_{B})

, where

S : C^{d} \otimes C^{d}

is the swap operator:

S (| x 〉 \otimes | y 〉) = | y 〉 \otimes | x 〉

(d) There exist orthonormal bases

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively, such that when

d_{A} \neq d_{B}

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

; When

d_{A} = d_{B} = d

, either

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

, or

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j

Proof.

(a) \Leftrightarrow (b)

: Suppose that

(a)

holds. Let

ρ^{A B} = \sum_{n = 1}^{m} c_{n} ρ_{n}^{A} \otimes ρ_{n}^{B}

be a convex combination of product states. It suffices to prove that

U (ρ_{n}^{A} \otimes ρ_{n}^{B}) U^{†}

is separable for each n. Fixed n and write

ρ_{n}^{A} = \sum_{s} λ_{s} | ψ_{s}^{A} 〉 〈 ψ_{s}^{A} |, ρ_{n}^{B} = \sum_{t} μ_{t} | ψ_{t}^{B} 〉 〈 ψ_{t}^{B} |,

then

U (ρ_{n}^{A} \otimes ρ_{n}^{B}) U^{†} = \sum_{s, t} λ_{s} μ_{t} U | ψ_{s}^{A} ψ_{t}^{B} 〉 〈 ψ_{s}^{A} ψ_{t}^{B} | U^{†},

which is separable using

(a)

. Hence,

U ρ^{A B} U^{†} = \sum_{n = 1}^{m} c_{n} U (ρ_{n}^{A} \otimes ρ_{n}^{B}) U^{†}

is separable. It follows from Theorem 1

(b)

that the density matrix

[〈 i_{A} j_{B} | U ρ^{A B} U^{†} | k_{A} ℓ_{B} 〉]

is separable, i.e.,

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is separable. Conversely, let

[〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

be separable. Then the density matrix

[〈 i_{A} j_{B} | U ρ^{A B} U^{†} | k_{A} ℓ_{B} 〉]

is separable and therefore the density operator

U ρ^{A B} U^{†}

is separable (Theorem 1

(b)

). Thus, we can write

U ρ^{A B} U^{†} = \sum_{n} c_{n} | α_{n}^{A B} 〉 〈 α_{n}^{A B} |

for some separable pure states

| α_{n}^{A B} 〉

C^{d_{A}} \otimes C^{d_{B}}

, where

c_{n} \geq 0

with

\sum_{n} c_{n} = 1

. Since

ρ^{A B} = \sum_{n} c_{n} U^{†} | α_{n}^{A B} 〉 〈 α_{n}^{A B} | U

and

U^{†} | α_{n}^{A B} 〉

is separable (using

(a)

) for all n, we see that

ρ^{A B}

is separable. Now,

(b)

follows. Clearly,

(b)

implies

(a)

. □

(a) \Rightarrow (c)

: Take ONBs

e_{A} = {| x_{i} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| y_{j} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively, and define unitary operators

U_{A} : H_{A} \to C^{d_{A}}

and

U_{B} : H_{B} \to C^{d_{B}}

U_{A} (\sum_{i = 1}^{d_{A}} c_{i} | x_{i} 〉) = {(c_{1}, c_{2}, \dots, c_{d_{A}})}^{T},

U_{B} (\sum_{j = 1}^{d_{B}} d_{j} | y_{j} 〉) = {(d_{1}, d_{2}, \dots, d_{d_{A}})}^{T},

and put

U_{A B} = U_{A} \otimes U_{B}, V = U U_{A B}^{- 1}

. Then we obtain a commutative diagram Figure 1:

Let

(a)

be valid. Since both U and

U_{A B}

preserve separability of pure states in both directions, so does V, i.e.,

V | ψ 〉

is separable if and only if

| ψ 〉

is separable. It follows from [47] that there are unitary operators

V_{A}

C^{d_{A}}

and

V_{B}

C^{d_{B}}

such that when

d_{A} \neq d_{B}

V = V_{A} \otimes V_{B}

; when

d_{A} = d_{B} = d

, either

V = V_{A} \otimes V_{B}

, or

V = S (V_{A} \otimes V_{B})

. Thus, condition

(c)

follows by letting

W_{X} = V_{X} U_{X}

(

X = A, B

). See Figure 2 for the last case.

(c) \Rightarrow (d)

: Suppose that condition

(c)

is satisfied. From the definition of U, we see that

U | e_{i j} 〉 = | i_{A} 〉 \otimes | j_{B} 〉

for all

i \in [d_{A}]

and

j \in [d_{B}]

where

{| i_{A} 〉}_{i = 1}^{d_{A}}

and

{| j_{B} 〉}_{j = 1}^{d_{B}}

are the canonical

{0, 1}

-bases for

C^{d_{A}}

and

C^{d_{B}}

, respectively. Put

| e_{i}^{A} 〉 = W_{A}^{- 1} | i_{A} 〉

and

| e_{j}^{B} 〉 = W_{B}^{- 1} | j_{B} 〉

for all

i, j

When

d_{A} \neq d_{B}

, we have

| e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 = W_{A}^{- 1} | i_{A} 〉 \otimes W_{B}^{- 1} | j_{B} 〉 = U^{- 1} (| i_{A} 〉 \otimes | j_{B} 〉) = | e_{i j} 〉

for all

i, j

;

When

d_{A} = d_{B} = d

, if

U = W_{A} \otimes W_{B}

, then we have

| e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 = W_{A}^{- 1} | i_{A} 〉 \otimes W_{B}^{- 1} | j_{B} 〉 = U^{- 1} (| i_{A} 〉 \otimes | j_{B} 〉) = | e_{i j} 〉

for all

i, j

; if

U = S (W_{A} \otimes W_{B})

, then we have

\begin{matrix} | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉 & = & W_{A}^{- 1} | j_{A} 〉 \otimes W_{B}^{- 1} | i_{B} 〉 = U^{- 1} S (| j_{A} 〉 \otimes | i_{B} 〉) = U^{- 1} (| i_{B} 〉 \otimes | j_{A} 〉) \\ = & U^{- 1} (| i_{A} 〉 \otimes | j_{B} 〉) = | e_{i j} 〉 \end{matrix}

for all

i, j

, here the fact that

| i_{B} 〉 = | i_{A} 〉

and

| j_{A} 〉 = | j_{B} 〉

were used. Now, condition

(d)

follows.

(d) \Rightarrow (a)

: Suppose that condition

(d)

is satisfied. Let

| ψ^{A} 〉

and

| ψ^{B} 〉

be any states of

H_{A}

and

H_{B}

, respectively. Then

| ψ^{A} 〉 = \sum_{i = 1}^{d_{A}} c_{i} | e_{i}^{A} 〉, | ψ^{B} 〉 = \sum_{j = 1}^{d_{B}} d_{j} | e_{j}^{B} 〉 .

When

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

, we have

U (| ψ^{A} 〉 \otimes | ψ^{B} 〉) = \sum_{i, j} c_{i} d_{j} U (| e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉) = \sum_{i, j} c_{i} d_{j} U | e_{i j} 〉 = \sum_{i, j} c_{i} d_{j} | i_{A} 〉 \otimes | j_{B} 〉 = | u 〉 \otimes | v 〉,

where

| u 〉 = \sum_{i} c_{i} | i_{A} 〉

and

| v 〉 = \sum_{j} d_{j} | j_{B} 〉 .

When

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j

, we have

d_{A} = d_{B} = d

and write

| ψ^{A} 〉 = \sum_{j = 1}^{d} d_{j} | e_{j}^{A} 〉, | ψ^{B} 〉 = \sum_{i = 1}^{d} c_{i} | e_{i}^{B} 〉 .

Thus,

U (| ψ^{A} 〉 \otimes | ψ^{B} 〉) = \sum_{i, j} c_{i} d_{j} U (| e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉) = \sum_{i, j} c_{i} d_{j} U | e_{j i} 〉 = \sum_{i, j} c_{i} d_{j} | j_{A} 〉 \otimes | i_{B} 〉 = | u 〉 \otimes | v 〉,

where

| u 〉 = \sum_{i} c_{i} | i_{A} 〉

and

| v 〉 = \sum_{j} d_{j} | j_{B} 〉 .

This shows that U maps any separable pure state as a separable pure state.

Conversely, we assume that

U | ψ^{A B} 〉 = | u 〉 \otimes | v 〉

for some states

| u 〉

and

| v 〉

C^{d_{A}}

and

C^{d_{B}}

, respectively. Let us show that

| ψ^{A B} 〉

is separable. To do this, we let

| ψ^{A B} 〉 = \sum_{i, j} t_{i j} | e_{i j} 〉, | u 〉 = \sum_{i} c_{i} | i_{A} 〉, | v 〉 = \sum_{j} d_{j} | j_{B} 〉 .

Then

\sum_{i, j} c_{i} d_{j} | i_{A} 〉 | j_{B} 〉 = | u 〉 \otimes | v 〉 = U | ψ^{A B} 〉 = \sum_{i, j} t_{i j} | i_{A} 〉 | j_{B} 〉

and so

t_{i j} = c_{i} d_{j}

for all

i, j .

Hence,

| ψ^{A B} 〉 = \sum_{i, j} c_{i} d_{j} | e_{i j} 〉

. Since either

| e_{i j} 〉 = | d_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

, or

d_{A} = d_{B} = d

and

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j

, we have either

| ψ^{A B} 〉 = (\sum_{i = 1}^{d} c_{i} | e_{i}^{A} 〉) \otimes (\sum_{j = 1}^{d} d_{j} | e_{j}^{B} 〉),

| ψ^{A B} 〉 = (\sum_{j = 1}^{d} d_{j} | e_{j}^{A} 〉) \otimes (\sum_{i = 1}^{d} c_{i} | e_{i}^{B} 〉) .

This shows that

| ψ^{A B} 〉

is separable. Thus, U preserves separability of pure states in both directions and so condition

(a)

is satisfied. The proof is completed.

3. Basis Dependent Bell Locality

In this section, we will discuss relationship between Bell locality and basis. To describe and discuss Bell locality, the following mathematical definitions were given by [14] abstracted from the literatures [2,20].

3.1. Concepts and Notations

To describe Bell locality of a bipartite state

ρ^{A B}

, we use x and y to denote the labels of POVMs of Alice and Bob and use a and b to denote their measurement outcomes, respectively. Thus, their POVM choices are denoted by

M^{x} = {M_{a | x}}_{a = 1}^{o_{A}}

and

N^{y} = {N_{b | y}}_{b = 1}^{o_{B}},

respectively, where

x \in [m_{A}], y \in [m_{B}]

. These POVMs form measurement assemblages (MA) of A and B:

M_{A} = {M^{x}}_{x = 1}^{m_{A}}

and

N_{B} = {N^{y}}_{y = 1}^{m_{B}},

respectively.

The following concepts were given in [14].

(1) A state

ρ^{A B}

is said to be Bell local for a given measurement assemblage

(M_{A}, N_{B})

if there exists a probability distribution (PD)

{π_{λ}}_{λ = 1}^{d}

such that, for each

(λ, x)

and

(λ, y)

, there exist PDs

{P_{A} (a | x, λ)}_{a = 1}^{o_{A}}

and

{P_{B} (b | y, λ)}_{b = 1}^{o_{B}}

, respectively, for which it holds that

tr [(M_{a | x} \otimes N_{b | y}) ρ^{A B}] = \sum_{λ = 1}^{d} π_{λ} P_{A} (a | x, λ) P_{B} (b | y, λ)

(7)

for all

a, b, x, y

Equation (7) is said to be a local hidden variable model (LHVM) of

ρ^{A B}

w.r.t. MA

(M_{A}, N_{B})

and

λ

is said to be an LHV with PD

{π_{λ}}_{λ = 1}^{d}

(2) A state

ρ^{A B}

is said to be Bell nonlocal for

(M_{A}, N_{B})

if it is not Bell local for

(M_{A}, N_{B})

(3) A state

ρ^{A B}

is said to be Bell local if for every

(M_{A}, N_{B})

, there exists a PD

{π_{λ}}_{λ = 1}^{d}

such that Equation (7) holds.

(4) A state

ρ^{A B}

is said to be Bell nonlocal if it is not Bell local, i.e.,

ρ^{A B}

is not Bell local for some

(M_{A}, N_{B})

Let

BL (M_{A}, N_{B})

denote the set of all states that are Bell local for

M_{A} \otimes N_{B}

BNL (M_{A}, N_{B})

denote the set of all states that are Bell nonlocal for

(M_{A}, N_{B})

BL (A B)

the set of all Bell local states of

A B

;

BNL (A B)

denote the set of all states that are Bell nonlocal. Thus, we see from the definition that

BL (A B) = ⋂_{M_{A}, N_{B}} BL (M_{A}, N_{B}); BNL (A B) = ⋃_{M_{A}, N_{B}} BNL (M_{A}, N_{B}) .

3.2. Bell Locality Depending on Basis

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

for

H_{A}

and

H_{B}

, respectively. Define unitary operators:

U_{A} : H_{A} \to C^{d_{A}}, U_{A} (\sum_{i = 1}^{d_{A}} c_{i} | e_{i}^{A} 〉) = {(c_{1}, c_{2}, \dots, c_{d_{A}})}^{T},

(8)

U_{B} : H_{B} \to C^{d_{B}}, U_{B} (\sum_{j = 1}^{d_{B}} d_{j} | e_{j}^{B} 〉) = {(d_{1}, d_{2}, \dots, d_{d_{A}})}^{T},

(9)

and for every linear operator

T : H_{A} \otimes H_{B} \to H_{A} \otimes H_{B}

, define a

d_{A} d_{B} \times d_{A} d_{B}

matrix

\tilde{T} = [t_{i j, k ℓ}] with (i j, k ℓ) - entry t_{i j, k ℓ} = 〈 e_{i}^{A} e_{j}^{B} | T | e_{k}^{A} e_{ℓ}^{B} 〉 .

Since

〈 i_{A} j_{B} | (U_{A} \otimes U_{B}) T (U_{A}^{†} \otimes U_{B}^{†}) | k_{A} ℓ_{B} 〉 = 〈 e_{i}^{A} e_{j}^{B} | T | e_{k}^{A} e_{ℓ}^{B} 〉 = t_{i j, k ℓ} = 〈 i_{A} j_{B} | \tilde{T} | k_{A} ℓ_{B} 〉

for all

i, j, k ℓ

, we have

tr (\tilde{T}) = tr ((U_{A} \otimes U_{B}) T (U_{A}^{†} \otimes U_{B}^{†})) .

(10)

Let

M_{A} = {M^{x}}_{x = 1}^{m_{A}}

and

N_{B} = {N^{y}}_{y = 1}^{m_{B}}

be measurement assemblages (MAs) of

H_{A}

and

H_{B}

, respectively, where

M^{x} = {M_{a | x}}_{a = 1}^{o_{A}}

and

N^{y} = {N_{b | y}}_{b = 1}^{o_{B}}

are POVMs of systems A and B, respectively. Then

\tilde{M^{x}} : = {\tilde{M_{a | x}}}_{a = 1}^{o_{A}}

and

\tilde{N^{y}} = {\tilde{N_{b | y}}}_{b = 1}^{o_{B}}

are POVMs of systems

C^{d_{A}}

and

C^{d_{B}}

, respectively, where

\tilde{M_{a | x}} = U_{A} M_{a | x} U_{A}^{†}, \tilde{N_{b | y}} = U_{B} N_{a | x} U_{B}^{†} .

Hence,

\tilde{M_{A}} = {\tilde{M^{x}}}_{x = 1}^{m_{A}}

and

\tilde{N_{B}} = {\tilde{N^{y}}}_{y = 1}^{m_{B}}

be MAs of

C^{d_{A}}

and

C^{d_{B}}

, respectively. Using Equation (10) yields that for all

x, y, a, b,

tr [(\tilde{M_{a | x}} \otimes \tilde{N_{b | y}}) ρ_{e_{A B}}^{A B}] = tr [(M_{a | x} \otimes N_{b | y}) ρ^{A B}] .

Thus,

ρ^{A B} \in BL (M_{A}, N_{B})

if and only if

ρ_{e_{A B}}^{A B} \in BL (\tilde{M_{A}}, \tilde{N_{B}}) .

Since Bell locality and separability of pure states are the same [48] for pure states, we see from Theorem 1

(a)

that a pure state

| ψ^{A B} 〉

H_{A} \otimes H_{B}

is Bell local if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is Bell local.

As a conclusion, we obtain the following.

Theorem 4.

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is a Bell local state of

H_{A} \otimes H_{B}

if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is a Bell local state of

C^{d_{A}} \otimes C^{d_{B}}

(b) A density operator

ρ^{A B}

is a Bell local state of

H_{A} \otimes H_{B}

if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is a Bell local state of

C^{d_{A}} \otimes C^{d_{B}} = C^{d_{A} d_{B}}

with

(i j, k ℓ)

-entry

〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉

Similarly, one can check the following.

Theorem 5.

Let

d_{A} = d_{B} = d

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is a Bell local state of

H_{A} \otimes H_{B}

if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is a Bell local state of

C^{d_{A}} \otimes C^{d_{B}}

(b) A density operator

ρ^{A B}

is Bell local if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is Bell local with

(i j, k ℓ)

-entry

〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉

Using Theorem 3, Theorems 4 and 5, once can check the following.

Theorem 6.

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

be an orthonormal basis for

H_{A} \otimes H_{B}

and and U be fined by Equation (2). Then TFSAE.

(a) The coordinate mapping U preserves Bell locality of pure states in both directions, i.e.,

| ψ^{A B} 〉

is Bell local if and only if

U | ψ^{A B} 〉

is Bell local.

(b) A density operator

ρ^{A B}

is Bell local if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is Bell local with

(i j, k ℓ)

-entry

〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉

W_{X} : H_{X} \to C^{d_{X}} (X = A, B)

such that when

d_{A} \neq d_{B}

U = W_{A} \otimes W_{B}

; when

d_{A} = d_{B} = d

, either

U = W_{A} \otimes W_{B}

, or

U = S (W_{A} \otimes W_{B})

, where

S : C^{d} \otimes C^{d}

is the swap operator:

S (| x 〉 \otimes | y 〉) = | y 〉 \otimes | x 〉

(d) There exist orthonormal bases

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively, such that when

d_{A} \neq d_{B}

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

; When

d_{A} = d_{B} = d

, either

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

, or

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j

Proof.

Use the implications:

(a) \Leftrightarrow (d) (Theorem), (d) \Leftrightarrow (c) (Theorem), (d) \Rightarrow (b) (Theorems and), (b) \Rightarrow (a) .

□

4. Basis Dependent Unsteerability

In this section, we will discuss relationship between unsteerability and basis.

4.1. Concepts and Notations

Recall that [14] a state

ρ^{A B}

H_{A} \otimes H_{B}

is said to be unsteerable from A to B with an MA

M_{A} = {M^{x}}_{x = 1}^{m_{A}}

where

M^{x} = {M_{a | x}}_{a = 1}^{o_{A}}

is a POVM of system A if there exists a PD

{π_{λ}}_{λ = 1}^{n}

and a set

{σ_{λ}}_{λ = 1}^{n}

of system B such that

{tr}_{A} [(M_{a | x} \otimes I_{B}) ρ^{A B}] = \sum_{λ = 1}^{n} π_{λ} P_{A} (a | x, λ) σ_{λ}, \forall x, a,

(11)

where

{P_{A} (a | x, λ)}_{a = 1}^{o_{A}}

is a PD for all

x, λ

ρ^{A B}

is said to be unsteerable from A to B if it is unsteerable from A to B with any MA

M_{A}

ρ^{A B}

is said to be steerable from A to B if it is not unsteerable for some MA

M_{A}

. A pure state

| ψ^{A B} 〉

is said to be unsteerable from A to B if so is its density operator

| ψ^{A B} 〉 〈 ψ^{A B} |

. Similarly, one define unsteerability from B to A. It is easy to see that a state that is unsteerable either from A to B or from B to A is Bell local. Thus, a pure state is unsteerable either from A to B or from B to A is separable.

4.2. Unsteerability Depending on Basis

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

for

H_{A}

and

H_{B}

, respectively. Then a density operator

ρ^{A B}

H_{A} \otimes H_{B}

has the corresponding density matrix

ρ_{e_{A B}}^{A B}

ρ^{A B}

under the basis

e_{A B} = e_{A} \otimes e_{B}

ρ_{e_{A B}}^{A B} = (U_{A} \otimes U_{B}) ρ^{A B} (U_{A}^{†} \otimes U_{B}^{†}),

(12)

where

U_{A}

and

U_{B}

are unitary operators given by Equations (8) and (9). In this case, for any MA

M_{A}

of system A, we have

{tr}_{A} [(\tilde{M_{a | x}} \otimes E_{d_{B}}) ρ_{e_{A B}}^{A B}] = {tr}_{A} [(M_{a | x} \otimes I_{B}) ρ^{A B}],

where

E_{d_{B}}

is the

d_{B} \times d_{B}

unit matrix. Thus,

ρ^{A B}

is unsteerable from A to B (resp. with MA

M_{A} = {M^{x}}_{x = 1}^{m_{A}}

) if and only if its density matrix

ρ_{e_{A B}}^{A B}

is unsteerable from A to B (resp. with MA

\tilde{M_{A}} : = {\tilde{M^{x}}}_{x = 1}^{m_{A}}

Similarly, when

d_{A} = d_{B} = d

and

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j \in [d]

, we have

ρ_{e_{A B}}^{A B} = S (U_{A} \otimes U_{B}) ρ^{A B} (U_{A}^{†} \otimes U_{B}^{†}) S^{†},

(13)

where S is the swap operator on

C^{d} \otimes C^{d}

. Thus,

{tr}_{B} [(E_{d_{A}} \otimes \tilde{M_{a | x}}) ρ_{e_{A B}}^{A B}] = {tr}_{A} [(M_{a | x} \otimes I_{B}) ρ^{A B}] .

This shows that

ρ^{A B}

is unsteerable from A to B (resp. with MA

M_{A} = {M^{x}}_{x = 1}^{m_{A}}

) if and only if its density matrix

ρ_{e_{A B}}^{A B}

is unsteerable from B to A (resp. with MA

\tilde{M_{A}} : = {\tilde{M^{x}}}_{x = 1}^{m_{A}}

From these observations, we obtain the following.

Theorem 7.

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is unsteerable from A to B if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is unsteerable from A to B.

(b) A density operator

ρ^{A B}

is unsteerable from A to B if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is unsteerable from A to B.

Theorem 8.

Let

d_{A} = d_{B} = d

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is unsteerable from A to B if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is unsteerable from B to A.

(b) A density operator

ρ^{A B}

is unsteerable from A to B if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is unsteerable from B to A.

Using Theorem 1 and the fact that a pure state

| ψ^{A B} 〉

is unsteerable from A to B if and only if it is separable [48], combining Theorem 7, one can check the following.

Theorem 9.

Let

d_{A} \neq d_{B}

and

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

be an orthonormal basis for

H_{A} \otimes H_{B}

and and U be defined by Equation (2). Then TFSAE.

(a)

| ψ^{A B} 〉

is unsteerable from A to B if and only if

U | ψ^{A B} 〉

is unsteerable from A to B.

(b) A density operator

ρ^{A B}

is unsteerable from A to B if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is unsteerable from A to B.

W_{X} : H_{X} \to C^{d_{X}} (X = A, B)

such that when

d_{A} \neq d_{B}

U = W_{A} \otimes W_{B}

(d) There exist orthonormal bases

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

for

H_{A}

and

H_{B}

, respectively, such that

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

Using Theorem 1 and the fact that a pure state

| ψ^{A B} 〉

is unsteerable from A to B if and only if it is separable [48], combining Theorem 8, one can check the following.

Theorem 10.

Let

d_{A} = d_{B} = d

and

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

be an orthonormal basis for

H_{A} \otimes H_{B}

and U be defined by Equation (2). Then the following (a) and (b) are equivalent.

(a)

| ψ^{A B} 〉

is unsteerable from A to B if and only if

U | ψ^{A B} 〉

is unsteerable from A to B.

(b) A density operator

ρ^{A B}

is unsteerable from A to B if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is unsteerable from A to B.

If (a) or (b) holds, then

W_{X} : H_{X} \to C^{d_{X}} (X = A, B)

such that

U = W_{A} \otimes W_{B}

, or

U = S (W_{A} \otimes W_{B})

(d) There exist orthonormal bases

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively, such that

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

, or

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j

5. Basis Dependent Classical Correlation

In this section, we will discuss relationship between classical correlation and basis.

5.1. Concepts and Notations

Recall that [41,42,43] a state

ρ^{A B}

H_{A} \otimes H_{B}

is said to classically correlated (CC) if there exists a rank-1 projective measurement

Π = {Π_{s}^{A} \otimes Π_{t}^{B} : s \in [m], t \in [n]}

such that

Π (ρ^{A B}) : = \sum_{s = 1}^{m} \sum_{t = 1}^{n} (Π_{s}^{A} \otimes Π_{t}^{B}) ρ^{A B} (Π_{s}^{A} \otimes Π_{t}^{B}) = ρ^{A B};

(14)

otherwise,

ρ^{A B}

is said to be quantum correlated (QC). A pure state

| ψ 〉

is said to be CC (resp. QC) if its density operator

| ψ 〉 〈 ψ |

is CC (resp. QC).

Luo in [41] proved that a state

ρ^{A B}

H_{A} \otimes H_{B}

is CC if and only if it can be represented as

ρ^{A B} = \sum_{i = 1}^{n} \sum_{j = 1}^{m} p_{i j} | e_{i} 〉 〈 e_{i} | \otimes | f_{j} 〉 〈 f_{j} |,

(15)

where

{p_{i j}}

is a PD,

{| e_{i} 〉}

and

{| f_{j} 〉}

are some ONBs for

H_{A}

and

H_{B}

, respectively. This implies that every CC state is separable while a separable state is not necessarily CC. But a bipartite pure state is CC if and only if it is separable ([48] [Theorem 5.1]).

5.2. Classical Correlation Depending on Basis

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then a density operator

ρ^{A B}

H_{A} \otimes H_{B}

has the corresponding density matrix of

ρ^{A B}

under the basis

e_{A B} = e_{A} \otimes e_{B}

ρ_{e_{A B}}^{A B} = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉] = [〈 e_{i}^{A} e_{j}^{B} | ρ^{A B} | e_{k}^{A} e_{ℓ}^{B} 〉],

which is equal to the matrix representation of the operator

(U_{A} \otimes U_{B}) ρ^{A B} (U_{A}^{†} \otimes U_{B}^{†}) : C^{d_{A}} \otimes C^{d_{B}} \to C^{d_{A}} \otimes C^{d_{B}}

under the canonical

{0, 1}

-basis (with

m = d_{A}, n = d_{B}

)

{| i_{A} j_{B} 〉}_{i, j} = {| 1_{A} 1_{B} 〉, | 1_{A} 2_{B} 〉, \dots, | 1_{A} n_{B} 〉, \dots, | m_{A} 1_{B} 〉, | m_{A} 2_{B} 〉, \dots, | m_{A} n_{B} 〉}

for

C^{d_{A}} \otimes C^{d_{B}} = C^{d_{A} d_{B}}

. With this basis, a linear operator X on

C^{d_{A}} \otimes C^{d_{B}}

is usually identified with matrix representation

[〈 i_{A} j_{B} | X | k_{A} ℓ_{B} 〉]

. Thus,

ρ_{e_{A B}}^{A B} = (U_{A} \otimes U_{B}) ρ^{A B} (U_{A}^{†} \otimes U_{B}^{†}),

(16)

where

U_{A}

and

U_{B}

are unitary operators given by Equations (8) and (9).

Similarly, when

d_{A} = d_{B} = d

and

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j \in [d]

, we have

ρ_{e_{A B}}^{A B} = S (U_{A} \otimes U_{B}) ρ^{A B} (U_{A}^{†} \otimes U_{B}^{†}) S^{†},

(17)

where S is the swap operator on

C^{d} \otimes C^{d}

Using the characterization Equation (15) and Formula (16), we obtain the following.

Theorem 11.

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is CC if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is CC.

(b) A density operator

ρ^{A B}

is CC if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is CC with

(i j, k ℓ)

-entry

〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉

Similarly, using the characterization Equation (15) and Formula (17), we obtain the following.

Theorem 12.

Let

d_{A} = d_{B} = d

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

where

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉 (i \in [d_{A}], j \in [d_{B}])

for some ONBs

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively. Then

(a) A pure state

| ψ^{A B} 〉

is CC if and only if its coordinate state

| ψ^{A B} 〉_{e_{A B}}

is CC.

(b) A density operator

ρ^{A B}

is CC if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is CC with

(i j, k ℓ)

-entry

〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉

Using Theorems 11 and 12, once can check the following.

Theorem 13.

Let

e_{A B} = {| e_{i j} 〉 : i \in [d_{A}], j \in [d_{B}]}

be an orthonormal basis for

H_{A} \otimes H_{B}

and U be fined by Equation (2). Then TFSAE.

(a)

| ψ^{A B} 〉

is CC if and only if

U | ψ^{A B} 〉

is CC.

(b) A density operator

ρ^{A B}

is CC if and only if its density matrix

ρ_{e_{A B}}^{A B} : = [〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉]

is CC with

(i j, k ℓ)

-entry

〈 e_{i j} | ρ^{A B} | e_{k ℓ} 〉

W_{X} : H_{X} \to C^{d_{X}} (X = A, B)

such that when

d_{A} \neq d_{B}

U = W_{A} \otimes W_{B}

; when

d_{A} = d_{B} = d

, either

U = W_{A} \otimes W_{B}

, or

U = S (W_{A} \otimes W_{B})

, where

S : C^{d} \otimes C^{d}

is the swap operator:

S (| x 〉 \otimes | y 〉) = | y 〉 \otimes | x 〉

(d) There exist orthonormal bases

e_{A} = {| e_{i}^{A} 〉}_{i = 1}^{d_{A}}

and

e_{B} = {| e_{j}^{B} 〉}_{j = 1}^{d_{B}}

for

H_{A}

and

H_{B}

, respectively, such that when

d_{A} \neq d_{B}

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

; When

d_{A} = d_{B} = d

, either

| e_{i j} 〉 = | e_{i}^{A} 〉 \otimes | e_{j}^{B} 〉

for all

i, j

, or

| e_{i j} 〉 = | e_{j}^{A} 〉 \otimes | e_{i}^{B} 〉

for all

i, j

6. Conclusions

Usually, a density operator

ρ^{A B}

(quantum state ) of a bipartite system

A B

is represented as a matrix

{\hat{ρ}}_{e_{A B}} = [ρ_{i j}]

under a basis

e_{A B} = {e_{i j}}_{i \in [d_{A}], j \in [d_{B}]}

for the Hilbert space

H_{A} \otimes H_{B}

of the system

A B

, called the density matrix of the density operator

ρ^{A B}

. In this work, we have discussed the relationships between quantum locality and basis, and observed that all density operators and their density matrices under a basis

e_{A B}

have the same quantum locality if and only if the basis

e_{A B}

is a product basis of two bases of subsystems. Consequently, different choices of bases may induce different quantum locality; equivalently, different bases define different quantum nonlocality, which we called basis-dependent quantum nonlocality. Also, entangled basis can generate quantum nonlocality of a density matrix from a separable density operator.

Author Contributions

The work of this paper was accomplished by Kaifeng Hu, Zhihua Guo, Huaixin Cao and Ling Lu. Moreover, all authors have read the paper carefully and approved the research contents that were written in the final manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

Data Availability Statement

Not applicable.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 12271325, 11871318, and the Special Plan for Young Top-notch Talent of Shaanxi Province under Grant No. 1503070117.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The unitary operator V on

C^{d_{A}} \otimes C^{d_{B}}

Figure 1. The unitary operator V on

C^{d_{A}} \otimes C^{d_{B}}

Figure 2. Decomposition of U.

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Basis-Dependent Quantum Nonlocality

Abstract

1. Introduction

2. Basis Dependent Separability

2.1. Concepts and Notations

2.2. Separability Depending on Basis

3. Basis Dependent Bell Locality

3.1. Concepts and Notations

3.2. Bell Locality Depending on Basis

4. Basis Dependent Unsteerability

4.1. Concepts and Notations

4.2. Unsteerability Depending on Basis

5. Basis Dependent Classical Correlation

5.1. Concepts and Notations

5.2. Classical Correlation Depending on Basis

6. Conclusions

Author Contributions

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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