论文：Contextuality as a Precondition for Quantum Entanglement

Plávala, M., Gühne, O., 2024. Contextuality as a Precondition for Quantum Entanglement. Phys. Rev. Lett. 132, 100201. https://doi.org/10.1103/PhysRevLett.132.100201

Abstract

量子资源理论中包括对于不同量子信息处理任务来说很关键的资源。
量子纠缠作为一种资源，与非局域的方案相对应。
量子互文作为一种资源，与制备量子态并对其进行一系列测量量子态的方案相对应。
将非局域的态制备和连续测量的方案联系起来，就能得到这两种资源之间的关系：
本文证明了在非局域系统中，当且仅当在制备和测量方案中存在互文的时候存在纠缠。
若没有互文，则同样没有纠缠。
一个直接的结果就是，任何一种检验互文的不等式同样也可以用来检验纠缠。
而纠缠见证（entanglement witness）也可以被用来设计检验互文的不等式。

Two main scenarios：

• Nonlocal scenarios (NLS)
• Sequential scenario (SQS)

Remote prepareation

Suppose the Hilbert space , the density operators defined on is .

Two remote parties, Alice and Bob, have a bipartite state $\rho{AB} \in \mathcal{D}(\mathcal{H{AB}}).

• is entangled if it cannot be prepared by local operations and classical communications.

• Otherwise, $\rho{AB}$ is separable, i.e. $\rho{AB} = \sum_i p_i \sigma_i^A \otimes \sigma_i^B$.

Given a , if Bob apply a local operation , the prepared state is:

$$\Gamma_A(\rho_{AB}) = \{\sigma_A \in \mathcal{D}(\mathcal{H}_A): \exist E_B \geq 0 \quad s.t. \\\sigma_A = \operatorname{Tr}_B[(\mathbb{I}_A \otimes E_B)\rho_{AB}]\}.$$

means that is positive semidefinite.

PM contextuality

Kochen-Specker theoreom (1967) established the notion of contextuality which is usually termed as KS-contextuality.

Spekken (2005) generalizes the definition of contextuality, and introduces it into operator theories: preparation, transformation, and measurement.

The definition of Spekkens:
A noncontextual ontological model of an operational theory is one wherein if two experimental procedures are operationally equivalent, then they have equivalent representations in the ontological model.

An operational theory:
Probability for outcome is .

• : Measurement procedure.
• : Preparation procedure.
• : Transformation procedure.

Equivalence of preparation procedures:
is equivalent to if
.

In quantum theory, a equivalence class of preparation procedures is a density operator .

Equivalence of measurement procedures:
is equivalent to if
.

In quantum theory, a equivalence class of measurement procedure is a POVM .

Equivalence of measurement procedures:
is equivalent to if
.

In quantum theory, a equivalence class of transformation procedures is a CP (complete positive) map .

Context: the feature that cannot be determined by an equivalent class.

An ontic model:
A complete set of variables: .
The space of : .
Preparation procedure : A mormalized probability density over .

$$\mu_P: \Omega \to [0,1] s.t. \int \mu_P(\lambda)d\lambda = 1.$$

Measurement procedure with outcome :

$$\xi_{M,k}:\Omega\to [0,1] s.t. \sum_k \xi_{M,k}(\lambda) = 1 \text{ for all } \lambda.$$

Transformation procedure : A transition matrix.

$$\Gamma_T: \Omega \times \Omega \to [0,1] s.t. \int \Gamma_T(\lambda,\lambda')d\lambda' = 1 \text{ for all } \lambda.$$

The probability density of predition is:

$$p(k \mid P, T, M) = \int d\lambda' d\lambda \xi_{M,k}(\lambda')\Gamma_T(\lambda',\lambda)\mu_P(\lambda).$$

The PM noncontextual model:
There exists PM noncontextual model for if for every and every POVM , we have

$$\operatorname{Tr}(\rho E_k) = \sum_\lambda \operatorname{Tr}(\rho N_\lambda) \operatorname{Tr}(\omega_\lambda E_k).$$

where $\operatorname{Tr}(\rho N\lambda) \geq 0, \sum\lambda \operatorname{Tr}(\rho N\lambda) = 1\omega\lambda \in \mathcal{D}(\mathcal{H})$.

Relation between PM Contextuality and Separability

Entanglement in a nonlocal scenario can arise only if there is preparation and measurement contextuality

Theorem 1: Let $\rho{AB}\Gamma_A(\rho{AB})\rho_{AB}$ is separable.

Proof:

The absence of entanglement implies the absence of contextuality.

Theorem 2: Let $\rho{AB}\rho{AB} = \sum\lambda \omega\lambda \otimes K\lambda\omega\lambda \geq 0\operatorname{Tr}[\omega\lambda = 1]K\lambda \geq 0K\lambda\Lambda_B(\rho{AB})\lambda\LambdaA(\rho{AB})$.

Special cases:

• :
  • $\mathcal{H}A = \mathcal{H}_B = 2K\lambda$ correspond to PM noncontextuality.
  • Else: $\GammaB(\rho{AB})$ spans almost the entire space.