Operational Entanglement Properties

• Operational entanglement properties are attributes of quantum operations defined via geometric, cryptographic, and resource-theoretic frameworks that determine implementability under LOCC conditions.
• They are characterized through the Choi–Jamiolkowski representation and separable operator cones, enabling a convex geometric analysis of LOSE and LOSR operations.
• Their study reveals computational limits, such as NP-hardness in verifying operational membership, and clarifies boundaries between no-signaling and physically realizable protocols.

Operational entanglement properties comprise a suite of structural, geometric, cryptographic, and resource-theoretic features that delineate which quantum correlations, transformations, and measurement outcomes are accessible via physically implementable protocols, often constrained by local operations, classical communication (LOCC), and available entanglement resources. These properties fundamentally govern both the structure of quantum operations (channels, measurements, and state conversions) and the effective deployment of entanglement as a resource, especially in multipartite settings. Understanding these operational facets is essential in characterizing the hierarchy of quantum correlations, determining computational tractability, and setting quantifiable limits for real-world protocols in distributed quantum information processing.

1. Characterization via Choi-Jamiolkowski Representations and Cones of Separable Operators

Central to operational entanglement properties is a convex geometric characterization of local quantum operations with shared entanglement (LOSE) and with shared randomness (LOSR). A multi-party quantum operation Λ is represented by its Choi–Jamiolkowski operator J(Λ). To formalize “localness,” one constructs a cone $K$ of separable Hermitian operators:

• For each party $i$:

$$Q_{(i)} = \{ X \in H(\mathcal{A}_i \otimes \mathcal{X}_i) : \mathrm{Tr}_{\mathcal{A}_i}(X) = \lambda I_{\mathcal{X}_i},~\lambda \in \mathbb{R} \}$$

• The full cone:

$$K = \{ Y \in H(\otimes_i (\mathcal{A}_i \otimes \mathcal{X}_i)) : Y = \sum_k p_k \bigotimes_{i=1}^m X^{(k)}_i,~X^{(k)}_i \in Q_{(i)},~p_k \geq 0 \}$$

A quantum operation Λ is LOSR if and only if for every real linear functional ϕ that is positive on $K$,

$\varphi\big(J(\Lambda)\big) \geq 0.$

For LOSE operations, one requires "complete positivity" with respect to $K$, i.e., for all auxiliary spaces $E_1, ..., E_m$, the extended functional $(\varphi \otimes \mathrm{Id}_{E_1 ... E_m})$ must be positive on $(K \otimes H(E_1) \otimes ... \otimes H(E_m))$. That is,

$\varphi(J(\Lambda)) \geq 0$

for every such functional. This mirrors (but strictly generalizes) the Horodecki criterion for state separability, embedding operational “localness” in the geometry of operator cones.

2. Computational Complexity and the Weak Membership Problem

A fundamental operational property is the computational intractability of testing (even approximately) whether a quantum operation is LOSE. The critical result is that the "weak membership" problem for LOSE is strongly NP-hard. This is established by demonstrating the existence of a ball of LOSR (and thus LOSE) operations centered at the completely noisy channel (with Choi operator proportional to the identity) lying entirely within the convex set of no-signaling operations. The Yudin–Nemirovskiĭ theorem then allows a reduction from the strongly NP-hard weak validity problem to the weak membership problem:

• Given as input an operation Λ (by its Choi operator) and an accuracy parameter ε, it is computationally infeasible to decide whether Λ lies within ε (in the Frobenius norm) of the set of LOSE operations.
• No polynomial-time algorithm is expected unless NP ⊆ P.

This operational hardness frames the practical impossibility of fully characterizing or optimizing over LOSE in general, with implications for multi-party games, distributed computation, and entanglement verification.

3. No-Signaling Operations and Separation from LOSE

Operational entanglement is sharply distinguished from merely no-signaling (NS) properties. NS operations are characterized by constraints on the Choi operator:

$$\forall K \subseteq \{1, ..., m\},~\exists Q:~\mathrm{Tr}_{\mathcal{A}_K}(J(\Lambda)) = Q \otimes I_{\mathcal{X}_K}$$

This guarantees that outputs for the subset $K$ are independent of inputs at the complement $\bar{K}$. However, the operational content is subtler:

• Every LOSE operation is no-signaling, but not every NS operation is LOSE.
• The paper exhibits the Popescu–Rohrlich (PR) box (the “nonlocal box”) whose Choi operator is separable (i.e., a convex combination of product operators), but which lies outside the LOSE set. Explicitly, for this operation, the corresponding quantum correlations violate Tsirelson’s bound and are too strong to be reproduced by local operations even with unrestricted entanglement, despite respecting relativistic causality.

Table: Key Operational Classes

Class	Choi Representation	Operationally Accessible?
LOSE	Completely positive functionals on K	Physically via LOCC+entanglement
LOSR	Positive functionals on K	Physically via LOCC+randomness
No-Signaling	Choi in product of Q_{(i)}	Not always via LOSE

This establishes strict operational boundaries in the space of quantum operations.

4. Duality Between Operations and Linear Functionals

A unifying operational property is duality: every linear constraint satisfied by LOSE operations corresponds to positivity with respect to a family of functionals, as guaranteed by separation theorems in convex analysis. The functional-analytic viewpoint generalizes structural results on separable states to the space of super-operators. Specifically, the characterization via positivity or complete positivity over $K$ links the feasibility of implementing a given operation to the dual cone structure, encapsulating the resource-theoretic limitations inherent to local protocols, and relates to generalized notions of complete positivity adapted to multipartite cones.

5. Implications for Physical and Resource Theories

These operational entanglement properties yield broad foundational and practical consequences:

• Resource Characterization: Entanglement becomes a resource for implementing operations beyond those realizable by LOSR, but below the full set of no-signaling transformations.
• Device-Independent Boundaries: The example of the nonlocal box delineates the ultimate limitations of physical protocols constrained by local operations and shared entanglement, defining attainable quantum correlations.
• Complexity Constraints: The NP-hardness result quantifies the computational boundary between physically testable operational membership and postulated, but likely unreachable, extensions (e.g., super-quantum, no-signaling boxes).
• Convex Geometry and Operational Protocols: The convex geometry induced by the space of Choi operators and separable cones mirrors the operational pathways available in distributed quantum systems, where only certain tolerant forms of nonlocality can be generated and manipulated.

6. Summary and Further Directions

Properties of operational entanglement for local quantum operations with shared entanglement are determined by:

• The representation of operations via the Choi–Jamiolkowski isomorphism and their position relative to cones of separable Hermitian operators.
• Duality with linear functionals and complete positivity conditions directly generalizing separability criteria for quantum states to the operational level.
• The NP-hard boundary of operationally accessible verification, which severely constrains both the certification and utility of distributed quantum protocols reliant on LOSE.
• A strict separation between no-signaling operations and those implementable with shared entanglement, exemplified by physically unachievable but mathematically consistent super-quantum channels.

These connections synthesize convex geometry, computational complexity, and physical implementability, establishing the landscape and constraints for distributed quantum operations from an operational entanglement perspective (0805.2209).