Structure of quantum measurements implementable with one round of classical communication

TL;DR

The paper develops a convergent semidefinite-programming (SDP) hierarchy to characterize the convex hull of one-round LOCC measurements with a bounded classical communication budget, conv($1\textnormal{R}-\textnormal{LOCC}_{m}$), using constrained symmetric extensions. This framework explicitly encodes directionality, message size, and adaptivity, enabling exact or tightly bounded discrimination tasks in LOCC settings and distinguishing adaptive from non-adaptive strategies. Applying the method to iso-entangled Bell-basis bases, the double trine ensemble, and two-ququart states, the authors demonstrate directional LOCC advantages, non-projective measurement benefits, and minimal communication requirements for perfect discrimination. The results provide a practical, scalable tool for quantifying classical communication resources in LOCC tasks and suggest extensions to broader quantum-information processing tasks, including multi-round protocols and channel discrimination.

Abstract

Measurements that can be implemented via local operations and classical communication (LOCC) constitute a class of operations that is available in future quantum networks in which parties share entangled resource states. We characterise the different classes of measurements implementable with LOCC, where communication is restricted to a single round with a fixed direction. In particular, using the framework of constrained separability problems, we provide a complete characterisation of the class of LOCC measurements that require one round of classical communication with a limit on the transmitted information. Furthermore, we show how to distinguish between adaptive and non-adaptive measurements strategies. Using our techniques we present examples where the success probability of state discrimination depends on the direction of communication as well as on the message size. We also discuss explicit instances of state ensembles where non-projective measurements provide an advantage and where adaptive measurement strategies lead to improved success rates when compared to all non-adaptive strategies.

Figures (3)

• Figure 1: Schematic sketch of general bipartite LOCC measurements. The parties Alice and Bob perform local measurements described by instruments whose specific setting may depend on all preceding measurement results that are freely communicated between Alice and Bob. In this work, we consider the particular case of measurements that only require a single round of classical communication with messages of bounded size as depicted in the shaded region of the figure.
• Figure 2: Schematic comparison between (a) one-round LOCC protocols and (b) non-adaptive LOCC protocols with message size bounded by $m$. In one-round LOCC measurements, the measurement outcome $a$ of the first party can influence the measurement setting of the other party. In non-adaptive protocols, both the local measurements are independently performed before the individual outcomes are post-processed classically.
• Figure 3: Optimal state discrimination of the Bell-basis family using Alice-first and Bob-first LOCC measurements with on bit of communication, together with the upper bound from the PPT relaxation. It can be seen that there is preferred direction of communication (Bob $\rightarrow$ Alice). For a comparison between the optimal success probabilities and the entanglement present in the state ensemble, we also plotted the tangle $T$ of the respective basis vectors.

Theorems & Definitions (17)

• Definition 1
• Theorem 1: $1\textnormal{R}-\textnormal{LOCC}_{m}$ hierarchy
• proof
• Definition 2
• Proposition 1
• Theorem 2: $\textnormal{NA}-\textnormal{LOCC}_{m}$ hierarchy
• proof
• Example 1: Bell-basis family
• Example 2: Double trine double_trine
• Example 3: Maximally entangled ququart-ququart states