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IEEE International Symposium on Information Theory (ISIT)

Simulation of Separable Quantum Measurements on Bipartite States via Likelihood POVMs

Arun Padakandla, Naqueeb Ahmad Warsi

Published Jul 7, 2024• 1 citation
Publisher

Abstract

By developing a new framework of likelihood POVMs, analysis techniques and a new proof of the quantum covering lemma, we address the simulation of separable quantum measurement over bipartite states. In addition to a new one shot inner bound that naturally generalizes to the asymptotic case, we demonstrate the power, generality and universality of the developed techniques in the most general distributed measurement scenario by recovering all current known inner bounds. In addition to the above results, this framework is appealing in being the most natural and simple POVM simulation protocol.

DOI
10.1109/isit57864.2024.10619522

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By developing a new framework of likelihood POVMs, analysis techniques and a new proof of the quantum covering lemma, we address the simulation of separable quantum measurement over bipartite states. In addition to a new one shot inner bound that naturally generalizes to the asymptotic case, we demonstrate the power, generality and universality of the developed techniques in the most general distributed measurement scenario by recovering all current known inner bounds. In addition to the above results, this framework is appealing in being the most natural and simple POVM simulation protocol.
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Entanglement detection by local measurements, which can possibly be performed by far distant observers, are of particular interest for applications in quantum key distribution and quantum communication. In this paper sufficient conditions for arbitrary dimensional bipartite entanglement detection based on correlation matrices and joint probability distributions of such local measurements are investigated. In particular, their dependence on the nature of the local measurements is explored for typical bipartite quantum states and for measurements involving local orthonormal hermitian operators bases (LOOs) or generalized measurements based on informationally complete positive operator valued measures of the recently introduced (N,M)(N,M)(N,M)-type ((N,M)(N,M)(N,M)-POVMs) \cite{NMPOVM}. It is shown that symmetry properties of (N,M)(N,M)(N,M)-POVMs imply that sufficient conditions for bipartite entanglement detection exhibit peculiar scaling properties relating different equally efficient local entanglement detection scenarios. For correlation-matrix based bipartite local entanglement detection, for example, this has the consequence that LOOs and all informationally complete (N,M)(N,M)(N,M)-POVMs are equally powerful. With the help of a hit-and-run Monte-Carlo algorithm the effectiveness of local entanglement detection of typical bipartite quantum states is explored numerically. For this purpose Euclidean volume ratios between locally detectable entangled states and all bipartite quantum states are determined.
We present a protocol to simulate the quantum correlations of an arbitrary bipartite state, when the parties perform a measurement according to two traceless binary observables. We show that log2(d){\mathrm{log}}_{2}(d)log2​(d) bits of classical communication is enough on average, where ddd is the dimension of both systems. To obtain this result, we use the sampling approach for simulating the quantum correlations.
We focus on the information theoretic study of quantifying the information content in the outcome of a quantum measurement, also referred to as the measurement compression problem (MCP). The goal of this work is to develop a unified framework of tools to address this problem in diverse network scenarios both in the one-shot and asymptotic regimes. Designing a new measurement simulation protocol based on likelihood POVMs and developing analysis techniques based on a proxy state, we present a unified framework to derive the current known largest inner bounds for several network scenarios. Our findings include new one-shot inner bounds for the distributed MCP with and without quantum side information and the distributed quantum classical rate distortion problems.
In a recent paper [S. M. Cohen, Phys. Rev. A 84, 052322 (2011)], we showed how to construct a protocol for implementing a bipartite, separable quantum measurement using only local operations on subsystems and classical communication between parties (LOCC) within any fixed number of rounds of communication, whenever such a protocol exists. Here, we generalize that construction to one that applies for any number of parties. One important observation is that the construction automatically determines the ordering of the parties' measurements, overcoming a significant apparent difficulty in designing protocols for more than two parties. We also present various other results about LOCC, including showing that if, in any given measurement operator of the separable measurement under consideration, the local parts for two different parties are rank-1 operators that are not repeated in any other measurement operator of the measurement, then this separable measurement cannot be exactly implemented by LOCC in any finite number of rounds.
Three distributed parties, two transmitters (Txs) and a receiver (Rx), hold one component each of a tripartite quantum state ρA1A2Cρ^{A_1A_2C}ρA1​A2​C. The goal is to simulate the action of a separable instrument acting on the A1A_1A1​ and A2A_2A2​ components, with the Rx recovering the classical outcome. To enable this, each Tx kkk can transfer bits on a noiseless bit pipe and share randomness at rates RkR_kRk​ and CkC_kCk​, respectively, with the Rx. Undertaking a Shannon-theoretic study, we characterize two new sets of inner bounds. The first set, derived for the one-shot regime, is based on instrument simulation protocols built using unstructured IID codes, while the second set, derived for the asymptotic regime, relies on coset codes and new decoding POVMs. The first set of bounds recovers current known inner bounds for instrument and measurement simulation in all previously studied scenarios. Our protocols are based on likelihood POVMs, and our analysis leverages Sen's smooth multiparty covering and simultaneous decoding, while handling the distributed-component scenario via a compatible operator sliding trick.