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arXiv:2102.02333 · Physical Review A

Entanglement cost for steering assemblages

Thomas Cope

Published Oct 25, 2021• 1 citation
arXivHTMLPDF

Abstract

In this paper we answer the question of how one can quantify the amount of entanglement necessary to create a given steering assemblage when the original state and measurements are unknown. To do this, we extend the concepts of entanglement cost and entanglement of formation to steering assemblages and provide easy-to-calculate upper and lower bounds. We prove that the entanglement of formation for assemblages is not generally continuous and not a flat roof extension (in contrast with the state-based quantity), and we use numerical analysis to illustrate these properties. Finally, we discuss the consequences of these results for assemblage-to-assemblage conversion and suggest potential applications.

DOI
10.1103/physreva.104.042220

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In this paper we answer the question of how one can quantify the amount of entanglement necessary to create a given steering assemblage when the original state and measurements are unknown. To do this, we extend the concepts of entanglement cost and entanglement of formation to steering assemblages and provide easy-to-calculate upper and lower bounds. We prove that the entanglement of formation for assemblages is not generally continuous and not a flat roof extension (in contrast with the state-based quantity), and we use numerical analysis to illustrate these properties. Finally, we discuss the consequences of these results for assemblage-to-assemblage conversion and suggest potential applications.
Quantum entanglement is a fundamental feature of quantum mechanics, yet certain entangled states that are unsteerable can be classically simulated in steering scenarios, making them unable to exhibit quantum steering. Despite their significance, a systematic comparison of such entangled states has not been explored. In this work, we quantify the resource content of unsteerable quantum states in terms of the amount of shared randomness required to simulate the assemblages they generate in the steering scenario. We rigorously demonstrate that the simulation cost is unbounded even for certain unsteerable two-qubit states. Moreover, the simulation cost of entangled two-qubit states is always strictly larger than that for any separable state. A significant portion of our results rests on the relationship between the simulation cost of two-qubit Werner states and that of noisy spin measurements. Using noisy spin measurements as our central example, we also investigate the minimum number of outcomes a parent measurement requires to simulate a given set of compatible measurements. Although certain continuous measurement families admit a finite-outcome parent measurement, we identify scenarios where the simulation cost is unbounded. Our results establish previously unknown lower bounds and upper bounds on the shared randomness simulation cost, supported by connections between the simulation cost of noisy spin measurements and various geometric inequalities, including ones from the zonotope approximation problem in Banach space theory.
Entanglement of formation quantifies the entanglement of a state in terms of the entropy of entanglement of the least entangled pure state needed to prepare it. An analytical expression for this measure exists only for special cases, and finding a closed formula for an arbitrary state remains an open problem. In this work we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for the measure that get tight for several special cases. Further, we show that the problem of calculating the actual value of the entanglement of formation for arbitrary two-mode Gaussian states reduces to a trivial single parameter optimization process, and we provide an efficient algorithm for the numerical calculation of the measure.
Following recent advancements, we consider a scenario of multipartite postquantum steering and general no-signaling assemblages. We introduce the notion of the edge of the set of no-signaling assemblages and we present its characterization. Next, we use this concept to construct witnesses for no-signaling assemblages without a local hidden state model. Finally, in the simplest nontrivial case of steering with two untrusted subsystems, we discuss the possibility of quantum realization of assemblages on the edge. In particular, for three-qubit states, we obtain a no-go-type result, which states that it is impossible to produce an assemblage on the edge using measurements described by positive operator valued measures as long as the rank of a given state is greater than or equal to 3.
The newfound importance of ``entanglement as a resource'' in quantum computation and quantum communication compels us to quantify it in as many distinct ways as possible. Here we explore a new measure of entanglement for mixed quantum states of bipartite systems, which we name the Entanglement of Assistance. We show it to be the maximum average entanglement of all pure-state ensembles consistent with the given density matrix. In this sense, the Entanglement of Assistance is a quantity directly dual to the more standard Entanglement of Formation. With the help of lower and upper bounds, we calculate the Entanglement of Assistance for a few cases and use these results to show that it possesses the surprising property of superadditivity. We believe that this may shed some light on the question of additivity for the Entanglement of Formation.
We quantify the one-shot entanglement cost of an arbitrary bipartite state, that is, the minimum number of singlets needed by two distant parties to create a single copy of the state up to a finite accuracy, by using local operations and classical communication only. This analysis, in contrast to the traditional one, pertains to scenarios of practical relevance, in which resources are finite and transformations can be achieved only approximately. Moreover, it unveils a fundamental relation between two well-known entanglement measures, namely, the Schmidt number and the entanglement of formation. Using this relation, we are able to recover the usual expression of the entanglement cost as a special case.