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arXiv:1306.5992 · Journal of Mathematical Physics

Interpolatability distinguishes LOCC from separable von Neumann measurements

Andrew M. Childs, Debbie Leung, Laura Mančinska, Māris Ozols

Published Nov 1, 2013• 2 citations
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Abstract

Local operations with classical communication (LOCC) and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement. Separable operations are strictly more powerful than LOCC, but no simple explanation of this phenomenon is known. We show that, in the case of von Neumann measurements, the ability to interpolate measurements is an operational principle that sets apart LOCC and separable operations.

DOI
10.1063/1.4830335

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Local operations with classical communication (LOCC) and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement. Separable operations are strictly more powerful than LOCC, but no simple explanation of this phenomenon is known. We show that, in the case of von Neumann measurements, the ability to interpolate measurements is an operational principle that sets apart LOCC and separable operations.
In this thesis we study the subset of quantum operations that can be implemented using only local quantum operations and classical communication (LOCC). This restricted paradigm serves as a tool to study not only quantum correlations and other nonlocal quantum effects, but also resource transformations such as channel capacities. \n \nThe mathematical structure of LOCC is complex and difficult to characterize. In the first part of this thesis we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. This perspective allows us to measure the distance between two LOCC instruments and hence discuss the closure of LOCC in a rigorous way. While the set of LOCC is not topologically closed, we show that the operations that can be implemented using some fixed number rounds of communication constitute a compact subset of all quantum operations. We also exhibit a subset of LOCC measurements that is closed. Additionally we establish the existence of an open ball around the completely depolarizing map consisting entirely of LOCC implementable maps. \n \nIn the second part of this thesis we focus on the task of discriminating states from some known set S by LOCC. Building on the work in the paper "Quantum nonlocality without entanglement", we provide a framework for lower bounding the error probability of any LOCC protocol aiming at discriminating the states from S. We apply our framework to an orthonormal product basis known as the domino states. This gives an alternative and simplified bound quantifying how well these states can be discriminated using LOCC. We generalize this result for similar bases in larger dimensions, as well as the "rotated" domino states, resolving a long-standing open question. These results give new examples of quantitative gaps between the classes of separable and LOCC operations. \n \nIn the last part of this thesis, we ask what differentiates separable from LOCC operations. Both of these classes play a key role in the study of entanglement. Separable operations are known to be strictly more powerful than LOCC ones, but no simple explanation of this phenomenon is known. We show that, in the case of bipartite von Neumann measurements, the ability to interpolate is an operational principle that separates LOCC and separable operations.
Entanglement plays an important role in quantum computation and information. We can only manipulate an entangled quantum state shared among several distant parties by local operations with classical communication (LOCC). Thus, it is of great importance to characterize the conditions by which we can determine whether one pure quantum state can be transformed to another pure state by LOCC. It is well known that separable operations and LOCC are not equivalent. Dozens of results have illustrated this difference. Little do we know about this difference despite these recent results. It is shown in this paper that if the initial state $|\ensuremath{\psi}\ensuremath{\rangle}$ and final state $|\ensuremath{\phi}\ensuremath{\rangle}$ are genuine tripartite pure states in the Greenberger-Horne-Zeilinger (GHZ) class then $|\ensuremath{\psi}\ensuremath{\rangle}$ can be transformed to $|\ensuremath{\phi}\ensuremath{\rangle}$ by separable operations if and only if $|\ensuremath{\psi}\ensuremath{\rangle}$ can be transformed to $|\ensuremath{\phi}\ensuremath{\rangle}$ by deterministic LOCC. That is, SEP equals LOCC on tripartite GHZ states.
Quantum entanglement is fundamentally related to the operational setting of local quantum operations and classical communication (LOCC). A more general class of operations known as separable operations (SEP) is often employed to approximate LOCC, but the exact difference between LOCC and SEP is unknown. In this letter, we compare the two classes in performing particular tripartite to bipartite entanglement conversions and report a gap as large as 12.5% between SEP and LOCC, which is the first known appreciable gap between the classes. Our results rely on constructing a computable entanglement monotone with a clear operational meaning that, unlike all other such monotones previously studied, is not monotonic under SEP. Finally, we prove the curious fact that convergent sequences of LOCC protocols need not be LOCC feasible in the limit.
Local operations assisted by classical communication (LOCC) play a fundamental role in the resource-theoretic formulation of entanglement theory inducing an operationally meaningful ordering in the set of entangled states. The particular class within LOCC of local unitary (LU) transformations is obviously closed under inversion. Thus, LU-equivalent states are interconvertible by LOCC and have therefore the same entanglement. For this reason LU-equivalence has been thoroughly studied in the context of entanglement theory. However, as I note here with a simple example, the converse is not true: there exist entangled states that are interconvertible by LOCC but which are not LU-equivalent. This motivates studying under which conditions LOCC interconvertibility is characterized by LU-equivalence. I provide a sufficient condition for this to hold. When one considers multipartite states with full entanglement dimensionality (as quantified by the Schmidt number vector) LOCC interconvertibility can only hold under LU-equivalence.
We give a necessary condition that a separable measurement can be implemented by local quantum operations and classical communication (LOCC) in any finite number of rounds of communication, generalizing and strengthening a result obtained previously. That earlier result involved a bound that is tight when the number of measurement operators defining the measurement is relatively small. The present results generalize that bound to one that is tight for any finite number of measurement operators, and we also provide an extension which holds when that number is infinite. We apply these results to the famous example on a $3\ifmmode\times\else\texttimes\fi{}3$ system known as ``domino states,'' which were the first demonstration of nonlocality without entanglement. Our extended necessary condition provides another way of showing that these states cannot be perfectly distinguished by (finite-round) LOCC. It directly shows that this conclusion also holds for their related rotated domino states. We also introduce a class of problems involving the unambiguous discrimination of quantum states, each of which is an example where the states can be optimally discriminated by a separable measurement, but according to our condition, cannot be optimally discriminated by LOCC. These examples nicely illustrate the usefulness of the present results, since our earlier necessary condition, which the present result generalizes, is not strong enough to reach a conclusion in any of these cases.