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arXiv:quant-ph/0304123 · Physical Review A

Direct estimation of functionals of density operators by local operations and classical communication

Carolina Moura Alves, Paweł Horodecki, Daniel K. L. Oi, L. C. Kwek, ARTUR K. EKERT

Published Sep 15, 2003• 43 citations
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Abstract

We present a method of direct estimation of important properties of a shared bipartite quantum state, within the ``distant laboratories'' paradigm, using only local operations and classical communication. We apply this procedure to spectrum estimation of shared states, and locally implementable structural physical approximations to incompletely positive maps. This procedure can also be applied to the estimation of channel capacity and measures of entanglement.

DOI
10.1103/physreva.68.032306

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We present a method of direct estimation of important properties of a shared bipartite quantum state, within the ``distant laboratories'' paradigm, using only local operations and classical communication. We apply this procedure to spectrum estimation of shared states, and locally implementable structural physical approximations to incompletely positive maps. This procedure can also be applied to the estimation of channel capacity and measures of entanglement.
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.
We propose a more general method for detecting a set of entanglement measures, i.e., negativities, in an arbitrary tripartite quantum state by local operations and classical communication. To accomplish the detection task using this method, three observers do not need to perform partial transposition maps by the structural physical approximation; instead, they only need to collectively measure some functions via three local networks supplemented by a classical communication. With these functions, they are able to determine the set of negativities related to the tripartite quantum state.
We present a method for checking the Peres separability criterion in an arbitrary bipartite quantum state ${\ensuremath{\rho}}_{\mathrm{AB}}$ within local operations and classical communication scenario. The method does not require noise operation which is needed in making the partial transposition map physically implementable. The main task for the two observers, Alice and Bob, is to measure some specific functions of the partial transposed matrix. With these functions, they can determine the eigenvalues of ${\ensuremath{\rho}}_{\mathrm{AB}}^{{T}_{\mathrm{B}}}$, among which the minimum serves as an entanglement witness.
We consider the optimal measurement procedures for estimating an unknown maximally entangled state with local operations and classical communications (LOCC). By the ``optimal,'' we mean that there is no other LOCC measurements that can provide a higher estimation fidelity. In fact, we first use the positive partial transpose constraint to obtain an upper bound of the fidelity; then, we show there exists a kind of LOCC measurement that can attain such a bound. The estimation fidelity of the optimal LOCC estimation procedures is D+2(D+1)D2\frac{D+2}{(D+1){D}^{2}}(D+1)D2D+2​ with DDD denoting the dimension of the relevant quantum system.