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Physical Review A

Positive-operator-valued measure optimization of classical correlations

S. Hamieh, R. Kobes, Haitham Zaraket

Published Nov 29, 2004• 108 citations
Publisher

Abstract

We study the problem of optimization over positive-operator-valued measures to extract classical correlation in a bipartite quantum system. The proposed method is applied to binary states only. Moreover, to illustrate this method, an explicit example is studied in detail.
DOI
10.1103/physreva.70.052325

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Abstract We give a brief summary of the current status of the problem of extracting the accessible information when a quantum system is received in one of a finite number of pre-known quantum states. We review analytical methods as well as a numerical strategy. In particular, the group-covariant positive-operator-valued measures are discussed, and several explicit examples are worked out in detail. These examples include some that occur in the security analysis of schemes for quantum cryptography.
In this thesis, we start with giving a mathematical description of bipartite quantum correlations and how they are built up in the Tensor model. This is needed because we want to recover the state and the operators when only the bipartite quantum correlation is known. In the literature, there are see-saw algorithms to recover the state, but they are limited to only lower dimensions. In this thesis, we explore an alternative approach, where we directly minimize the function f(ψ,{Esa},{Ftb}) = ∑a,b,s,t(P(a,b|s,t)- ψ*(Esa ⊗ Ftb)ψ)2. Here, P(a,b|s,t) is the bipartite correlation, ψ is the state vector, and Esa and Ftb are the POVMs. Furthermore, ⊗ is the Kronecker product and * indicates the conjugate transpose of a vector. These variables are subject to constraints and some of them can easily be transformed into penalty functions. The matrices Esa and Ftb have to be Hermitian positive semidefinite, for which we parameterize them by their Cholesky decompositions. The gradient of this (now unconstrained) problem can be explicitly determined with the use of Wirtinger calculus. This offers an elegant way to determine the gradient of real-valued functions with complex variables. Also, a total description of Wirtinger calculus is also given, including a proof that the gradient indeed points towards the direction of the steepest incline. We use first-order methods like gradient descent with backtracking line search and momentum-based gradient descent to find a minimum solution of the equation. If the cost function converges towards zero, we assume that the variables converge to a correct state and measurement operators. These methods can find large correlations of approximately 3000 separate variables in 1.5 hours and are able to find many different other correlations and states. The algorithm had some problems finding the operators and state of a family of correlations that had four inputs and two outputs. For some correlations, the algorithms found states and operators of lower dimension than the correlations were build with.
We search a simplest and minimal way to determine whether a given quantum system is entangled or separable. For this end, we propose binary correlation measurements in which restricted knowledge of only zero or non-zero correlations is available. We consider the concrete investigation on a pure state for two particles, each particle having two basis states (the 2x2 system). We show that, even with this limited information from the binary correlation measurements, we can still reach the known minimum of three measurements for entanglement detection. We next consider the comparable problem applied to the mixed density matrix. The mixed quantum case appears to require more detailed information, which we illustrate by studying the concrete example of the Werner density matrix.