We present a universal technique for quantum state estimation based on the maximum-likelihood method. This approach provides a positive definite estimate for the density matrix from a sequence of measurements performed on identically prepared copies of the system. The method is versatile and can be applied to multimode radiation fields as well as to spin systems. The incorporation of physical constraints, which is natural in the maximum-likelihood strategy, leads to a substantial reduction of statistical errors. Numerical implementation of the method is based on a particular form of the Gauss decomposition for positive definite Hermitian matrices.Comment: 4 pages, 3 figures (5 eps files). Submitted to Phys. Rev. A as a Rapid Communicatio
We address the problem of discriminating with minimal error probability two given quantum operations. We show that the use of entangled input states generally improves the discrimination. For Pauli channels we provide a complete comparison of the optimal strategies where either entangled or unentangled input states are used.Quantum nonorthogonality is a basic feature of quantum mechanics that has deep implications in many areas, as quantum computation and communication, quantum entanglement, cloning, and cryptography. Nonorthogonality is strongly related to the concept of distinguishability, and many measures have been defined to compare quantum states [1] and quantum processes [2], according to some experimentally or theoretically meaningful criteria. Since the pioneering work of Helstrom [3] on quantum hypothesis testing, the problem of discriminating nonorthogonal quantum states has received a lot of attention [4], with some experimental verifications as well [5]. The most popular scenarios are the minimal-error probability discrimination, where each measurement outcome selects one of the possible states and the error probability is minimized, and the optimal unambiguous discrimination [6], where unambiguity is paid by the possibility of getting inconclusive results from the measurement. Stimulated by the rapid developments in quantum information theory, the problem of discrimination has been addressed also for bipartite quantum states, along with the comparison of global strategies where unlimited kind of measurements is considered, with the scenario of LOCC scheme, where only local measurements and classical communication are allowed [7].The concepts of nonorthogonality and distinguishability can be applied also to quantum operations, namely all physically allowed transformations of quantum states. Not very much work, however, has been devoted to the problem of discriminating general quantum operations, and major efforts have been directed at the case of unitary transformations [8]. In fact, the most elementary formulation of the problem can be recast to the evaluation of the norm of complete boundedness [9], which is in general a very hard task. We recall that such a norm entered the quantum information field as the diamond norm [10], and one of its most relevant application is found in the problem of quantifying quantum capacities of quantum information channels [11].In this Letter, we address the problem of discriminating with minimal error probability two given quantum operations. After briefly reviewing the case of quantum states, we formulate the problem for two quantum operations. Differently from the case of unitary transformations [8], we show that entangled input states generally improve the discrimination. We prove that the use of an arbitrary maximally entangled state turns out to be always an optimal input when we are asked to discriminate two quantum operations that generalize the Pauli channel in any dimension. In the case of qubits, we give a complete comparison of the strategies where eithe...
In a recent paper, Walgate et. al. [1] demonstrated that any two orthogonal multipartite pure states can be optimally distinguished using only local operations. We utilise their result to show that this is true for any two multiparty pure states, in the sense of inconclusive discrimination. There are also certain regimes of conclusive discrimination for which the same also applies, although we can only conjecture that the result is true for all conclusive regimes. We also discuss a class of states that can be distinguished locally according to any discrimination measure, as they can be locally recreated in the possession of one party. A consequence of this is that any two maximally entangled states can always be optimally discriminated locally, according to any figure of merit.
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