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International Conference on Quantum Information

Linearly independent states decomposition: quantum state discrimination

Luis Roa, Alejandra Maldonado-Trapp, M. Alid-Vaccarezza

Published Jan 1, 2011• 0 citations
Publisher

Abstract

We put the pure-state decomposition mathematical property of a mixed state to a physical test. We begin by characterizing all the possible decompositions of a rank-two mixed state by means of the complex overlap between two involved states. The physical test proposes a scheme of quantum state recognition of one of the two linearly independent states which arise from the decomposition.

DOI
10.1364/icqi.2011.qmi27

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We put the pure-state decomposition mathematical property of a mixed state to a physical test. We begin by characterizing all the possible decompositions of a rank-two mixed state by means of the complex overlap between two involved states. The physical test proposes a scheme of quantum state recognition of one of the two linearly independent states which arise from the decomposition.
We put the pure-state decomposition mathematical property of a mixed state to a physical test. We present a protocol for preparing two known nonorthogonal quantum states with well-defined a priori probabilities. Hence we characterize all the possible decompositions of a rank-two mixed state by means of the complex overlap between the two involved states. The physical test proposes a scheme for quantum state recognition of one of the two linearly independent states that arise from the decomposition. We find that the two states associated with the balanced pure-state decomposition have the smaller overlap modulus and therefore the smallest probability of being discriminated conclusively, while in the nonconclusive process they have the highest probability of having an error. In addition, we design an experimental scheme that allows discriminating conclusively and optimally two nonorthogonal states prepared with different a priori probabilities. Thus we propose a physical implementation for this linearly independent pure-state decomposition and state discrimination test by using twin photons generated in the process of spontaneous parametric down-conversion. The information state is encoded in a one-photon polarization state, whereas the second single photon is used for heralded detection.
The linearity of quantum operations puts many fundamental constraints on the information processing tasks we can achieve on a quantum system whose state is not exactly known, just as we observe in quantum cloning and quantum discrimination. In this paper we show that in a probabilistic manner, linearity is in fact the only one that restricts the physically realizable tasks. To be specific, if a system is prepared in a state secretly chosen from a linearly independent pure state set, then any quantum state separation can be physically realized with a positive probability. Furthermore, we derive a lower bound on the average failure probability of any quantum state separation.
Recently, J. A. Bergou et al. proposed sequential state discrimination as a new quantum state discrimination scheme. In the scheme, by the successful sequential discrimination of a qubit state, receivers Bob and Charlie can share the information of the qubit prepared by a sender Alice. A merit of the scheme is that a quantum channel is established between Bob and Charlie, but a classical communication is not allowed. In this report, we present a method for extending the original sequential state discrimination of two qubit states to a scheme of N linearly independent pure quantum states. Specifically, we obtain the conditions for the sequential state discrimination of N = 3 pure quantum states. We can analytically provide conditions when there is a special symmetry among N = 3 linearly independent pure quantum states. Additionally, we show that the scenario proposed in this study can be applied to quantum key distribution. Furthermore, we show that the sequential state discrimination of three qutrit states performs better than the strategy of probabilistic quantum cloning.
We consider the discrimination of two quantum channels acting on bosonic modes, with a probe state of fixed mean photon number and no ancilla. We prove that pure probes are optimal: for every mixed probe there is a pure probe with exactly the same mean photon number that performs at least as well. The result holds for arbitrary channels and for every figure of merit built from the Chernoff quantities $\Tr[ρ_1^sρ_2^{1-s}]$, including the quantum Chernoff exponent, as well as for the single-shot error probability. The proof relies on a decomposition of any state of mean photon number $\nb$ into pure states of mean photon number exactly $\nb$, which extends a matrix rank-one decomposition of Huang and Zhang to density operators of infinite rank.
Abstract This chapter gives a detailed overview of entanglement for pure and mixed states. We begin with the Schmidt-decomposition theorem, which allows us to associate the subsystem mixedness of pure state with entanglement. We discuss the purification of mixed quantum states, the convexity of the set of separable states, and entanglement as a form of correlation. As an example we examine the Bloch decomposition for two qubits. For the detection of entanglement the Peres-Horodecki criterion is introduced, before we analyse the relation between entanglement and non-locality via the CHSH-Operator Criterion and the family of Werner states. We further consider Tsirelson’s bound and hidden non-locality. We then discuss important mathematical tools for entanglement detection, including the positive-map theorem, the Størmer-Woronowicz theorem, and the reduction criterion. In this context we examine the family of isotropic states. Finally, the Weyl states provide an example for which we study the geometry of two-qubit entanglement