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arXiv:1011.1871 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

PT -symmetric quantum state discrimination

Carl M. Bender, Dorje C. Brody, João Luiz Caldeira, Uwe Günther, Bernhard K. Meister, Boris F. Samsonov

Published Mar 19, 2013• 39 citations

Abstract

The objective of this paper is to explain and elucidate the formalism of PT quantum mechanics by applying it to a well-known problem in conventional Hermitian quantum mechanics, namely the problem of state discrimination. Suppose that a system is known to be in one of two quantum states, |ψ(1)> or |ψ(2)>. If these states are not orthogonal, then the requirement of unitarity forbids the possibility of discriminating between these two states with one measurement; that is, determining with one measurement what state the system is in. In conventional quantum mechanics, there is a strategy in which successful state discrimination can be achieved with a single measurement but only with a success probability p that is less than unity. In this paper, the state-discrimination problem is examined in the context of PT quantum mechanics and the approach is based on the fact that a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states. It is shown that it is always possible to choose this inner product so that the two states |ψ(1)> and |ψ(2)> are orthogonal. Using PT quantum mechanics, one cannot achieve a better state discrimination than in ordinary quantum mechanics, but one can instead perform a simulated quantum state discrimination, in which with a single measurement a perfect state discrimination is realized with probability p.

DOI: 10.1098/rsta.2012.0160

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The objective of this paper is to explain and elucidate the formalism of PT quantum mechanics by applying it to a well-known problem in conventional Hermitian quantum mechanics, namely the problem of state discrimination. Suppose that a system is known to be in one of two quantum states, |ψ(1)> or |ψ(2)>. If these states are not orthogonal, then the requirement of unitarity forbids the possibility of discriminating between these two states with one measurement; that is, determining with one measurement what state the system is in. In conventional quantum mechanics, there is a strategy in which successful state discrimination can be achieved with a single measurement but only with a success probability p that is less than unity. In this paper, the state-discrimination problem is examined in the context of PT quantum mechanics and the approach is based on the fact that a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states. It is shown that it is always possible to choose this inner product so that the two states |ψ(1)> and |ψ(2)> are orthogonal. Using PT quantum mechanics, one cannot achieve a better state discrimination than in ordinary quantum mechanics, but one can instead perform a simulated quantum state discrimination, in which with a single measurement a perfect state discrimination is realized with probability p.

\mathcal{PT}

-Symmetric Quantum Discrimination of Three States

If the system is known to be in one of two non-orthogonal quantum states,

|ψ_1\rangle

or

|ψ_2\rangle

, it is not possible to discriminate them by a single measurement due to the unitarity constraint. In a regular Hermitian quantum mechanics, the successful discrimination is possible to perform with the probability

p < 1

, while in

\mathcal{PT}

-symmetric quantum mechanics a \textit{simulated single-measurement} quantum state discrimination with the success rate

p

can be done. We extend the

\mathcal{PT}

-symmetric quantum state discrimination approach for the case of three pure quantum states,

|ψ_1\rangle

,

|ψ_2\rangle

and

|ψ_3\rangle

without any additional restrictions on the geometry and symmetry possession of these states. We discuss the relation of our approach with the recent implementation of

\mathcal{PT}

symmetry on the IBM quantum processor.

As a fundamental and challenging problem in quantum information processing, quantum state discrimination (QSD) has recently been revisited from the perspective of non-Hermitian (NH) physics. The existing work about QSD in NH systems is mostly limited to PT-symmetric or pseudo-Hermitian systems with real spectra, while generic NH Hamiltonians possess complex spectra. To explore the underlying physics of quantum state discrimination in NH systems, we first demonstrate that the unambiguous discrimination of non-orthogonal quantum states can be realized with P-pseudo-Hermitian and PT-symmetric Hamiltonians in the broken phase and the required evolution time can be arbitrarily small. We further extend the regime to generic non-Hermitian Hamiltonians and demonstrate the feasibility of unambiguous discrimination. Under the same energy constraint, we establish the criteria for constructing P-pseudo-Hermitian Hamiltonians that enable unambiguous discrimination of quantum states with smaller angular separation or within shorter evolution time than any fixed PT-symmetric system. Furthermore, we also discuss the potential impact of exceptional points (EPs) on QSD. Our results suggest that the non-orthogonal eigenstates of NH Hamiltonians, rather than PT symmetry or pseudo-Hermitian symmetry, are fundamental to unambiguous discrimination and information flow in NH systems.

Discriminating an Arbitrary Number of Pure Quantum States by the Combined

\mathcal{CPT}

and Hermitian Measurements

If the system is known to be in one of two non-orthogonal quantum states,

|ψ_1\rangle

or

|ψ_2\rangle

,

\mathcal{PT}

-symmetric quantum mechanics can discriminate them, \textit{in principle}, by a single measurement. We extend this approach by combining

\mathcal{PT}

-symmetric and Hermitian measurements and show that it's possible to distinguish an arbitrary number of pure quantum states by an appropriate choice of the parameters of

\mathcal{PT}

-symmetric Hamiltonian.

Quantum state discrimination (QSD) is a fundamental task in quantum information processing, improving the computation efficiency and communication security. Non-Hermitian (NH) PT-symmetric systems were found to be able to discriminate two quantum states better than the Hermitian strategy. In this work, we propose a QSD approach based on P-pseudo-Hermitian systems with real spectra. We theoretically prove the feasibility of realizing QSD in the real-spectrum phase of a P-pseudo-Hermitian system, i.e., two arbitrary non-orthogonal quantum states can be discriminated by a suitable P-pseudo-Hermitian Hamiltonian. In detail, we decide the minimal angular separation between two non-orthogonal quantum states for a fixed P-pseudo-Hermitian Hamiltonian, and we find the orthogonal evolution time is able to approach zero under suitable conditions, while both the trace distance and the quantum relative entropy are employed to judge their orthogonality. We give a criterion to choose the parameters of a P-pseudo-Hermitian Hamiltonian that evolves the two initial orthogonal states faster than a fixed arbitrary PT-symmetric one with an identical energy difference. Our work expands the NH family for QSD, and can be used to explore real quantum systems in the future.

If the system is known to be in one of two non-orthogonal quantum states,

|\psi_1\rangle

or

|\psi_2\rangle

,

\mathcal{PT}

-symmetric quantum mechanics can discriminate them, \textit{in principle}, by a single measurement. We extend this approach by combining

\mathcal{PT}

-symmetric and Hermitian measurements and show that it's possible to distinguish an arbitrary number of pure quantum states by an appropriate choice of the parameters of

\mathcal{PT}

-symmetric Hamiltonian.

Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem associated with such Hamiltonians have shown that in many cases the entire energy spectrum is real and positive and that the eigenfunctions form an orthogonal and complete basis. Furthermore, the quantum theories determined by such Hamiltonians have been shown to be consistent in the sense that the probabilities are positive and the dynamical trajectories are unitary. However, the geometrical structures that underlie quantum theories formulated in terms of such Hamiltonians have hitherto not been fully understood. This paper studies in detail the geometric properties of a Hilbert space endowed with a parity structure and analyses the characteristics of a PT-symmetric Hamiltonian and its eigenstates. A canonical relationship between a PT-symmetric operator and a Hermitian operator is established. It is shown that the quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. The indefiniteness of the norm can be circumvented by introducing a symmetry operator C that defines a positive definite inner product by means of a CPT conjugation operation.