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International Journal of Quantum Information

Proof of the orthogonal measurement conjecture for qubit states

Andreas Keil

Published Apr 30, 2024• 2 citations
Publisher

Abstract

The accessible information of general signal states is obtained by performing a generalized measurement. In the case that the signal alphabet consists of two states of a qubit system, it is proved that a von Neumann (orthogonal) measurement is sufficient to reach the accessible information.
DOI
10.1142/s0219749924400082

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We prove a new result limiting the amount of accessible information in a quantum channel. This generalizes Kholevo's theorem and implies it as a simple corollary. Our proof uses the strong subadditivity of the von Neumann entropy functional $S(\ensuremath{\rho})$ and a specific physical analysis of the measurement process. The result presented here has application in information obtained from ``weak'' measurements, such as those sometimes considered in quantum cryptography.
Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .
We have proposed a quantum system with equally-distant partially-entangled alphabet which has the minimal mutual overlap and the highly distinguishability, these quantum are used as the signal states of the quantum communication. We have also constructed the positive operator-valued measure for these signal states and discussed their entanglement properties and measurement of entanglement. We calculate the accessible information for these alphabet and show that the accessible information is closely related to the entanglement of the signal states: the higher the entanglement of the signal states, the better the accessible information of the quantum system, and the accessible information reaches its maximal value when the alphabet have their maximal entanglement.
We prove Shor's conjecture: an orthogonal measurement maximizes the information obtainable from any two quantum states in finite dimension. We then show that an error constraint can require three outcomes. For a class of full-rank qubit pairs, the best orthogonal basis switches abruptly as the error penalty increases. Near the switch, three outcomes yield more information at the same error than any orthogonal measurement or random mixture of them. The extra outcome distinguishes confidence levels in one guess; every outcome is used.