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Lobachevskii Journal of Mathematics

On Restoring a Quantum State after the Measurement

A. O. Alekseev, Grigori Gennadievich Amosov

Published Jun 1, 2023• 1 citation
Publisher

Abstract

Abstract Earlier we built POVM, which is absolutely continuous relative to the scalar measure with a projector-valued density, and studied the quantum channel generated by it. In this paper, we consider the possibility of restoring states measured by the Davis–Lewis instrument. An experiment with two outcomes is considered, and we compare two cases: with post-selection and without post-selection. We study how states are transformed in these cases and whether it is possible to restore them.

DOI
10.1134/s1995080223060069

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Abstract Earlier we built POVM, which is absolutely continuous relative to the scalar measure with a projector-valued density, and studied the quantum channel generated by it. In this paper, we consider the possibility of restoring states measured by the Davis–Lewis instrument. An experiment with two outcomes is considered, and we compare two cases: with post-selection and without post-selection. We study how states are transformed in these cases and whether it is possible to restore them.
In this paper, we will discuss the possibility of increasing the Holevo value when measuring with post-selection. By measurement with post-selection we mean that some of the outcomes of the measurement are discarded, and then we work with the resulting states. Using the example of two mixed states, it will be shown that mutual information after performing post-selection is greater than their initial Holevo value. An increase in the Holevo value follows from the monotonicity property of the quantum relative entropy.
Abstract A quantum measurement (e.g., positive-operator valued measure, or more generally, instrument) performed on a system state yields a post-measurement ensemble (a family of outcome states together with the corresponding outcome probabilities), which may have quantumness due to the impossibility to perfectly discriminate the constituent states. In order to characterize information contents and structure of a measurement, it is desirable to investigate the post-measurement ensemble. In this work, we study post-measurement quantumness (i.e., quantumness of the post-measurement ensemble) by introducing a Gram matrix summarizing the interaction between a measurement and a state, and explore its implications and consequences. Gram matrix of an ensemble can be regarded as a quantum state (density operator) in an auxiliary system with a computational basis labelling the measurement outcomes. Our basic idea is to exploit coherence of the Gram matrix (with respect to the computational basis of the auxiliary system) and employ it to quantify post-measurement quantumness . We propose several quantifiers of quantumness and reveal their basic properties. We further evaluate explicitly quantumness of measurements arising from several important quantum channels, and illustrate a comparison between them. This provides a method for investigating measurements from the perspective of coherence of Gram matrix.
The standard approach to quantum measurement discrimination is to perform the given unknown measurement on a probe state, possibly entangled with an auxiliary system, and make a decision based on the measurement outcome obtained. In this work, we go beyond the standard aforementioned scenarios by consider not only the classical measurement outcome of a measurement, but also its the post-measurement quantum state. More specifically, instead of considering only the positive-operator valued measure (POVM) operators, we consider their associated Lüders' instrument associated with them. We prove that, when the post-measurement quantum states are available, the task of discriminating two qubit projective measurements is equivalent to discriminating two copies of quantum states associated to each projector pair, extending previous results known for the case where probe states are separable. Then, we proceed by showing that the advantage of considering post-measurement states in measurement discrimination can be large. We formalise this claim by presenting a family of pairs of measurements where the ratio between the discrimination bias of the measurement discrimination task with and without post-measurement states can be arbitrarily large. This shows that, while the post-measurement state was neglected in most of the previous literature, its use can significantly improve the performance of quantum measurement discrimination.
Studying sequential measurements is of the utmost importance to both the foundational aspects of quantum theory and the practical implementations of quantum technologies, with both of these applications being abstractly described by the concatenation of quantum instruments into a sequence of a certain length. In general, the choice of instrument at any given step in the sequence can be conditionally chosen based on the classical results of all preceding instruments. For two instruments in a sequence we consider the conditional second instrument as an effective way of postprocessing the first instrument into a new one. This is similar to how a measurement described by a positive operator-valued measure (POVM) can be postprocessed into another by way of classical randomization of its outcomes using a stochastic matrix. In this work we study the postprocessing relation of instruments and the partial order it induces on their equivalence classes. We characterize the greatest and the least element of this order, give examples of postprocessings between different types of instruments, and draw connections between postprocessings of some of these instruments and their induced POVMs.
The problem of constructing a measure on a discrete set X taking values in a positive cone of bounded operators in a Hilbert space is considered. It is assumed that a projectionvalued function defined on a subset of X0 of the original set X is initially given. The aim of the study is to find such a scalar measure μ on the set X and the continuation of a projector-valued function from X0 to X, which results in an operator-valued measure having a projector-valued density relative to μ. In general, the problem is solved for |X| = 4 and |X0| = 2. As an example, we consider a function on X0 that takes values in a set of projections on coherent states. For this case, the question of the information completeness of the measurement determined by the constructed measure is investigated. In other words, is it possible to reconstruct a quantum state (a positive unit trace operator) from the values of the matrix trace from the product of a measure with a quantum state. It is shown that for the constructed measure it is possible to restore the quantum state only if it is a projection. A restriction on the probability distribution is also found, at which it can be obtained as a result of measuring a certain quantum state.
It is important problem to clarify the class of implementable quantum measurements from both fundamental and applicable viewpoints. Positive-Operator-Valued Measure (POVM) measurements are implementable by the indirect measurement methods, and the class is the largest class determined by the mathematical structure of Hilbert space. However, if we assume probabilistic consistency in our operations instead of the structure of Hilbert space, we can deal with Non-Positive-Operator-Valued Measure (N-POVM) measurements in the framework of General Probabilistic Theories (GPTs). N-POVM measurements are not considered as implementable, but this paper gives a constructive way to implement N-POVM measurements by POVM measurements and post-selection in quantum theory when we restrict the domain of target states. Besides, we show that a post-selected POVM measurement is regarded as an N-POVM measurement in a restricted domain. These results provide a new relationship between N-POVM measurements in GPTs and post-selection.