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arXiv:1804.10792

Optimal weak measurements: Mixed States

N. D. Hari Dass, R. Rajath Krishna

Published Apr 28, 2018• 0 citations
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Abstract

In an earlier publication we had given an exhaustive analysis of the criteria for weak value measurements of pure states to be optimal in the sense considered by Wootters and Fields. We had proved, for arbitrary spin cases, that the measurements are optimal when the post-selected state is mutually unbiased wrt the eigenstates of the observable being measured.Here we extend the discussion to mixed states. For these, weak value measurements have several problems which we illustrate with the protocol proposed by Shengjun Wu. We discuss tomography of mixed states based on weak measurements and show that while the principal results of Wootters and Fields hold, namely, the set of observables needed for complete tomography are such that their eigenstates form a mutually unbiased bases, weak tomography removes a serious lacuna from the Wootters and Fields analysis i.e the need to consider only state averaged error volumes or information. We also consider another proposal for weak tomography of mixed states by Lundeen and Bamber, and reach similar conclusions about MUB.

DOI
10.48550/arxiv.1804.10792

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In an earlier publication we had given an exhaustive analysis of the criteria for weak value measurements of pure states to be optimal in the sense considered by Wootters and Fields. We had proved, for arbitrary spin cases, that the measurements are optimal when the post-selected state is mutually unbiased wrt the eigenstates of the observable being measured.Here we extend the discussion to mixed states. For these, weak value measurements have several problems which we illustrate with the protocol proposed by Shengjun Wu. We discuss tomography of mixed states based on weak measurements and show that while the principal results of Wootters and Fields hold, namely, the set of observables needed for complete tomography are such that their eigenstates form a mutually unbiased bases, weak tomography removes a serious lacuna from the Wootters and Fields analysis i.e the need to consider only state averaged error volumes or information. We also consider another proposal for weak tomography of mixed states by Lundeen and Bamber, and reach similar conclusions about MUB.
We apply the notion of \emph{optimality} of measurements for state determination(tomography) as originally given by Wootters and Fields to \emph{weak value tomography} of \emph{pure states}. They defined measurements to be optimal if they 'minimised' the effects of statistical errors. For technical reasons they actually maximised the state averaged information, precisely quantified as the negative logarithm of 'error volume'. In this paper we optimise both the state averaged information as well as error volumes. We prove, for Hilbert spaces of arbitrary (finite) dimensionality, that varieties of weak value measurements are optimal when the post-selected bases are \emph{mutually unbiased} with respect to the eigenvectors of the observable being measured. We prove a number of important results about the geometry of state spaces when expressed through the weak values as coordinates. We derive an expression for the Kaëhler potential for the N-dimensional case with the help of which we give an exact treatment of the arbitrary-spin case.
Three recent results on weak measurements are presented. They are: i) repeated measurements on a single copy can not provide any information on it and further, that in the limit of very large such measurements, weak measurements have exactly the same characterstics as strong measurements, ii) the apparent non-invasiveness of weak measurements is \emph{illusory} and they are no more advantageous than strong measurements even in the specific context of establishing Leggett-Garg inequalities, when errors are properly taken into account, and, finally, iii) weak value measurements are optimal, in the precise sense of Wootters and Fields, when the post-selected states are mutually unbiased with respect to the eigenstates of the observable whose weak values are being measured. Notion of weak value coordinates for state spaces are introduced and elaborated.
We study the determination of an unknown mixed state of a ddd-dimensional quantum system by means of unambiguous state discrimination. We show that optimal and nonoptimal unambiguous state discrimination can be used to reconstruct unknown states of a qubit. This result is extended to the case of a qudit by a sequence of reconstructions in two-dimensional subspaces. The total number of projections scales approximately as 2d22{d}^{2}2d2 for ddd large; this is twice as much as in the case of tomography based on mutually unbiased bases or symmetric informationally complete positive-operator-valued measures, and less than the d3{d}^{3}d3 projections required by standard tomography.
When an optimal measurement of (Sx,Sy,Sz)({S}_{x},{S}_{y},{S}_{z})(Sx​,Sy​,Sz​) is made on a qubit and what we call a mutually unbiased mixture of the resulting ensembles is taken, then the post-measurement density matrix is shown to be related to the premeasurement density matrix through a simple linear relation. It is also shown that the form of this relation is the same for all quantum systems. It is shown that for a general quantum system such a relation holds only when the measurements are made in mutually unbiased bases. The post-measurement density matrix is shown to be a normalized incoherent superposition of the identity map and the pin map of Gorini and Sudarshan. A pin map is one which maps all density matrices to a fixed density matrix which in our case turns out to be the unit matrix. The result is shown to be true irrespective of whether the initial state is pure or mixed. For spin-1∕21∕21∕2 systems it is also shown explicitly that nonorthogonal measurements fail to give such a linear relation no matter how the ensembles are mixed.