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arXiv:quant-ph/0406169 · Physical Review A

Optimal measurements and fixed-point maps

Chirag Dhara, N. D. Hari Dass

Published Aug 12, 2005• 0 citations
arXivHTMLPDF

Abstract

When an optimal measurement of $({S}_{x},{S}_{y},{S}_{z})$ 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-$1∕2$ systems it is also shown explicitly that nonorthogonal measurements fail to give such a linear relation no matter how the ensembles are mixed.

DOI
10.1103/physreva.72.024102

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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.
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.
The structure of the set of density matrices, its linear transformations, generalized linear measurements, and entanglement are studied. The set of density matrices is shown to be a convex and stratified set with simplex and group symmetries. Generalized measurements for density matrices are shown to be reducible to one unitary transformation and one von Neumann measurement carried out with an ancillary system of fixed size. Linear maps of density matrices are considered and the volume of the set of maps is derived. Positive but not completely positive maps are studied in consideration of obtaining a test for entanglement in density matrices. Using the Jamiolkowski representation and Schmidt decomposition of the map eigen matrices, several properties of these maps are shown. An algebraic approach to constructing these positive but not completely positive maps is partially formulated. The positivity of the linear map describing the evolution of an open system and its dependence on the initialized to a zero-discord state, the evolution is shown to be given by a completely positive map. In quantum process tomography, the results obtained from a open system that is initially prepared using von Neumann measurements is shown to be described by a bi-linear map, not a linear map. A method for quantum process tomography is derived for qubit bi-linear maps. The difference between preparing states for an experiment by measurement and by stochastic process is analyzed, and it is shown that the two different methods will give fundamentally different outcomes.
In standard optical tomographic methods, the off-diagonal elements of a density matrix ρ are measured indirectly. Thus, the reconstruction of ρ, even if it is based on linear inversion, typically magnifies small errors in the experimental data. Recently, an optimal tomography solution measuring all the elements of ρ one-by-one without error magnification has been theoretically proposed. We implemented this method for two-qubit polarization states. For comparison, we also experimentally implemented other well-known tomographic protocols, either based solely on local measurements (of, e.g., the Pauli operators and James-Kwiat-Munro-White projectors) or with mutually unbiased bases requiring both local and global measurements. We reconstructed seventeen separable, partially and maximally entangled two-qubit polarization states. Our experiments show that our method has the highest stability against errors in comparison to other quantum tomographies. In particular, we demonstrate that each optimally-reconstructed state is embedded in an uncertainty circle of the smallest radius, both in terms of trace distance and disturbance. We explain how to experimentally estimate uncertainty radii for all the implemented tomographies and show that, for each reconstructed state, the relevant uncertainty circles intersect indicating the approximate location of the corresponding physical density matrix.
We propose an efficient scheme for qutrit state tomography based on optimal unambiguous discrimination of three nonorthogonal pure qutrit states. In our scheme, we first introduce an ancillary qubit initially prepared in a known state and apply a joint unitary transformation on the whole system composed of a qutrit state and an ancillary qubit state. Afterwards, we make a local von Neumann measurement (VNM) on the ancillary qubit. If it is projected onto the state |0〉, we finally perform mutually unbiased measurements (MUMs) on the collapsed qutrit state arbitrarily chosen from a complete set of four mutually unbiased bases (MUBs). With only six properly-chosen projectors, the real and imaginary parts of non-diagonal elements can be entirely obtained. Together with the diagonal elements obtained directly from one VNM in the computational bases, the density matrix of an arbitrary unknown qutrit state can be fully determined. Compared with the previous work (R. Salazar and A. Delgado 2012), our scheme consumes much less resources. In comparison with MUBs-based tomography scheme (H. Yuan, et al., 2016), our scheme requires more measurements and less projectors, but pays the price of a loss of certain fraction identically prepared states. Our proposal can be extended to the case of other d-dimensional quantum states if their d MUBs are exactly known.