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arXiv:1008.3858 · Physical Review A

Probing light polarization with the quantum Chernoff bound

Iulia Ghiu, Gunnar G. Bjork, Paulina Marian, Tudor A. Marian

Published Aug 6, 2010• 17 citations
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Abstract

We recall the framework of a consistent quantum description of polarization of light. Accordingly, the degree of polarization of a two-mode state $\mathrm{\ensuremath{\rho}\ifmmode \hat{}\else \^{}\fi{}}$ of the quantum radiation field can be defined as a distance of a related state ${\mathrm{\ensuremath{\rho}\ifmmode \hat{}\else \^{}\fi{}}}_{b}$ to the convex set of all SU(2)-invariant two-mode states. We explore a distance-type polarization measure in terms of the quantum Chernoff bound and derive its explicit expression. A comparison between the Chernoff and Bures degrees of polarization leads to interesting conclusions for some particular states chosen as illustrative examples.

DOI
10.1103/physreva.82.023803

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We recall the framework of a consistent quantum description of polarization of light. Accordingly, the degree of polarization of a two-mode state \ensuremathρ\ifmmode^\elseˆ\fi\mathrm{\ensuremath{\rho}\ifmmode \hat{}\else \^{}\fi{}}\ensuremathρ\ifmmode^\elseˆ\fi of the quantum radiation field can be defined as a distance of a related state \ensuremathρ\ifmmode^\elseˆ\fib{\mathrm{\ensuremath{\rho}\ifmmode \hat{}\else \^{}\fi{}}}_{b}\ensuremathρ\ifmmode^\elseˆ\fib​ to the convex set of all SU(2)-invariant two-mode states. We explore a distance-type polarization measure in terms of the quantum Chernoff bound and derive its explicit expression. A comparison between the Chernoff and Bures degrees of polarization leads to interesting conclusions for some particular states chosen as illustrative examples.
It is well established that unpolarized light is invariant with respect to any SU(2) polarization transformation. This requirement fully characterizes the set of density matrices representing unpolarized states. We introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We use two different candidates of distance, namely the Hilbert-Schmidt and the Bures metric, showing that they induce fundamentally different degrees of polarization. We apply these notions to relevant field states and we demonstrate that they avoid some of the problems arising with the classical definition.
The degree of polarization of a quantum field can be defined as its distance to an appropriate set of states. When we take unpolarized states as this reference set, the states optimizing this degree for a fixed average number of photons N‾\overline{N}N present a fairly symmetric, parabolic photon statistic, with a variance scaling as N‾2{\overline{N}}^{2}N2. Although no standard optical process yields such a statistic, we show that, to an excellent approximation, a highly squeezed vacuum can be taken as maximally polarized. We also consider the distance of a field to the set of its SU(2) transformed, finding that certain linear superpositions of SU(2) coherent states make this degree to be unity.
We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.