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arXiv:quant-ph/0407054 · Journal of Mathematical Physics

Implementation of group-covariant positive operator valued measures by orthogonal measurements

Thomas Decker, Dominik Janzing, Martin Rötteler

Published Dec 24, 2004• 20 citations
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Abstract

We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find a quantum circuit implementation of a given group-covariant POVM which uses the symmetry of the POVM. Based on representation theory of the symmetry group we develop a general approach for the implementation of group-covariant POVMs which consist of rank-one operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl–Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.

DOI
10.1063/1.1827924

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We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find a quantum circuit implementation of a given group-covariant POVM which uses the symmetry of the POVM. Based on representation theory of the symmetry group we develop a general approach for the implementation of group-covariant POVMs which consist of rank-one operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl–Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.
In a previous paper we have presented a general scheme for the implementation of symmetric generalized measurements (POVMs) on a quantum computer. This scheme is based on representation theory of groups and methods to decompose matrices that intertwine two representations. We extend this scheme in such a way that the measurement is minimally disturbing, i.e., it changes the state vector ∣Ψ⟩ of the system to Π∣Ψ⟩ where Π is the positive operator corresponding to the measured result. Using this method, we construct quantum circuits for measurements with Heisenberg–Weyl symmetry. A continuous generalization leads to a scheme for optimal simultaneous measurements of position and momentum of a Schrödinger particle moving in one dimension such that the outcomes satisfy ΔxΔp⩾ℏ. The particle to be measured collides with two probe particles, one for the position and the other for the momentum measurement. The position and momentum resolution can be tuned by the entangled joint state of the probe particles which is also generated by a collision with hard-core potential. The parameters of the POVM can then be controlled by the initial widths of the wave functions of the probe particles. We point out some formal similarities and differences to simultaneous measurements of quadrature amplitudes in quantum optics.
In the quantum theory of measurement, the positive operator-valued measure (POVM) is an important concept, and its implementation can be useful. A POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and mention of the history of POVMs in quantum theory, a particular implementation of a POVM for use in the measurement of nonorthogonal photon polarization states is reviewed. The implementation consists simply of a Wollaston prism, a mirror, two beam splitters, a polarization rotator and three phototubes arranged in an interferometric configuration, and it is shown analytically that the device faithfully represents the POVM. Based on Neumark's extension theorem, the two-dimensional Hilbert space of the POVM implementation can be embedded in the three-dimensional Hilbert space of an ordinary projective-valued measure. Also, analytical expressions are given for the maximum Renyi information loss from the device to a disturbing probe, and for the error and inconclusive rates induced by the probe. Various aspects of the problem of probe optimization are elaborated.
We study the local implementation of positive-operator-valued measures (POVMs) when we require only the faithful reproduction of the statistics of the measurement outcomes for all initial states. We first demonstrate that any POVM with separable elements can be implemented by a separable superoperator, and develop techniques for calculating the extreme points of POVMs under a certain class of constraint that includes separability and positive partial transposition. As examples we consider measurements that are invariant under various symmetry groups (Werner, isotropic, Bell diagonal, local orthogonal), and demonstrate that in these cases separability of the POVM elements is equivalent to implementability via local operations and classical communication (LOCC). We also calculate the extrema of these classes of measurement under the groups that we consider, and give explicit LOCC protocols for attaining them. These protocols are hence optimal methods for locally discriminating between states of these symmetries. One of many interesting consequences is that the best way to locally discriminate Bell-diagonal mixed states is to perform a two-outcome POVM using local von Neumann projections. This is true regardless of the cost function, the number of states being discriminated, or the prior probabilities. Our results give the first cases of local mixed-state discrimination that can be analyzed quantitatively in full, and may have application to other problems such as demonstrations of nonlocality, experimental entanglement witnesses, and perhaps even entanglement distillation.