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arXiv:quant-ph/0011037 · Journal of Optics B Quantum and Semiclassical Optics

An expectation value expansion of Hermitian operators in a discrete Hilbert space

Roberth Asplund, Gunnar G. Bjork, Mohamed Bourennane

Published May 14, 2001• 11 citations
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Abstract

We discuss a real-valued expansion of any Hermitian operator defined in a Hilbert space of finite dimension N , where N is a prime number, or an integer power of a prime. The expansion has a direct interpretation in terms of the operator expectation values for a set of complementary bases. The expansion can be said to be the complement of the discrete Wigner function. We expect the expansion to be of use in quantum information applications since qubits typically are represented by a discrete, and finite-dimensional, physical system of dimension N = 2 p , where p is the number of qubits involved. As a particular example we use the expansion to prove that an intermediate measurement basis (a Breidbart basis) cannot be found if the Hilbert space dimension is three or four.

DOI
10.1088/1464-4266/3/3/314

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We discuss a real-valued expansion of any Hermitian operator defined in a Hilbert space of finite dimension N , where N is a prime number, or an integer power of a prime. The expansion has a direct interpretation in terms of the operator expectation values for a set of complementary bases. The expansion can be said to be the complement of the discrete Wigner function. We expect the expansion to be of use in quantum information applications since qubits typically are represented by a discrete, and finite-dimensional, physical system of dimension N = 2 p , where p is the number of qubits involved. As a particular example we use the expansion to prove that an intermediate measurement basis (a Breidbart basis) cannot be found if the Hilbert space dimension is three or four.
The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a $2N\ifmmode\times\else\texttimes\fi{}2N$ discrete phase space for a system with NNN orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having NNN elements. There exists such a field if and only if NNN is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our $N\ifmmode\times\else\texttimes\fi{}N$ phase space also leads naturally to a method of constructing a complete set of N+1N+1N+1 mutually unbiased bases for the state space.
We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality NNN. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator PPP and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.
We derive a direct reconstruction algorithm for the discrete Wigner function through different types of measurements. For a system described in a Hilbert space of dimension N=N1…Np,{N=N}_{1}\dots{}{N}_{p},N=N1​…Np​, where the numbers Ni{N}_{i}Ni​ are prime, the reconstruction is accomplished with (N1+1)…(Np+1){(N}_{1}+1)\dots{}{(N}_{p}+1)(N1​+1)…(Np​+1) factorable (local) von Neumann measurements. For the special case where the dimension is a power of a prime, the reconstruction can be performed in a much more efficient way using N+1N+1N+1 complementary measurements. If the system is composed of a number of smaller subsystems, these measurements will then in general be nonseparable.
Mutually unbiased bases and discrete Wigner functions are closely but not uniquely related. Such a connection becomes more interesting when the Hilbert space has a dimension that is a power of a prime N=dn, which describes a composite system of n qudits. Hence, entanglement naturally enters the picture. Although our results are general, we concentrate on the simplest nontrivial example of dimension N=8=23. It is shown that the number of fundamentally different Wigner functions is severely limited if one simultaneously imposes translational covariance and that the generating operators consist of rotations around two orthogonal axes, acting on the individual qubits only.
We study the class of discrete Wigner functions proposed by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to describe quantum states using a discrete phase space based on finite fields. We find the extrema of such functions for small Hilbert-space dimensions and present a quantum-information application: a construction of quantum random-access codes. These are constructed using the complete set of phase-space point operators to find encoding states and to obtain the codes' average success rates for Hilbert-space dimensions 2, 3, 4, 5, 7, and 8.
A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.