Discrete phase space based on finite fields

By K. Gibbons, M. J. Hoffman, W. Wootters. 2004

381 citations recorded by the source corpus.

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 2Nx2N discrete phase space for a system with N 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 N elements. There exists such a field if and only if N 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 NxN phase space also leads naturally to a method of constructing a complete set of N+1 mutually unbiased bases for the state space.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/fd1b5e4059f1b9a7c537086725221f61752ac495

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Citations in the corpus

Quasi-probability representations of quantum theory with applications to quantum information science (2010, methodology): “The construction presented in Gibbons et al (2004) is physically significant for a system of n objects (called particles) having a p-dimensional Hilbert space.”
Symmetric informationally complete–positive operator valued measures and the extended Clifford group (2004, neutral): “ar X iv :q ua nt -p h/ 04 12 00 1v 2 1 7 Ja n 20 05 SIC-POVMs AND THE EXTENDED CLIFFORD GROUP”
On SIC-POVMs and MUBs in Dimension 6 (2004, background): “, [4]), the link between SIC-POVMs and finite affine planes is on a more abstract level.”
Corrigendum: Negative quasi-probability as a resource for quantum computation (2012, background): “Conveniently, for quantum systems with odd Hilbert space dimension such a representation is already known to exist: this is the discrete Wigner function picked out by Gross [25, 26] from the broad class defined by Gibbons et al [22].”
Wigner Function Negativity and Contextuality in Quantum Computation on Rebits (2014, neutral): “[12], to define many different Wigner functions.”
The resource theory of stabilizer quantum computation (2013, background): “We are now ready to state a few salient facts about the discrete Wigner representation[15, 22]:”