Hudson's theorem for finite-dimensional quantum systems

By David Gross. 2006

422 citations recorded by the source corpus.

We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/ed80e67433f8424854064e389b0b7c0716ef9f51

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Citations in the corpus

Many-Body Quantum Magic (2020, background): “It is known that pure stabilizer states are precisely those pure states that have non-negative Wigner functions (discrete Hudson’s theorem) [16].”
Negative quasi-probability as a resource for quantum computation (2012, methodology): “An explicit example of such a state for the qutrit is given by Gross (2006).”
Geometry of Quantum States: Index (2007, methodology): “8 For more details see Vourdas [942], Gross [366], and Appleby [41].”
Contextuality supplies the ‘magic’ for quantum computation (2014, neutral): “By the results of [24, 30], the set of states PSIM coincides exactly with the set of states that are nonnega- tively represented within a distinguished quasiprobability representation—a discrete Wigner function (DWF) [31, 32].”
Quantum resource theories (2018, background): “The proof of this result involves connecting the task of magic state distillation to the discrete Wigner function of a quantum state (Gross, 2006; Wootters, 1987).”
Corrigendum: Negative quasi-probability as a resource for quantum computation (2012, background): “This first step is easy given that such a representation already exists; this distinguished representation is the discrete Wigner function picked out by Gross [25, 26] from the broad class defined by Gibbons et al [22].”
The resource theory of stabilizer quantum computation (2013, methodology): “In section §IV, we will need the discrete Wigner function[22, 50], which is defined for quantum systems with finite, odd Hilbert space dimension.”
Wigner Function Negativity and Contextuality in Quantum Computation on Rebits (2014, background): “Properties of the qudit Wigner function It can be shown [14] that the Wigner function has the following properties”
Stim: a fast stabilizer circuit simulator (2021, background): “Second, describing an arbitrary n-qubit stabilizer state takes at least (1)2n 2 bits [14, 22, 18].”
Quasi-probability representations of quantum theory with applications to quantum information science (2010, background): “…on discrete phase spaces which are not reviewed here (Hannay and Berry 1980, Cohen and Scully 1986, Feynman 1987, Galetti and de Toledo Piza 1988, Opatrný et al 1996, Hakioglu 1998, Luis and Perina 1998, Rivas 1999, Mukunda 2004, Vourdas 2004, Chaturvedi et al 2005, 2006, Gross 2006, 2007).”
Recovering Low-Rank Matrices From Few Coefficients in Any Basis (2009, background): “[15], [16], [17]; for a brief introduction see [18]).”