dorsal/arxiv
View SchemaHudson's Theorem for finite-dimensional quantum systems
| Authors | D. Gross |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0602001 |
| URL | https://arxiv.org/abs/quant-ph/0602001 |
| DOI | 10.1063/1.2393152 |
| Journal | J. Math. Phys. 47, 122107 (2006) |
Abstract
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 counter-example. 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.
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"abstract": "We show that, on a Hilbert space of odd dimension, the only pure states to\npossess a non-negative Wigner function are stabilizer states. The Clifford\ngroup is identified as the set of unitary operations which preserve positivity.\nThe result can be seen as a discrete version of Hudson\u0027s Theorem. Hudson\nestablished that for continuous variable systems, the Wigner function of a pure\nstate has no negative values if and only if the state is Gaussian. Turning to\nmixed states, it might be surmised that only convex combinations of stabilizer\nstates give rise to non-negative Wigner distributions. We refute this\nconjecture by means of a counter-example. Further, we give an axiomatic\ncharacterization which completely fixes the definition of the Wigner function\nand compare two approaches to stabilizer states for Hilbert spaces of\nprime-power dimensions. In the course of the discussion, we derive explicit\nformulas for the number of stabilizer codes defined on such systems.",
"arxiv_id": "quant-ph/0602001",
"authors": [
"D. Gross"
],
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"doi": "10.1063/1.2393152",
"journal_ref": "J. Math. Phys. 47, 122107 (2006)",
"title": "Hudson\u0027s Theorem for finite-dimensional quantum systems",
"url": "https://arxiv.org/abs/quant-ph/0602001"
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