Interpreting symplectic linear transformations in a two-qubit phase space

• URL: http://arxiv.org/abs/2402.09922v4
• Date: Thu, 21 Mar 2024 17:20:16 GMT
• Title: Interpreting symplectic linear transformations in a two-qubit phase space
• Authors: William K. Wootters,
• Abstract summary: For certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. A symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a $d \times d$ phase space when the Hilbert-space dimension $d$ is odd. It does not hold for a $d \times d$ phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.

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