aiqbench analysis
This paper provides a rigorous classification of unitary dynamics that preserve the positivity of Kirkwood-Dirac quasiprobability distributions, linking these structures to classical simulation limits. The authors identify three distinct gate types and demonstrate that while real inputs remain efficiently simulable under certain conditions, non-real inputs can incur exponential sampling overhead for specific dimensions. This work clarifies the computational boundaries of KD-based quantum information protocols and highlights fundamental constraints on resource theories involving total non-positivity.
Classification of Kirkwood-Dirac positivity-preserving unitaries
arXiv:2502.11784v4 Announce Type: replace Abstract: Kirkwood-Dirac (KD) quasiprobability distributions are increasingly used across quantum information science, yet their computational significance remains unclear. We classify unitary dynamics that preserve KD positivity and connect this structure to classical simulation. In contrast to the discrete Wigner setting, we show that KD positivity preservation, stochastic evolution of quasiprobabilities, and preservation of total non-positivity do not coincide. We identify three classes of positivity-preserving unitaries: type I gates, which are exactly the KD-stochastic unitaries; type II gates, which are non-stochastic yet preserve total non-positivity by permuting and conjugating KD entries; and, for Fourier-conjugate bases in dimension $d=pq$, type III gates, which preserve KD positivity but can change the total non-positivity of non-real distributions. The classification is complete for Haar-random bases and for Fourier-conjugate dimensions $d=p^k$ and $d=pq$, with $p, q$ distinct primes. Adapting the sampling algorithm of Pashayan et al. [PRL 115, 070501], we simulate all of these positivity-preserving circuits efficiently on KD-positive inputs; type III gates, however, create a sharp distinction between real inputs, which remain efficiently simulable, and non-real inputs, whose sampling overhead can grow exponentially. Consequently, for $d=pq$ no resource theory can both treat the KD total non-positivity as a monotone and admit every efficiently simulable positivity-preserving unitary as a free operation.
Abstract:Kirkwood-Dirac (KD) quasiprobability distributions are increasingly used across quantum information science, yet their computational significance remains unclear. We classify unitary dynamics that preserve KD positivity and connect this structure to classical simulation. In contrast to the discrete Wigner setting, we show that KD positivity preservation, stochastic evolution of quasiprobabilities, and preservation of total non-positivity do not coincide. We identify three classes of positivity-preserving unitaries: type I gates, which are exactly the KD-stochastic unitaries; type II gates, which are non-stochastic yet preserve total non-positivity by permuting and conjugating KD entries; and, for Fourier-conjugate bases in dimension $d=pq$, type III gates, which preserve KD positivity but can change the total non-positivity of non-real distributions. The classification is complete for Haar-random bases and for Fourier-conjugate dimensions $d=p^k$ and $d=pq$, with $p, q$ distinct primes. Adapting the sampling algorithm of Pashayan et al. [PRL 115, 070501], we simulate all of these positivity-preserving circuits efficiently on KD-positive inputs; type III gates, however, create a sharp distinction between real inputs, which remain efficiently simulable, and non-real inputs, whose sampling overhead can grow exponentially. Consequently, for $d=pq$ no resource theory can both treat the KD total non-positivity as a monotone and admit every efficiently simulable positivity-preserving unitary as a free operation.
arXiv:2502.11784 [quant-ph]
(or arXiv:2502.11784v4 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2502.11784 Focus to learn more arXiv-issued DOI via DataCite
Phys. Rev. A 114, 042446 (2026)
https://doi.org/10.1103/rsy9-12fx Focus to learn more DOI(s) linking to related resources
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