Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel
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| Format: | Preprint |
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2025
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| _version_ | 1866913950108483584 |
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| author | Calixto, Manuel Guerrero, Julio |
| author_facet | Calixto, Manuel Guerrero, Julio |
| contents | For a symmetric $N$-quDit system described by a density matrix $ρ$, we construct a one-parameter $s$ family $\mathcal{F}^{(s)}_ρ$ of quasi-probability distributions through generalized Fano multipole operators and Stratonovich-Weyl kernels. The corresponding phase space is the complex projective ${C}P^{D-1}=U(D)/U(D-1)\times U(1)$, related to fully symmetric irreducible representations of the unitary group $U(D)$. For the particular cases $D=2$ (qubits) and $D=3$ (qutrits), we analyze the phase-space structure of Schrödinger $U(D)$-spin cat (parity adapted coherent) states and we provide plots of the corresponding Wigner $\mathcal{F}^{(0)}_ρ$ function. We examine the connection between non-classical behavior and the negativity of the Wigner function. We also compute the generalized heat kernel relating two quasi-probability distributions $\mathcal{F}^{(s)}_ρ$ and $\mathcal{F}^{(s')}_ρ$, with $t=(s'-s)/2$ playing the role of ``time'', together with their twisted Moyal product in terms of a trikernel. In the thermodynamic limit $N\to\infty$, we recover the usual Gaussian smoothing for $s'>s$. A diagramatic interpretation of the phase-space construction in terms of Young tableaux is also provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_14866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel Calixto, Manuel Guerrero, Julio Quantum Physics Mathematical Physics For a symmetric $N$-quDit system described by a density matrix $ρ$, we construct a one-parameter $s$ family $\mathcal{F}^{(s)}_ρ$ of quasi-probability distributions through generalized Fano multipole operators and Stratonovich-Weyl kernels. The corresponding phase space is the complex projective ${C}P^{D-1}=U(D)/U(D-1)\times U(1)$, related to fully symmetric irreducible representations of the unitary group $U(D)$. For the particular cases $D=2$ (qubits) and $D=3$ (qutrits), we analyze the phase-space structure of Schrödinger $U(D)$-spin cat (parity adapted coherent) states and we provide plots of the corresponding Wigner $\mathcal{F}^{(0)}_ρ$ function. We examine the connection between non-classical behavior and the negativity of the Wigner function. We also compute the generalized heat kernel relating two quasi-probability distributions $\mathcal{F}^{(s)}_ρ$ and $\mathcal{F}^{(s')}_ρ$, with $t=(s'-s)/2$ playing the role of ``time'', together with their twisted Moyal product in terms of a trikernel. In the thermodynamic limit $N\to\infty$, we recover the usual Gaussian smoothing for $s'>s$. A diagramatic interpretation of the phase-space construction in terms of Young tableaux is also provided. |
| title | Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2507.14866 |