On Quantum Channel Learning

• URL: http://arxiv.org/abs/2407.04406v1
• Date: Fri, 5 Jul 2024 10:43:24 GMT
• Title: On Quantum Channel Learning
• Authors: Mikhail Gennadievich Belov, Victor Victorovich Dubov, Alexey Vladimirovich Filimonov, Vladislav Gennadievich Malyshkin,
• Abstract summary: It is shown that a quadratic on $mathcalU$ fidelity can be constructed by considering $sqrtrho(l) to sqrtvarrho(l)$ mapping, and on a general quantum channel of Kraus rank $N_s$. The approach can be applied to study decoherence effects, spontaneous coherence, synchronizing, etc.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $\rho^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} \omega^{(l)} F\left(\varrho^{(l)},\sum_s B_s \rho^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,\sigma)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed to find the global maximum. The result comprises in $N_s$ operators $B_s$ that collectively form an $IN$ to $OUT$ quantum channel $A^{OUT}=\sum_s B_s A^{IN} B_s^{\dagger}$. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a general quantum channel. This marks a crucial advancement from the commonly studied unitary mapping of pure states $\phi_l=\mathcal{U} \psi_l$ to a general quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} \rho^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{\rho^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a general quantum channel of Kraus rank $N_s$, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then built as a hierarchy of unitary mappings. The approach can be applied to study decoherence effects, spontaneous coherence, synchronizing, etc.

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