Coherifying quantum channels

Abstract

Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel $\Phi$, i.e., we look for channels ${\Phi }^{\mathcal{C}}$ that induce the same classical transitions $T$, but are "more coherent". To quantify the coherence of a channel $\Phi$ we measure the coherence of the corresponding Jamiołkowski state $J_{\Phi }$. We show that the classical transition matrix $T$ can be coherified to reversible unitary dynamics if and only if $T$ is unistochastic. Otherwise the Jamiołkowski state $J_{\Phi }^{\mathcal{C}}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process ${\Phi }^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy and purity of $J_{\Phi }^{\mathcal{C}}$, and relate the latter to the unitarity of ${\Phi }^{\mathcal{C}}$. We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel $\Phi$ and reduces its rank (the minimal number of required Kraus operators) from $d^2$ to $d$.

20 pages, 8 figures. Published version

References in corpus (6)

• Description of quantum coherence in thermodynamic processes requires constraints beyond free energy
• Quantum coherence, time-translation symmetry and thermodynamics
• Random Quantum Operations
• Quartic quantum theory: an extension of the standard quantum mechanics
• Diagonal unitary entangling gates and contradiagonal quantum states
• Physical Purification of Quantum States

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