![]() We prove that this definition satisfies all of the axioms, recently put forward by Gour, required for a channel entropy function. Motivated by the fact that the entropy of a state ρ can be formulated as the difference of the number of physical qubits and the “relative entropy distance” between ρ and the maximally mixed state, here we define the entropy of a channel N as the difference of the number of physical qubits of the channel output with the “relative entropy distance” between N and the completely depolarizing channel. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Sci.The von Neumann entropy of a quantum state is a central concept in physics and information theory, having a number of compelling physical interpretations. Wigner, E.P., Yanase, M.M.: Information content of distributions. Wehrl, A.: General properties of entropy. Vershynina, A., Carlen, E., Lieb, E.: Strong subadditivity of quantum entropy. Umegaki, H.: Conditional expectation in an operator algebra. Simon, B.: Loewner’s Theorem on Monotone Matrix Functions (Springer, 2019) Ruskai, M.B.: Remarks on Kim’s strong subadditivity matrix inequality: extensions and equality conditions’’. Ruskai, M.B.: Another short and elementary proof of strong subadditivity of quantum entropy. Ruskai, M.B.: “Lieb’s simple proof of concavity of \( \) and remarks on related inequalities” Int. B.: “Inequalities for quantum entropy: a review with conditions for equality” J. Petz, D.: Monotone metrics on matrix spaces. Petz, D.: Quasi-entropies for finite quantum systems. Ohya, M., Petz, D.: Quantum Entropy and Its Use (Springer, 1993) Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information (Cambridge University Press, 2000) ![]() Lieb, E.H., Ruskai, M.B.: Some operator inequalities of the Schwarz type. Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. Lieb, E., Ruskai, M.B.: A fundamental property of quantum-mechanical entropy. ![]() Lieb, E.H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Lesniewski, A., Ruskai, M.B.: Monotone Riemannian metrics and relative entropy on noncommutative probability spaces. Lanford, O., III., Robinson, D.W.: Mean entropy of states in quantum statistical mechanics. Kim, I.: Operator extension of strong subadditivity of entropy. Kiefer, J.: Optimum experimental designs. Jenčová, A., Ruskai, M.B.: A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality. Hiai, F., Ruskai, M.B.: Contraction coefficients for noisy quantum channels. Hasegawa, H.: \(\alpha \)-divergence of the noncommutative information geometry” Rep. 529, 73–140 (AMS, 2010)Įffros, E.G.: “A matrix convexity approach to some celebrated quantum inequalities’’. : Trace inequalities and quantum entropy: an introductory course Contemp. : Topics on Operator Inequalities Sapporo Lecture Notes (1978)Īndo, T.: Concavity of certain maps on positive definite matrices and applications to Hadamard products.
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