This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Koßmann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy of states, enabling us to handle maximization tasks for the relative entropy of channels. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.
This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Koßmann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy of states, enabling us to handle maximization tasks for the relative entropy of channels. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.
Abstract Finding the minimal relative entropy of two quantum states under semidefinite constraints is a pivotal problem located at the mathematical core of various applications in quantum information theory. An efficient method for providing provable upper and lower bounds is the central result of this work. Our primordial motivation stems from the essential task of estimating secret key rates for QKD from the measurement statistics of a real device. Further applications include the computation of channel capacities, the estimation of entanglement measures and many more. We build on a recently introduced integral representation of quantum relative entropy by [Frenkel, Quantum 7, 1102 (2023)] and provide reliable bounds as a sequence of semidefinite programs (SDPs). Our approach ensures provable sublinear convergence in the discretization, while also maintaining resource efficiency in terms of SDP matrix dimensions. Additionally, we can provide gap estimates to the optimum at each iteration stage.
The quantum relative entropy is a fundamental quantity in quantum information science, characterizing the distinguishability between two quantum states. However, this quantity is not additive in general for correlated quantum states, necessitating regularization for precise characterization of the operational tasks of interest. Recently, we proposed the study of the regularized relative entropy between two sequences of sets of quantum states in [arXiv:2411.04035], which captures a general framework for a wide range of quantum information tasks. Here, we show that given suitable structural assumptions and efficient descriptions of the sets, the regularized relative entropy can be efficiently approximated within an additive error by a quantum relative entropy program of polynomial size. This applies in particular to the regularized relative entropy in adversarial quantum channel discrimination. Moreover, we apply the idea of efficient approximation to quantum resource theories. In particular, when the set of interest does not directly satisfy the required structural assumptions, it can be relaxed to one that does. This provides improved and efficient bounds for the entanglement cost of quantum states and channels, entanglement distillation and magic state distillation. Numerical results demonstrate improvements even for the first level of approximation.
Abstract The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum–classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semidefinite programming (SDP), by making use of variational formulas for the measured relative entropies of states and semidefinite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semidefinite programs provide the optimal values of the measured relative entropies of states and channels, but also the numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.
Inspired by a conjecture related to numerical investigations for a quantum channel in a system of large size, we shall study relations between two different entropies of a quantum channel. One of them is defined by Gour and Wilde as the generalization of quantum relative entropy and maximally mixed state. The second one is defined as the difference between the entropy of the Choi state of the quantum channel and a constant number. At first, we show that the latter entropy is an upper boundary of the former entropy. Moreover, we also give a sufficient and necessary condition that the two entropies are equal. That is, if and only if the relative entropy reaches its supremum on the maximal entangled pure state in the former entropy. At last, to study the difference between the two entropies, we expect to find some quantum channels in which their related entropies do not reach their supremum on the maximal entangled pure states for the former entropy. By calculating the two entropies, we find that amplitude-damping channels are desired examples.
Abstract Quantum relative entropy optimization refers to a class of convex problems in which a linear functional is minimized over an affine section of the epigraph of the quantum relative entropy function. Recently, the self-concordance of a natural barrier function was proved for this set, and various implementations of interior-point methods have been made available to solve this class of optimization problems. In this paper, we show how common structures arising from applications in quantum information theory can be exploited to improve the efficiency of solving quantum relative entropy optimization problems using interior-point methods. First, we show that the natural barrier function for the epigraph of the quantum relative entropy composed with positive linear operators is self-concordant, even when these linear operators map to singular matrices. Compared to modeling problems using the full quantum relative entropy cone, this allows us to remove redundant log-determinant expressions from the barrier function and reduce the overall barrier parameter. Second, we show how certain slices of the quantum relative entropy cone exhibit useful properties which should be exploited whenever possible to perform certain key steps of interior-point methods more efficiently. We demonstrate how these methods can be applied to applications in quantum information theory, including quantifying quantum key rates, quantum rate-distortion functions, quantum channel capacities, and the ground state energy of Hamiltonians. Our numerical results show that these techniques improve computation times by up to several orders of magnitude, and allow previously intractable problems to be solved.
Discriminating between noisy quantum processes is a central primitive for quantum communication, metrology, and computing. While discrimination limits for finite-dimensional channels are well understood, the continuous-variable setting, particularly under experimentally relevant energy constraints, remains significantly less developed. In this work, we establish an energy-constrained chain rule for the Belavkin-Staszewski channel divergence, which yields a fundamental upper bound on the error exponents achievable by fully adaptive, energy-constrained quantum channel discrimination protocols. We then derive efficiently computable bounds on asymmetric error exponents for energy-constrained discrimination of bosonic dephasing and loss-dephasing channels. Specifically, we show that three operationally relevant quantities -- the measured relative entropy, the Umegaki relative entropy, and the geometric Renyi divergence -- admit semidefinite program (SDP) formulations when the input energy is bounded and the Hilbert space is suitably truncated. Applying these tools, we demonstrate that optimal probes for these channels under energy constraints are Fock-diagonal, and we also enable numerically precise evaluation of bounds on achievable error exponents across discrimination strategies ranging from separable to fully adaptive. The resulting SDPs provide practical benchmarks for quantum-limited sensing in low-energy bosonic platforms.