We provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution. A large class of entropy measures in considered, which includes Rényi and Tsallis entropies. The result is derived from a general bound on the trace-norm of a commutator, which can be expected to find other implementations. We apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with Rényi and Tsallis entanglement entropies taken as measures of entanglement in the system.
We provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution. A large class of entropy measures in considered, which includes Rényi and Tsallis entropies. The result is derived from a general bound on the trace-norm of a commutator, which can be expected to find other implementations. We apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with Rényi and Tsallis entanglement entropies taken as measures of entanglement in the system.
Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of a quantum observable. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for [Formula: see text] observables respectively.
In a closed system, the total number of particles is fixed. We ask how much does this conservation law restricts the amount of entanglement that can be created. We derive a tight upper bound on the bipartite entanglement entropy in closed systems, and find what a maximally entangled state looks like in such a system. Finally, we illustrate numerically on an isolated system of one-dimensional fermionic gas, that the upper bound can be reached during its unitary evolution, when starting in a pure state that emulates a thermal state with high enough temperature. These results are in accordance with current experiments measuring R\'enyi-2 entanglement entropy, all of which employ a particle-conserving Hamiltonian, where our bound acts as a loose bound, and will become especially important for bounding the amount of entanglement that can be spontaneously created, once a direct measurement of entanglement entropy becomes feasible.
We provide an upper bound on the maximal rate at which irreversible quantum dynamics can generate entanglement in a bipartite system. The generator of irreversible dynamics consists of a Hamiltonian and dissipative terms in Lindblad form. The relative entropy of entanglement is chosen as a measure of entanglement in an ancilla-free system. We provide an upper bound on the entangling rate which has a logarithmic dependence on a dimension of a smaller system in a bipartite cut. We also investigate the rate of change of quantum mutual information in an ancilla-assisted system and provide an upper bound independent of dimension of ancillas.
Abstract We study systematically resource measures of coherence and entanglement based on Rényi relative entropies, which include the logarithmic robustness of coherence, geometric coherence, and conventional relative entropy of coherence together with their entanglement analogues. First, we show that each Rényi relative entropy of coherence is equal to the corresponding Rényi relative entropy of entanglement for any maximally correlated state. By virtue of this observation, we establish a simple operational connection between entanglement measures and coherence measures based on Rényi relative entropies. We then prove that all these coherence measures, including the logarithmic robustness of coherence, are additive. Accordingly, all these entanglement measures are additive for maximally correlated states. In addition, we derive analytical formulas for Rényi relative entropies of entanglement of maximally correlated states and bipartite pure states, which reproduce a number of classic results on the relative entropy of entanglement and logarithmic robustness of entanglement in a unified framework. Several nontrivial bounds for Rényi relative entropies of coherence (entanglement) are further derived, which improve over results known previously. Moreover, we determine all states whose relative entropy of coherence is equal to the logarithmic robustness of coherence. As an application, we provide an upper bound for the exact coherence distillation rate, which is saturated for pure states.
The speed limit on entanglement is defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary and arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For the unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.
In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order α, dimension d, and additive error ε,
1. For 0<α<1, the sample complexity is O(d1+1/α/ε1/α+d1/α−1/ε2) for Rényi entropy and O(d1+1/α/ε1/α+d2−2α/ε2) for Tsallis entropy. In particular, for 0<α≤1/2, the sample complexity for both entropies is O(d1+1/α/ε1/α).
2. For non-integer α>1, the sample complexity is O(d2/ε1/α+d1−1/α/ε2) for Rényi entropy.
Our upper bounds improve the quantum Rényi entropy estimators due to Acharya, Issa, Shende, and Wagner (2017) and the quantum Tsallis entropy estimators due to Chen, Liu, and Wang (2026), and match the lower bounds recently established by Wang (2026).