The Renyi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, Renyi entropy is closely related to free energy. Suppose we start with a system in thermal equilibrium and then suddenly divide the temperature by q. Then the maximum amount of work the system can do as it moves to equilibrium at the new temperature, divided by the change in temperature, equals the system's Renyi entropy in its original state. This result applies to both classical and quantum systems. Mathematically, we can express this result as follows: the Renyi entropy of a system in thermal equilibrium is minus the "1/q-derivative" of its free energy with respect to temperature. This shows that Renyi entropy is a q-deformation of the usual concept of entropy.
It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched Rényi relative $α$-entropy with $α>1$ is also equivalent to this property. For the proof, we use an interpolating family of $L_p$-norms with respect to a state.
The Rényi and von Neumann entropies of the thermal state in the generalized uncertainty principle (GUP)-corrected single harmonic oscillator system are explicitly computed within the first order of the GUP parameter $α$. While the von Neumann entropy with $α= 0$ exhibits a monotonically increasing behavior in external temperature, the nonzero GUP parameter makes the decreasing behavior of the von Neumann entropy at the large temperature region. As a result, the von Neumann entropy is maximized at the finite temperature if $α\neq 0$. The Rényi entropy $S_γ$ with nonzero $α$ also exhibits simila…
In recent years, the Renyi entropy has repeatedly been discussed for characterization of quantum critical states and entanglement. Here, time evolution of the Renyi entropy is studied. A compact general formula is presented for the lower bound on the entropy rate.
Dimensional regularization is a common method used to regulate the UV divergence of field theoretic quantities. When it is used in the context of Renyi entropy, however, it is important to consider whether such a procedure eliminates the statistical interpretation thereof as a measure of entanglement of states living on a Hilbert space. We therefore examine the dimensionally regularized Renyi entropy of a 4d unitary CFT and show that it admits no underlying Hilbert space in the state-counting sense. This gives a concrete proof that dimensionally regularized Renyi entropy cannot always be obtai…
We show that the minimum Renyi entropy output of a quantum channel is locally additive for Renyi parameter alpha>1. While our work extends the results of [10] (in which local additivity was proven for alpha=1), it is based on several new techniques that incorporate the multiplicative nature of p-norms, in contrast to the additivity property of the von-Neumann entropy. Our results demonstrate that the counterexamples to the Renyi additivity conjectures exhibit global effects of quantum channels. Interestingly, the approach presented here can not be extended to Renyi entropies with parameter alp…
We explore the role of sandwiched Renyi relative entropy in AdS/CFT and in finite-dimensional models of holographic quantum error correction. In particular, in the context of operator algebra quantum error correction, we discuss a suitable generalization of sandwiched Renyi relative entropy over finite-dimensional von Neumann algebras. It is then shown that the equality of bulk and boundary sandwiched relative Renyi entropies is equivalent to algebraic encoding of bulk and boundary states, the Ryu-Takayanagi formula, the equality of bulk and boundary relative entropy, and subregion duality. Th…
In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_α$. We rigorously prove that $R_α$ with Renyi index $α>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.
We investigate the time-dependent Rényi entanglement entropy after a quantum quench starting from the Mott-insulating and charge-density-wave states in a one-dimensional free boson system. The second Rényi entanglement entropy is found to be the negative of the logarithm of the permanent of a matrix consisting of time-dependent single-particle correlation functions. From this relation and a permanent inequality, we obtain rigorous conditions for satisfying the volume-law entanglement growth. We also succeed in calculating the time evolution of the Rényi entanglement entropy in unprecedentedly …
A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge/gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Renyi entropies. Here we show that all Renyi entropies satisfy a similar area law in holographic theories and are given by the areas of du…
This paper introduces "swiveled Renyi entropies" as an alternative to the Renyi entropic quantities put forward in [Berta et al., Phys. Rev. A 91, 022333 (2015)]. What distinguishes the swiveled Renyi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A consequence of this extra degree of freedom is that the swiveled Renyi entropies are ordered, which is an important property of the Renyi family of entropies. The swiveled Renyi entropies are however generally dis…
