A formula for the relative entropy in chiral CFT

Letters in Mathematical Physics - We prove the QNEC on the Virasoro nets for a class of unitary states extending the coherent states, that is states obtained by applying an exponentiated stress...     

A note on relative entropy

The quantum-mechanical concept of relative entropy is discussed from an information-theoretic point of view. We show that not all definitions found in the recent literature are equally suitable for the purpose of statistical inference by entropy maximization.     

On the relative entropy

ForA any subset of () (the bounded operators on a Hilbert space) containing the unit, and and restrictions of states on () toA, ent _A (|)—the entropy of relative to given the information inA—is defined and given an axiomatic characterisation. It is compared with ent _A ^A (|)—the relative entropy introduced by Umegaki and generalised by various authors—which is defined only forA an algebra. It is proved that ent and ent ^S agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for ent ^S : monotonicity, concavity,w* upper semicontinuity, etc.     

A new approach for protein design based on the relative entropy

A new effective and fast minimization approach completely based on the physical theory is proposed for protein design. The sequence space is essentially searched according to the Boltzmann distribution. In this approach, the relative entropy is used as a minimization object function. The method has been tested on an off-lattice model of proteins and the results are better than those obtained from other similar work. Therefore, it can be applied as a uniform frame for both folding and inverse folding of proteins.     

On variational expressions for quantum relative entropies

Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback–Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki’s quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz’ conclusion remains true if we allow general positive operator-valued measures. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for \(\alpha \in (\frac{1}{2}, \infty )\) and strictly smaller for \(\alpha \in [0,\frac{1}{2})\). The latter statement provides counterexamples for the data processing inequality of the sandwiched Rényi relative entropy for \(\alpha < \frac{1}{2}\). Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.     

Free energy and the relative entropy

Under the assumption of an identity determining the free energy of a state of a statistical mechanical system relative to a given equilibrium state by means of the relative entropy, it is shown: first, that there is in any physically definable convex set of states a unique state of minimum free energy measured relative to a given equilibrium state; second, that if a state has finite free energy relative to an equilibrium state, then the set of its time translates is a weakly relatively compact set; and third, that a unique perturbed equilibrium state exists following a change in Hamiltonian that is bounded below.     

Maximum relative entropy of coherence for quantum channels

Based on the resource theory for quantifying the coherence of quantum channels, we introduce a new coherence quantifier for quantum channels via maximum relative entropy. We prove that the maximum relative entropy for coherence of quantum channels is directly related to the maximally coherent channels under a particular class of superoperations, which results in an operational interpretation of the maximum relative entropy for coherence of quantum channels. We also introduce the conception of subsuperchannels and sub-superchannel discrimination. For any quantum channels, we show that the advantage of quantum channels in sub-superchannel discrimination can be exactly characterized by the maximum relative entropy of coherence for quantum channels. Similar to the maximum relative entropy of coherence for channels, the robustness of coherence for quantum channels has also been investigated. We show that the maximum relative entropy of coherence for channels provides new operational interpretations of robustness of coherence for quantum channels and illustrates the equivalence of the dephasing-covariant superchannels,incoherent superchannels, and strictly incoherent superchannels in these two operational tasks.     

Unified scheme for correlations using linear relative entropy

A linearized variant of relative entropy is used to quantify in a unified scheme the different kinds of correlations in a bipartite quantum system. As illustration, we consider a two-qubit state with parity and exchange symmetries for which we determine the total, classical and quantum correlations. We also give the explicit expressions of its closest product state, closest classical state and the corresponding closest product state. A closed additive relation, involving the various correlations quantified by linear relative entropy, is derived.     

Physics behind the minimum of relative entropy measures for correlations

The relative entropy of a correlated state and an uncorrelated reference state is a reasonable measure for the degree of correlations. A key question is, however, which uncorrelated state to compare to. The relative entropy becomes minimal for the uncorrelated reference state that has the same one-particle density matrix as the correlated state. Hence, this particular measure, coined nonfreeness, is unique and reasonable. We demonstrate that for relevant physical situations, such as finite temperatures or a correlation enhanced orbital splitting, other choices of the uncorrelated state, even educated guesses, overestimate correlations.     

Some variational formulas for Hausdorff dimension,topological entropy,and SRB entropy for hyperbolic dynamical systems

We derive several variational formulas for the topological entropy and SRB entropy of Axiom A flows on compact manifolds and for the Hausdorff dimension of basic sets for Axiom A diffeomorphisms on compact surfaces.     

Markov entropy decomposition: a variational dual for quantum belief propagation

We present a lower bound for the free energy of a quantum many-body system at finite temperature. This lower bound is expressed as a convex optimization problem with linear constraints, and is derived using strong subadditivity of von Neumann entropy and a relaxation of the consistency condition of local density operators. The dual to this minimization problem leads to a set of quantum belief propagation equations, thus providing a firm theoretical foundation to that approach. The minimization problem is numerically tractable, and we find good agreement with quantum Monte Carlo calculations for spin-1/2 Heisenberg antiferromagnet in two dimensions. This lower bound complements other variational upper bounds. We discuss applications to Hamiltonian complexity theory and give a generalization of the structure theorem of [P. Hayden et al., Commun. Math. Phys. 246, 359 (2004).] to trees in an appendix.     

Maximizing relative entropy at given energies

We consider maximization of the relative entropy (with respect to a fixed normal state) in a von Neumann algebra among the states having fixed expectation for finitely many self-adjoint elements.     

A new expression for the combinatorial entropy of mixing in liquid mixtures

In the classic expression for the combinatorial entropy in terms of the mole fractions x_A and x_B′ the space unoccupied by the B molecules appears for them as divided in numerable parts, each belonging to an A molecule. In the Flory-Huggins expression based on the volume fractions φ_A and φ_B the space unoccupied by the B molecules is for them a continuum, not parcelled out in portions based on A standards. However in a solution containing n_B moles of B for a total number of moles n, the border of a B molecule is separated from that of the nearest B on the average by a distance equal to ((n/n_B)^1/2?1) times the mean diameter of the A molecules. Thus, there exists still in the liquid, directions for the B molecules which can be divided according to A standards. Based on this consideration, a new expression is derived for the combinatorial entropy of liquid mixtures which contains a “numerable” part besides a “continuous” part. on applying this equation for the prediction of the solubility of solid n-alkanes in liquid alkanes and neglecting non-ideality effects, the predicted values agree much better with the experimental ones than those deduced from the classic or from the Flory-Huggins expression.     

Entanglement entropy and variational methods: Interacting scalar fields

We develop a variational approximation to the entanglement entropy for scalar ?⁴${?}^4$ theory in 1+1$1+1$, 2+1$2+1$, and 3+1$3+1$ dimensions, and then examine the entanglement entropy as a function of the coupling. We find that in 1+1$1+1$ and 2+1$2+1$ dimensions, the entanglement entropy of ?⁴${?}^4$ theory as a function of coupling is monotonically decreasing and convex. While ?⁴${?}^4$ theory with positive bare coupling in 3+1$3+1$ dimensions is thought to lead to a trivial free theory, we analyze a version of ?⁴${?}^4$ with infinitesimal negative bare coupling, an asymptotically free theory known as precarious  ?⁴${?}^4$ theory, and explore the monotonicity and convexity of its entanglement entropy as a function of coupling. Within the variational approximation, the stability of precarious ?⁴${?}^4$ theory is related to the sign of the first and second derivatives of the entanglement entropy with respect to the coupling.     

Improvement of the relative entropy based protein folding method

The “relative entropy” has been used as a minimization function to predict the tertiary structure of a protein backbone, and good results have been obtained. However, in our previous work, the ensemble average of the contact potential was estimated by an approximate calculation. In order to improve the theoretical integrity of the relative-entropy-based method, a new theoretical calculation method of the ensemble average of the contact potential was presented in this work, which is based on the thermodynamic perturbation theory. Tests of the improved algorithm were performed on twelve small proteins. The root mean square deviations of the predicted versus the native structures from Protein Data Bank range from 0.40 to 0.60 nm. Compared with the previous approximate values, the average prediction accuracy is improved by 0.04 nm. Contributed equally to this work Supported by the National Natural Science Foundation of China (Grant No. 30670497), the Beijing Natural Science Foundation (Grant No. 5072002), and the Specialized Research Foundation for the Doctoral Program of Higher Education (Grant No. 200800050003)