We present a formula for the relative entropy S(rho||sigma) of two n mode gaussian states rho, sigma in the boson Fock space. It is shown that the relative entropy has a classical and a quantum part: The classical part consists of a weighted linear combination of relative Shannon entropies of n pairs of Bernouli trials arising from the thermal state composition of the gaussian states rho and sigma. The quantum part has a sum of n terms, that are functions of the annihilation means and the covariance matrices of 1-mode marginals of the gaussian state $ρ'$, which is equivalent to ρunder a disentangling unitary gaussian symmetry operation of the state $σ$. A generalized formula for the Petz-Renyi relative entropy S_alpha(rho||sigma) for gaussian states ρ, σis also presented. Furthermore it is shown that the Petz-Renyi relative entropy converges to the limit S(rho||sigma) as alpha increases to 1.
The most useful measure of a bipartite entanglement is the von Neumann entropy of either of the reduced density matrices. For a particular class of continuous-variable states, the Gaussian states, the entropy of entanglement can be expressed rather elegantly in terms of the symplectic eigenvalues, elements that characterize a Gaussian state and depend on the correlations of the canonical variables. We give a pedagogical step-by-step derivation of this result and provide some insights that can be useful in practical calculations.
We emphasize in these pedagogical notes the that the theory of the Radon transform and its applications is best understood using the theory of the metaplectic group and the quadratic Fourier transforms generating metaplectic operator.. Doing this we hope that these notes will be useful to a larger audience, including researchers in time-frequency analysis.
Lecture Notes on Quantum Many-Body Theory: A Pedagogical Introduction
Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli（意大利国家核物理研究所（INFN）那不勒斯分部）；CNR - SPIN – Istituto Superconduttori, Materiali Innovativi e Dispositivi（国家研究委员会 - SPIN 超导、创新材料与器件研究所）
Incremental information, as measured by the quantum entropy, is increasing when two ensembles are united. This result was proved by Lieb and Ruskai, and it is the foundation for the proof of strong subadditivity of quantum entropy. We present a truly elementary proof of this fact in the context of the broader family of matrix entropies introduced by Chen and Tropp.
One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss few channel characteristics expressed by means of generalized entropies. Such characteristics can often be dealt in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the $q$-average output entropy of degree $q\geq1$ is bounded from above by the $q$-entropy of the input density matrix. Concavity properties of the $(q,s)$-entropy exchange are considered. Fano type quantum bounds on the $(q,s)$-entropy exchange ar…
We study pseudo entropy for a particular linear combination of entangled states in qubit systems, two-dimensional free conformal field theories (CFT), and two-dimensional holographic CFT. We observe phenomena that the pseudo entropy can be parametrically large compared with the logarithm of the dimension of Hilbert space. We call these phenomena pseudo entropy amplification. The pseudo entropy amplification is analogous to the amplification of the weak value. In particular, our result suggests the holographic CFT does not lead the amplification as long as the non-perturbative effects are negli…
We study the time evolution of Rényi entropy in a system of two coupled Brownian SYK clusters evolving from an initial product state. The Rényi entropy of one cluster grows linearly and then saturates to the coarse grained entropy. This Page curve is obtained by two different methods, a path integral saddle point analysis and an operator dynamics analysis. Using the Brownian character of the dynamics, we derive a master equation which controls the operator dynamics and gives the Page curve for purity. Insight into the physics of this complicated master equation is provided by a complementary p…
We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have Rényi 2-entropy at least as great as $K_{1,n-1}$ and for $α>1$, $K_n$ maximizes Rényi $α$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.
Using the information content of correlations between multipartite systems, together with the notion of partitioning, we show that some general results about the evolution of correlations in quantum systems can be derived with only elementary methods. In particular, we show that for 2 quantum systems A and B, each comprised of a number of sub-systems, in which a partition of A interacts unitarily with a partition of B, then the total correlation can only increase (or remain unchanged) and is given simply by the sum of the initial correlation and the correlation that develops as a result of the…
