Exact correspondence between Renyi entropy flows and physical flows

Abstract

We present a universal relation between the flow of a Renyi entropy and the full counting statistics of energy transfers. We prove the exact relation for a flow to a system in thermal equilibrium that is weakly coupled to an arbitrary time-dependent and nonequilibrium system. The exact correspondence, given by this relation, provides a simple protocol to quantify the flows of Shannon and Renyi entropies from the measurements of energy transfer statistics. Exact correspondences between seemingly different concepts play an important role in all fields of physics. An example is the fluctuation-dissipation theorem, which states that the linear response of a system to externally applied forces corresponds to the system fluctuations [1,2]. In the last decade, the fluctuation-dissipation theorem has initiated important developments in quantum transport, quantum computation, and other similar phenomenological theories [3]. This theorem can be extended to nonlinear responses [4] and to full counting statistics (FCS) [5], giving more extended sets of such relations similar to Crooks' formula [6]. In this paper we present a relation similar to the fluctuation-dissipation theorem that provides an exact correspondence between the flows of Renyi entropy and FCS of energy transfers. DOIIn transport theory, stationary flow of a physical quantity can take place from a system into an infinitely large system. In the case a quantity is locally conserved in each system, its flow is determined only by interaction between the two systems [7]. The traditional examples include electric current, which is the flow of charge, and energy flow. Moreover, there are other conserved quantities which are not physical in a strict sense. An example is the generalization of entropy by Renyi into S M = n p M n , with p n being the probability to be in state n and arbitrary M > 0 [8]. Quantum generalization of the Renyi entropy is obviously conserved in a system under Hamiltonian evolution, with the Hamiltonian involving only the degrees of freedom of this system [9]. For a system in thermal equilibrium at temperature T this entropy corresponds to the difference of free energies, i.e., ln S M = F (T ) − F (T /M). In nonequilibrium thermodynamics, the Renyi entropies have already been considered [10]. They have been studied in strongly interacting systems [11,12], in particular spin chains [13,14].The Renyi entropies in quantum physics are considered unphysical, or nonobservable, due to their nonlinear dependence on density matrix. So is the Shannon entropy, which is derived from the Renyi entropy S = lim M→1 ∂S M /∂M [9]. Such quantities cannot be determined from immediate measurements; instead their quantification seems to be equivalent to determining the density matrix. This requires re-initialization of the density matrix between many successive measurements [15]. Therefore, the flows of Renyi entropy between systems F M ≡ −d ln S M /dt are the conserved measures of nonphysical quantities. The same pertains to Shannon entropy flow [9]. An interest...

Entanglement entropy fluctuation and distribution for open systems

“…In addition, total quantum dimension of topological order can be measured by topological entropy which is a linear combination of entanglement entropy of different regions of the system [8,9]. Moreover, the entanglement spectrum (the entanglement Renyi entropies) has been studied extensively [12][13][14][15][16][17][18].…”

Report on 1808.09395v1

“…These are not well-defined unless the quantum state ρ(t) is positive for all times. In this context the Keldysh approach has proven useful by its extension to 'parallel-worlds' [25] to compute Renyi entropies. Our operator-sum formulation of the Keldysh approach identifies a variety of approximations that give meaningful entropic quantities, including also the exchange entropy [67,68] associated with the environment [48] which requires the evolution Π(t) to be completely positive.…”

“…However, the experimental importance of effects due to strong coupling and non-Markovianity-which are tied together [9]has spurred progress in a variety of other approaches: Inclusion of parts of the environment into the system [10], time-convolutionless master equations [11,12], stochastic descriptions such as quantum trajectories [13], pathintegral [14], QMC [15], and hierarchical methods [16,17], multilayer multiconfiguration time-dependent Hartree method [18], perturbative expansions [19][20][21][22][23][24][25], resummation [26][27][28] and related techniques [29][30][31], projection techniques [32,33] and real-time renormalization-group methods [34][35][36][37][38][39][40][41]. These are applicable to more general reduced dynamics derived from unitary evolution U (t) of initially uncorrelated states ρ(0) of the system (S) and ρ E of the environment (E):…”

Density-operator evolution: Complete positivity and the Keldysh real-time expansion

“…However, the experimental importance of effects due to strong coupling and non-Markovianity-which are tied together [9]-has spurred progress in a variety of other approaches: Inclusion of parts of the environment into the system [10], time-convolutionless master equations [11,12], stochastic descriptions such as quantum trajectories [13], pathintegral [14], quantum Monte Carlo [15], and hierarchical methods [16,17], multilayer multiconfiguration time-dependent Hartree method [18], perturbative expansions [19][20][21][22][23][24][25], resummation [26][27][28] and related techniques [29][30][31], projection techniques [32,33] and real-time renormalization-group methods [34][35][36][37][38][39][40][41]. These are applicable to more general reduced dynamics derived from unitary evolution U (t) of initially uncorrelated states ρ(0) of the system (S) and ρ E of the environment (E):…”

“…This is a fundamental difference between the Liouville-von Neumann and Schrödinger equation which by our considerations ties in with the state-evolution correspondence 25. In special limits where the evolution is a dynamical semi-group this goes unnoticed: there one can easily apply the operator-sum theorem directly to Π(t + dt) = Π(dt)Π(t), leading to the GKSL form of the quantum master equation[72].…”