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arXiv:2207.11459 · Physical Review A

Capacity of entanglement for a nonlocal Hamiltonian

Divyansh Shrimali, Swapnil Bhowmick, Vivek Pandey, Arun Kumar Pati

Published Oct 14, 2022• 23 citations
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Abstract

The notion of capacity of entanglement is the quantum information theoretic counterpart of the heat capacity which is defined as the second cumulant of the entanglement spectrum. Given any bipartite pure state, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the reduced state of any of the subsystems. Here, we study the dynamics of this quantity under a nonlocal Hamiltonian. Specifically, we address the following question: Given an arbitrary nonlocal Hamiltonian, what is the capacity of entanglement that the system can possess? As a useful application, we show that the quantum speed limit for creating the entanglement is not only governed by the fluctuation in the nonlocal Hamiltonian, but also depends inversely on the time average of the square root of the capacity of entanglement. Furthermore, we discuss this quantity for a general self-inverse Hamiltonian and provide a bound on the rate of capacity of entanglement. Towards the end, we generalize the capacity of entanglement for bipartite mixed states based on the relative entropy of entanglement and show that the above definition reduces to the capacity of entanglement for pure bipartite states. Our results can have several applications in diverse areas of physics.

DOI
10.1103/physreva.106.042419

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The notion of capacity of entanglement is the quantum information theoretic counterpart of the heat capacity which is defined as the second cumulant of the entanglement spectrum. Given any bipartite pure state, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the reduced state of any of the subsystems. Here, we study the dynamics of this quantity under a nonlocal Hamiltonian. Specifically, we address the following question: Given an arbitrary nonlocal Hamiltonian, what is the capacity of entanglement that the system can possess? As a useful application, we show that the quantum speed limit for creating the entanglement is not only governed by the fluctuation in the nonlocal Hamiltonian, but also depends inversely on the time average of the square root of the capacity of entanglement. Furthermore, we discuss this quantity for a general self-inverse Hamiltonian and provide a bound on the rate of capacity of entanglement. Towards the end, we generalize the capacity of entanglement for bipartite mixed states based on the relative entropy of entanglement and show that the above definition reduces to the capacity of entanglement for pure bipartite states. Our results can have several applications in diverse areas of physics.
Many quantum information theoretic quantities are similar to and/or inspired by thermodynamic quantities, with entanglement entropy being a well-known example. In this paper, we study a less well-known example, capacity of entanglement, which is the quantum information theoretic counterpart of heat capacity. It can be defined as the second cumulant of the entanglement spectrum and can be loosely thought of as the variance in the entanglement entropy. We review the definition of capacity of entanglement and its relation to various other quantities such as fidelity susceptibility and Fisher information. We then calculate the capacity of entanglement for various quantum systems, conformal and nonconformal quantum field theories in various dimensions, and examine their holographic gravity duals. Resembling the relation between response coefficients and order parameter fluctuations in Landau-Ginzburg theories, the capacity of entanglement in field theory is related to integrated gravity fluctuations in the bulk. We address the question of measurability, in the context of proposals to measure entanglement and R\'enyi entropies by relating them to U(1)U(1)U(1) charges fluctuating in and out of a subregion, for systems equivalent to non-interacting fermions. From our analysis, we find universal features in conformal field theories, in particular the area dependence of the capacity of entanglement appears to track that of the entanglement entropy. This relation is seen to be modified under perturbations from conformal invariance. In quenched 1+11+11+1 dimensional CFTs, we compute the rate of growth of the capacity of entanglement. The result may be used to refine the interpretation of entanglement spreading being carried by ballistic propagation of entangled quasiparticle pairs created at the quench.
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.
We investigate the speed at which entanglement can be generated by an interaction parameter encoded in a two-qubit Hamiltonian, quantified by the derivative of concurrence with respect to the coupling parameter. For arbitrary pure two-qubit states evolving under a general nonlocal interaction, we derive a bound relating this entanglement speed to the quantum Fisher information (QFI). Specifically, we show that ∣∂gC∣≤FQ(g)1−C2|\partial_g C| \le \sqrt{F_Q^{(g)}} \sqrt{1-C^2}∣∂g​C∣≤FQ(g)​​1−C2​, where FQ(g)F_Q^{(g)}FQ(g)​ is the QFI associated with estimation of the parameter. This establishes FQ1−C2\sqrt{F_Q}\sqrt{1-C^2}FQ​​1−C2​ as an upper bound on the speed of entanglement generation in parameter space. We further derive the saturation conditions and identify the states and dynamical regimes for which equality is attained. At saturation, concurrence evolves at the maximum rate permitted by the distinguishability of the underlying quantum state. These results reveal a direct connection between quantum metrology and entanglement generation, showing that the same information-theoretic quantity that governs parameter-estimation precision also limits the speed at which entanglement resources can be created.
How fast an observable can evolve in time is answered by so-called observable speed limit. Here, we prove a stronger version of the observable speed limit and show that the previously obtained bound is a special case of the new bound. The stronger quantum speed limit for the state also follows from the stronger quantum speed limit for observables (SQSLO). We apply this to prove a stronger bound for the entanglement rate using the notion of capacity of entanglement (the quantum information theoretic counterpart of the heat capacity) and show that it outperforms previous bounds. Furthermore, we apply the SQSLO for the rate of modular Hamiltonian and in the context of interacting qubits in a quantum battery. These illustrative examples reveal that the speed limit for the modular energy and the time required to charge the battery can be exactly predicted using the new bound. This shows that for estimating the charging time of quantum battery SQSLO is actually tight, i.e., it saturates. Our findings can have important applications in quantum thermodynamics, the complexity of operator growth, predicting the time rate of quantum correlation growth, and quantum technology, in general.
The heat capacity of a system quantifies how it energetically responds to changes in temperature at equilibrium. Whilst this quantity is positive and even additive for non-interacting systems, self-gravitating systems such as stars or subsystems of strongly interacting quantum systems are known to have negative or anomalous specific heat capacities. In this work, we investigate how the presence of entanglement at equilibrium can influence how an interacting system responds energetically to changes in temperature. We examine the local heat capacity of interacting quantum systems providing an analytical understanding for when anomalies occur. Most interestingly, we find a connection between local heat capacity anomalies and entanglement by deriving a separability bound based on the fluctuations of local and interaction energies. We illustrate our results with two examples (i) a nearest neighbour spin-1/2 chain and (ii) two coupled quantum harmonic oscillators. Lastly, we provide an information-theoretic formula connecting mutual information and athermality to the non-additivity of the heat capacity. Our results provide model-independent thermodynamic entanglement detection bounds and insight into the relationship between quantum correlations and the heat capacity of quantum systems.
We investigate the fundamental limits of entanglement generation under bipartite Hamiltonian dynamics when only finite physical resources-specifically, bounded energy variance-are available. Using the relative entropy of entanglement, we derive a closed analytical expression for the instantaneous entanglement generation rate for arbitrary pure states and Hamiltonians expressed in the Schmidt basis. We find that constraints based solely on the mean energy of the Hamiltonian are insufficient to bound the entanglement generation rate, whereas imposing a variance constraint ensures a finite and well-defined maximum. We fully characterize the Hamiltonians that achieve this optimal rate, establishing a direct relation between their imaginary components in the Schmidt basis and the structure of the optimal initial states. For systems without ancillas, we obtain a closed-form expression for the maximal rate in terms of the surprisal variance of the Schmidt coefficients and identify the family of optimal states and Hamiltonians. We further extend our analysis to scenarios where Alice and Bob may employ local ancillary systems: using a matrix-analytic framework and a refined description of the Hamiltonians allowed by the physical constraints, we derive an explicit optimization formula and characterize the attainable enhancement in entanglement generation.