We argue that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when we excite the system. In relativistic setups, its effective temperature is proportional to the inverse of the subsystem size. This provides a universal relationship between the energy and the amount of quantum information. We derive the results using holography and confirm them in two-dimensional field theories. We will also comment on an example with negative specific heat and suggest a connection between the second law of thermodynamics and the strong subadditivity of entanglement entropy.
We argue that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when we excite the system. In relativistic setups, its effective temperature is proportional to the inverse of the subsystem size. This provides a universal relationship between the energy and the amount of quantum information. We derive the results using holography and confirm them in two-dimensional field theories. We will also comment on an example with negative specific heat and suggest a connection between the second law of thermodynamics and the strong subadditivity of entanglement entropy.
We study thermodynamics of entanglement entropy for weakly excited states in certain nonconformal field theories, whose gravity duals are given by nonconformal Dp-branes. We observe that the entanglement entropy of a sufficiently small system in nonconformal backgrounds still obeys a first-law-like relation, just as the anti--de Sitter counterparts investigated in [J. Bhattacharya, M. Nozaki, T. Takayanagi, and T. Ugajin, Phys. Rev. Lett. 110, 091602 (2013)]. The effective temperature is proportional to the inverse of the size of the subsystem. The proportionality is a dimensionless constant which is only determined by the shape of the entangling region and independent of any coupling. This universality is confirmed by working with the 10-dimensional string frame metric as well as the lower-dimensional effective metric. When the entangling region is a strip and translational invariance is broken by metric fluctuations, we derive a first-law-like relation where additional components of the stress energy tensor are involved.
A class of (2+1)-dimensional quantum many body system characterized by an anisotropic scaling symmetry (Lifshitz symmetry) near their quantum critical point can be described by a (3+1)-dimensional dual gravity theory with negative cosmological constant along with a massive vector field, where the scaling symmetry is realized by the metric as an isometry. We calculate the entanglement entropy of an excited state of such a system holographically, i.e., from the asymptotic perturbation of the gravity dual using the prescription of Ryu and Takayanagi, when the subsystem is sufficiently small. With suitable identifications, we show that this entanglement entropy satisfies an energy conservation relation analogous to the first law of thermodynamics. The non-trivial massive vector field here plays a crucial role and contributes to an additional term in the energy relation.
Entanglement temperature is an interesting quantity which relates the increased amount of entanglement entropy to that of energy for a weakly excited state in the first-law of entanglement entropy, it is proportional to the inverse of the size of entanglement subsystem and only depends on the shape of the entanglement region. We find the explicit formula of entanglement temperature for the general hyperscaling violation backgrounds with a strip-subsystem. We then investigate the entanglement temperature for a round ball-subsystem, we check that the entanglement temperature has a universal form when the hyperscaling violation exponent is near zero.
The second law of thermodynamics is discussed and reformulated from a quantum information theoretic perspective for open quantum systems using relative entropy. Specifically, the relative entropy of a quantum state with respect to equilibrium states is considered and its monotonicity property with respect to an open quantum system evolution is used to obtain second lawlike inequalities. We discuss this first for generic quantum systems in contact with a thermal bath and subsequently turn to a formulation suitable for the description of local dynamics in a relativistic quantum field theory. A local version of the second law similar to the one used in relativistic fluid dynamics can be formulated with relative entropy or even relative entanglement entropy in a space-time region bounded by two light cones. We also give an outlook toward isolated quantum field theories and discuss the role of entanglement for relativistic fluid dynamics.
We construct upper bounds on entanglement entropies of many-body quantum states that have fixed energy expectation values with respect to local Hamiltonians. Our focus is on entanglement entropies of subsystems that make up approximately half of the full system. The upper bound on the von Neumann entanglement entropy is half the sum of the thermal entropies of two fictitious systems at the same temperature as one another, with an additional area-law contribution in some systems. The effective temperature is chosen such that the sum of the thermal energies of the two fictitious systems matches the constraint on the energy of the state in the original problem; at subextensive energies, this temperature decreases with increasing system size. Our upper bounds on Rényi entanglement entropies take an analogous form. These inequalities show that, in systems that are frustration-free (FF) and whose zero-temperature subsystem thermal entropies scale with surface areas (rather than subsystem volumes), ground states are area-law entangled. For physical models of quantum matter, which have well-defined specific heat capacities (and are not necessarily FF), our bounds provide a way to convert this thermodynamic data into constraints on pure-state entanglement at both subextensive and extensive energies. We also show that our general upper bounds on half-system entanglement entropies are optimal, up to subleading corrections, in systems with reflection symmetry and above a subextensive energy threshold. A key application of our results is to constrain the entanglement of quantum states relevant to simulations of zero-temperature response functions, which have energies of order unity. We prove that, in wide varieties of FF systems in two dimensions and higher, the contributions of excitations to von Neumann entanglement entropy are subleading compared with the contribution from the ground state.
Thermal equilibrium in quantum many-body systems can be fully described by pure quantum states as far as local observables are concerned, while the conventional description of it is based on mixed-state ensembles.The thermodynamic entropy, in that case, is identified with the entanglement entropy of small subsystems.It scales with the size of the subsystem, which is called the volume law of entanglement.When the size of the subsystem increases, however, the correspondence between thermal and entanglement entropies fails and there appears a quantum correction to the simple volume-law scaling.Explicating the scaling of the entanglement entropy with the subsystem size is hence of great importance in connecting quantum physics to thermodynamics as well as analyzing recent experiments on ultracold atoms.In this thesis, we study the volume law of entanglement for pure quantum states representing thermal equilibrium.We derive an analytic formula of the volume law of entanglement for a certain class of pure states called canonical Thermal Pure Quantum (cTPQ) states which represent thermal equilibrium.We illustrate an advantage of the formula by numerically calculating the entanglement entropy of the cTPQ states and applying the formula to it.Furthermore, we argue that our formula universally applies to any sufficiently scrambled pure state, even if the state is not thermal.We consider two examples of such scrambled states, namely, stationary pure states after quantum quench and energy eigenstates of general Hamiltonians with and without integrability.The entanglement entropy of the former states, including actual experimental data in ultracold atoms, is in an excellent agreement with our formula as long as the states are scrambled.For the latter states, we find that our formula works as a good fitting function in non-integrable models to extract information in the thermodynamic limit from finite size systems.We also show that our formula can distinguish eigenstates of integrable models from non-integrable ones.4.3.1 Review of result by Lu and Grover for eigenstates of non-integrable models . . . . . . . . . . . .