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arXiv:quant-ph/0508193 · Physical Review B

Heat capacity as an indicator of entanglement

Marcin Wieśniak, Vlatko Vedral, Časlav Brukner

Published Aug 12, 2008• 64 citations
arXivHTMLPDF

Abstract

We demonstrate that the presence of entanglement in macroscopic bodies (e.g., solids) in thermodynamical equilibrium could be revealed by measuring heat capacity. The idea is that if the system was in a separable state, then for certain Hamiltonians heat capacity would not tend asymptotically to zero as the temperature approaches absolute zero. Since this would contradict the third law of thermodynamics, one concludes that the system must contain entanglement. The separable bounds are obtained by minimalization of the heat capacity over separable states and using its universal low-temperature behavior. Our results open up a possibility to use standard experimental techniques of solid-state physics---namely, heat-capacity measurements---to detect entanglement in macroscopic samples.

DOI
10.1103/physrevb.78.064108

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We demonstrate that the presence of entanglement in macroscopic bodies (e.g., solids) in thermodynamical equilibrium could be revealed by measuring heat capacity. The idea is that if the system was in a separable state, then for certain Hamiltonians heat capacity would not tend asymptotically to zero as the temperature approaches absolute zero. Since this would contradict the third law of thermodynamics, one concludes that the system must contain entanglement. The separable bounds are obtained by minimalization of the heat capacity over separable states and using its universal low-temperature behavior. Our results open up a possibility to use standard experimental techniques of solid-state physics---namely, heat-capacity measurements---to detect entanglement in macroscopic samples.
Thermodynamics is a meta-theory describing rules of natural behavior to which all microscopic physical theories have to conform. Physicists have used it to derive many, at first sight unrelated, results, such as Einstein's general relativity \\cite{jacobson} or the quantum hypothesis \\cite{einstein}. It is natural to ask whether quantum entanglement - a phenomenon in which two or more quantum systems have to be described with reference to each other \\cite{schroedinger} - can be based on thermodynamics. Here we show that the universal validity of the Third Law of Thermodynamics \\cite{nernst} relies on the existence of quantum entanglement. This is because without it heat capacity would diverge in general in the limit of very low temperatures, in contrast to the Third Law of Thermodynamics. Our result implies that heat capacity, although a macroscopic property, can be used to detect entanglement. This can be of a great importance for emerging quantum information technology \\cite{nielsen}.}
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.
A new experimental realization of heat capacity as an entanglement witness is reported. Entanglement properties of a low–dimensional quantum spin system are investigated by heat capacity measurements performed down to very low temperatures (400 mK), for various applied magnetic field values. The experimentally extracted results for the value of heat capacity at zero field matches perfectly with the theoretical estimates of entanglement from model Hamiltonians. The studied sample is a spin half antiferromagnetic system that shows a clear signature of quantum phase transition at very low temperatures when the heat capacity is varied as a function of fields at a fixed temperature. The variation of entanglement as a function of field is then explored in the vicinity of the quantum phase transition to capture the sudden loss of entanglement.
We propose a witness of temporal quantum entanglement using the imaginary component of the complex heat capacity—a measurable thermodynamic quantity in temperature-modulated calorimetry. By establishing a direct correspondence between complex heat capacity and the pseudodensity matrix formalism, our approach enables the characterization of both spatial and temporal quantum correlations without demanding additional state-level manipulation beyond initial tomography. We analytically demonstrate this connection for an open quantum system modeled by a qubit coupled to a thermal bath and show how both pseudodensity matrix negativity and violations of a temporal Clauser-Horne-Shimony-Holt inequality emerge as indicators of nonclassical temporal correlations. We further identify bounds on the imaginary heat capacity that guarantee temporal entanglement, providing an experimentally accessible criterion at the macroscopic scale. This framework offers a feasible route for probing temporal quantum effects in condensed-matter systems and opens a viable path toward experimental realization.
We study the scaling of entanglement in low-energy states of quantum many-body models on lattices of arbitrary dimensions. We allow for unbounded Hamiltonians such that systems with bosonic degrees of freedom are included. We show that, if at low enough temperatures the specific heat capacity of the model decays exponentially with inverse temperature, the entanglement in every low-energy state satisfies an area law (with a logarithmic correction). This behavior of the heat capacity is typically observed in gapped systems. Assuming merely that the low-temperature specific heat decays polynomially with temperature, we find a subvolume scaling of entanglement. Our results give experimentally verifiable conditions for area laws, show that they are a generic property of low-energy states of matter, and constitute proof of an area law for unbounded Hamiltonians beyond those that are integrable.
Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.