Survival of entanglement in thermal states

We present a general sufficiency condition for the presence of multipartite entanglement in thermal states stemming from the ground state entanglement. The condition is written in terms of the ground state entanglement and the partition function and it gives transition temperatures below which entanglement is guaranteed to survive. It is flexible and can be easily adapted to consider entanglement for different splittings, as well as be weakened to allow easier calculations by approximations. Examples where the condition is calculated are given. These examples allow us to characterize a minimum gapping behavior for the survival of entanglement in the thermodynamic limit. Further, the same technique can be used to find noise thresholds in the generation of useful resource states for one-way quantum computing. In recent years there has been much effort to investigate the role of entanglement in general physics problems. Entanglement is known to be a key resource for quantum information, essential for faithful teleportation and allowing an absolute secure key distribution among other things [1], and the study of entanglement has been mainly from this perspective. However, entanglement is also a key foundational issue in quantum mechanics, and has recently been associated to various phenomena in different areas of physics, for example Hawking radiation in cosmology [2], symmetry breaking in high energy physics [3] and in particular, to many areas of condensed matter physics such as critical phenomena [4]. In addition, entanglement theory has also been helpful in finding the ground state for difficult many-body systems [5]. All of these results and connections are very intriguing, and lead us to ask when and where else entanglement exists, and what is its role in the associated phenomena. To address these issues, in the first instance, it would be very useful to have a general, easy test to see if a system contains entanglement. There are several difficulties in this, especially in many-body physics. First, in nature, systems are in thermal states, and calculating the density matrix involves the diagonalisation of large Hamiltonians which, in general, proves impossible. Second, given this state density matrix, it is difficult to test if it is entangled or not. One situation that helps simplify the problem is given by systems where the ground state is highly entangled. This is often the case for symmetric or interacting many-body systems. There, at low enough temperatures, the properties are governed by the ground state, and the system is entangled. Along these lines, for example, in [6, 7], by taking the minimum expectation of the energy allowed for separable states, conditions for entanglement are found on the average energy and associated thermodynamic quantities, so observing energy below this minimum value means the system is entangled. In this paper we present an explicit connection between the ground state entanglement and the entanglement of the thermal state. We give a condition for the existence of entanglement based on minimum knowledge of the ground state and statistical properties of the full state (the partition function), so that we do not need to calculate the full density matrix. This is a kind of coarse grain approach to the existence of entanglement. Weaker, more easily