In this paper, we intend to show that macroscopic entanglement is possible at high temperatures. We have analysed multipartite entanglement produced by the η-pairing mechanism, which features strongly in the fermionic lattice models of high T c superconductivity. This problem is shown to be equivalent to calculating multipartite entanglement in totally symmetric states of qubits. It is demonstrated that we can conclusively calculate the relative entropy of entanglement within any subset of qubits in the overall symmetric state. Three main results are then presented. First, the condition for superconductivity, namely existence of the off-diagonal long-range order (ODLRO), is dependent not on two-site entanglement but just classical correlations as the sites become more and more distant. Secondly, the entanglement that does survive in the thermodynamical limit is the entanglement of the total lattice and, at half-filling, it scales with the log of the number of sites. It is this entanglement that will exist at temperatures below the superconducting critical temperature, which can currently be as high as 160 K. Finally, it is proved that a complete mixture of symmetric states does not contain any entanglement in the macroscopic limit. On the other hand, a mixture of symmetric states possesses the same two qubit entanglement features as the pure states involved, in the sense that the mixing does not destroy entanglement for a finite number of qubits, albeit it does decrease it. Furthermore, maximal mixing of symmetric states does not destroy ODLRO and classical correlations. We discuss generalizations to the subsystems of any dimensionality (i.e. higher than spin-half).
The relationship between the mean-field approximations in various interacting models of statistical physics and measures of classical and quantum correlations is explored. We present a method that allows us to find an upper bound for the total amount of correlations (and hence entanglement) in a physical system in thermal equilibrium at some temperature in terms of its free energy and internal energy. This method is first illustrated by using two qubits interacting through the Heisenberg coupling, where entanglement and correlations can be computed exactly. It is then applied to the one-dimensional (1D) Ising model in a transverse magnetic field, for which entanglement and correlations cannot be obtained by exact methods. We analyse the behaviour of correlations in various regimes and identify critical regions, comparing them with already known results. Finally, we present a general discussion of the effects of entanglement on the macroscopic, thermodynamical features of solid-state systems. In particular, we exploit the fact that a d -dimensional quantum system in thermal equilibrium can be made to correspond to a ( d +1)-dimensional classical system in equilibrium to substitute all entanglement for classical correlations.
We study macroscopic entanglement of various pure states of a one-dimensional
N-spin system with
N⪢1. Here, a quantum state is said to be macroscopically entangled if it is a superposition of macroscopically distinct states. To judge whether such superposition is hidden in a general state, we use an essentially unique index
p: A pure state is macroscopically entangled if
p=2, whereas it may be entangled but not macroscopically if
p<2. This index is directly related to fundamental stabilities of many-body states. We calculate the index
p for various states in which magnons are excited with various densities and wave numbers. We find macroscopically entangled states
(p=2) as well as states with
p=1. The former states are unstable in the sense that they are unstable against some local measurements. On the other hand, the latter states are stable in the senses that they are stable against any local measurements and that their decoherence rates never exceed
O(N) in any weak classical noises. For comparison, we also calculate the von Neumann entropy
SN∕2(N) of a subsystem composed of
N∕2 spins as a measure of bipartite entanglement. We find that
SN∕2(N) of some states with
p=1 is of the same order of magnitude as the maximum value
N∕2. On the other hand,
SN∕2(N) of the macroscopically entangled states with
p=2 is as small as
O(logN)⪡N∕2. Therefore larger
SN∕2(N) does not mean more instability. We also point out that these results are partly analogous to those for interacting many bosons. Furthermore, the origin of the huge entanglement, as measured either by
p or
SN∕2(N), is discussed to be due to spatial propagation of magnons.
Non-classical correlations arise in various quantum mechanical systems. Characterization and quantification of these correlations is an important and active branch of research in the field of quantum information theory. Investigation of non-classical correlations in condensed matter systems gives important insights about the characteristics of these systems. In particular, systems possessing a quantum critical point in their phase diagrams have attracted much attention due to the peculiar behavior of correlations near these points. In this thesis, we have investigated two distinct quantum spin models from the perspective of correlations and, we have discussed the correlation content of an important subclass of bipartite states. We start by an analytical calculation of the quantum discord for a system composed of spin-j and spin-1/2 subsystems possessing rotational symmetry. We have compared our results with the quantum discord of states having similar symmetries and seen that in rotationally invariant states the amount of quantum discord is much higher. Moreover, using the well known entanglement properties of these states, we have compared their quantum discord with entanglement and seen that quantum discord is higher than the entanglement. Next, we have investigated the thermal quantum correlations and entanglement in spin-1 Bose-Hubbard model with two and three particles. We have demonstrated that the energy level crossings in the ground state of the system are signalled by both the behavior of thermal quantum correlations and entanglement. Finally, we have investigated various thermal quantum and total correlations in the anisotropic XY spin-chain with transverse magnetic field. We have shown that the ability of the considered measures to estimate the critical points of this system at finite temperature strongly depends on the anisotropy parameter of the Hamiltonian. Furthermore, we have studied the effect of temperature on long-range correlations of the XY chain.
One of the most correlated quantum phenomena achievable is superconductivity. In superconductivity pairs of electrons condense, in a manner analogous to Bose-Einstein condensation, into the same quantum state with their subsequent entanglement permitting frictionless conductivity. Quantum computers have the potential to predict complex, entangled quantum states at non-exponential cost, unlike their classical counterparts, with possible applications to hard problems from physics to economics. Here, using pairs of qubits to represent the electron pairs, we create superconductivity on a transmon-qubit quantum computer. A series of maximally entangled superconducting states with different numbers of electron pairs is prepared, and then the occupation of the superconducting state is measured by tomography. Each occupation on the order of the number of electrons
O(N) confirms the formation of a maximally entangled superconducting state. Moreover, the large eigenvalue provides experimental validation of C. N. Yang's derivation of large occupation being associated with off-diagonal long-range order. The work provides a critically important step in predicting strongly entangled quantum physics as well as mathematically related non-local phenomena in the pure and social sciences on quantum computers.
This paper summarises the results of our research on macroscopic entanglement in spin systems and free Bosonic gases. We explain how entanglement can be observed using entanglement witnesses which are themselves constructed within the framework of thermodynamics and thus macroscopic observables. These thermodynamical entanglement witnesses result in bounds on macroscopic parameters of the system, such as the temperature, the energy or the susceptibility, below which entanglement must be present. The derived bounds indicate a relationship between the occurrence of entanglement and the establishment of order, possibly resulting in phase transition phenomena. We give a short overview over the concepts developed in condensed matter physics to capture the characteristics of phase transitions in particular in terms of order and correlation functions. Finally we want to ask and speculate whether entanglement could be a generalised order concept by itself, relevant in (quantum induced) phase transitions such as BEC, and that taking this view may help us to understand the underlying process of high-T superconductivity.
Off-diagonal long-range order (ODLRO) which is believed to be one characteristic of superconductivity is a quantum phenomenon not describable in classical mechanical terms. The quantum state constructed by η-pairing demonstrates ODLRO. Entanglement is a key concept of the quantum information processing and has no classical counterpart. We study the entanglement property of the η-pairing quantum state by concurrence and entropy which are two measures of the entanglement. We show that the concurrence of entanglement between one-site and the rest sites is exactly the correlation function of the ODLRO for the η-pairing state in the thermodynamic limit. So, when the η-pairing state is entangled, it demonstrates ODLRO and is thus in the superconducting phase, if it is a separable state, there is no ODLRO. In the thermodynamical limit, the entanglement between the M -site and other sites of the η-pairing state does not vanish. Other types of ODLRO of the η-pairing state are presented. We show that the behaviour of the ODLRO correlation functions is equivalent to that of the entanglement of the η-pairing state. The scaling of the entropy of the entanglement for the η-pairing state is studied.