Quantum entanglement bears some resemblance to ordinary classical correlation, but there are important differences. In particular, whereas any number of classical variables can in principle be perfectly correlated with each other–the temperature fluctuations in ten different cities could conceivably be exactly parallel–any entanglement that exists between two quantum objects has the effect of limiting the entanglement that either of these objects can have with the rest of the world. In this talk I will discuss a number of examples of quantum systems in which this restriction on the sharing of entanglement can be expressed quantitatively. In the special case of three spin-1/2 particles A, B, and C, there is a convenient measure of entanglement called the concurrence in terms of which the trade-off between the AB entanglement and the AC entanglement takes a particularly simple form: the sum of the squares cannot exceed unity. For larger spins, preliminary results suggest that the limitation on entanglement sharing is less stringent. Finally, for a collection of many spin-1/2 particles arranged in a line, the problem of sharing entanglement points to an interesting feature of antiferromagnetic chains, namely, that they favor states with high nearest-neighbor entanglement.
It is a central trait of quantum information theory that there exist limitations to the free sharing of quantum correlations among multiple parties. Such monogamy constraints have been introduced in a landmark paper by Coffman, Kundu and Wootters, who derived a quantitative inequality expressing a trade-off between the couplewise and the genuine tripartite entanglement for states of three qubits. Since then, a lot of efforts have been devoted to the investigation of distributed entanglement in multipartite quantum systems. In this paper we report, in a unifying framework, a bird's eye view of the most relevant results that have been established so far on entanglement sharing in quantum systems. We will take off from the domain of N qubits, graze qudits, and finally land in the almost unexplored territory of multimode Gaussian states of continuous variable systems.
Quantum mechanical objects can exhibit correlations with one another that are fundamentally at odds with the paradigm of classical physics; one says that the objects are ‘entangled’. In the last few years, entanglement has come to be studied not only as a marvel of nature but as a potential resource, particularly as a resource for certain unusual kinds of communication. This paper reviews two such uses of entanglement, called ‘teleportation’ and ‘dense coding’. Teleportation is the direct, though not instantaneous, transfer of a quantum state from one object to another over a distance. Dense coding is the effective doubling of the information–carrying capacity of a quantum particle through prior entanglement with a particle at the receiving end. The final section of the paper presents various quantitative measures of entanglement and considers novel features that arise when entanglement is shared among three objects.
Entanglement represents purely quantum effect involving two or more particles. Spin systems are good candidates for studying this effect and its relation with other collective phenomena ruled by quantum mechanics. While the presence of entangled states can be easily verified, the quantitative estimate of this property is still under investigation. One of the most useful tools in this framework is the concurrence whose definition, albeit limited to S = 1/2 systems, can be related to the correlators. We consider quantum spin systems defined along chains and square lattices, and described by Heisenberg-like Hamiltonians: our goal is to clarify the relation between entanglement and quantum phase transitions, as well as that between the concurrence the and the specific quantum state of the system.
In this paper, the low-temperature (${k}_{B}T\ensuremath{\ll}U$) frequency-dependent $\ensuremath{\sigma}(\ensuremath{\omega})$ conductivity in a half-filled Hubbard chain is calculated to the lowest order in the parameter
Ut and in closed form to all orders in $\frac{t}{\ensuremath{\omega}}\ensuremath{-}U$. Here
t is the nearest-neighbor hopping integral and
U the intra-atomic Coulomb interaction. The problem is formulated in terms of the number of random walks of an electron and a hole which leave the spin configuration unchanged, at temperatures such that ${k}_{B}T\ensuremath{\ll}t\ensuremath{\ll}U$ (where
kB is Boltzmann's constant). The conductivity then depends on spin configuration, and for both the antiferromagnetic (AF) (
T=0 K) ground state and the equiprobable or random arrangement (
R) (${k}_{B}T\ensuremath{\gg}\frac{{t}^{2}}{U}$) an exact expression is found. For the partially disordered regime ($T\ensuremath{\simeq}{T}_{N}$) the thermodynamical averaging is expressed in terms of a phenomenological spin-spin correlation length
l which interpolates between the AF and
R exact limits. This solution is such that for $l=\ensuremath{\infty}$ the AF line shape is recovered except at the center of the spectrum (i.e., at $\ensuremath{\omega}=U$), where the absorption always diverges logarithmically. For a ferromagnetic (
l=0) arrangement induced by a magnetic field, the absorption then vanishes. The line shapes obtained have sharp edges at $|\ensuremath{\omega}|=U\ifmmode\pm\else\textpm\fi{}4t$ for all temperatures. At the critical value of
4t=U the AF ground state is found to undergo a metallic transition. A closed-form analytical expression in terms of the three elliptical integrals is given for the absorption lines. Except for the AF case, which has a square-root edge singularity, the absorption is found to vanish linearly. Finally, a comparison is made with some recent calculations.
Entanglement sharing among pairs of spins in Heisenberg antiferromagnets is investigated using the concurrence measure. For a nondegenerate
S=0 ground state, a simple formula relates the concurrence to the diagonal correlation function. The concurrence length is seen to be extremely short. Various finite clusters are studied numerically, to see the trend in two dimensions. It is argued that the concurrences vanish between pairs of spins which are not nearest neighbors for the linear-chain and square-lattice antiferromagnet ground states. For the triangular lattice and Kagome' lattice, nearest-neighbor concurrences also vanish. The concurrences in the maximal-spin states are explicitly calculated, where the concurrence averaged over all pairs is larger than the
S=0 states.
Entanglement can be understood as an extraordinary degree of correlation between states of quantum systems – a correlation that cannot be given an explanation in terms of something like a common cause. Entanglement can occur between two or more quantum systems, and the most interesting case is when these correlations occur between systems that are space-like separated, meaning that changes made to one system are immediately correlated with changes in a distant system even though there is no time for a signal to travel between them. In this case one says that quantum entanglement leads to non-local correlations, or non-locality. More precisely, entanglement can be defined in the following way. Consider two particles, A and B, whose (pure) states can be represented by the state vectors ψA and ψB. Instead of representing the state of each particle individually, one can represent the composite two-particle system by another wavefunction, Ψ AB. If the two particles are unentangled , then the composite state is just the tensor product of the states of the components: Ψ AB = ψ A ⊗ ψ B; the state is then said to be factorable (or separable). If the particles are entangled, however, then the state of the composite system cannot be written as such a product of a definite state for A and a definite state for B. This is how an entangled state is defined for pure states: a state is entangled if and only if it cannot be factored: Ψ AB ≠ ψ A ⊗ ψ B.