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arXiv:quant-ph/0208179 · Physical Review A

Classicality of spin-coherent states via entanglement and distinguishability

Damian Markham, Vlatko Vedral

Published Apr 28, 2003• 43 citations
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Abstract

We trace the resistance to entanglement generation of spin-coherent states when passed through a beam splitter as we vary S through $S=1/\stackrel{\ensuremath{\rightarrow}}{2}\ensuremath{\infty}.$ In the infinite S limit the spin-coherent states are equivalent to the high-amplitude limit of the optical coherent states. These states generate no entanglement and are completely distinguishable. This transition is discussed in terms of the classicality of the states. The decline of the generated entanglement, and in this sense increase in classicality with S, is very slow and dependent on the amplitude z of the state. Surprisingly, we find that, for $|z|>1,$ there is an initial increase in entanglement followed by an extremely gradual decline to zero. Other aspects of classicality are also discussed over the transition in S, including the distinguishability, which decreases quickly and monotonically. We illustrate the distinguishability of spin-coherent states using the representation of Majorana.

DOI
10.1103/physreva.67.042113

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We trace the resistance to entanglement generation of spin-coherent states when passed through a beam splitter as we vary S through S=1/2\ensuremath→\ensuremath∞.S=1/\stackrel{\ensuremath{\rightarrow}}{2}\ensuremath{\infty}.S=1/2\ensuremath→​\ensuremath∞. In the infinite S limit the spin-coherent states are equivalent to the high-amplitude limit of the optical coherent states. These states generate no entanglement and are completely distinguishable. This transition is discussed in terms of the classicality of the states. The decline of the generated entanglement, and in this sense increase in classicality with S, is very slow and dependent on the amplitude z of the state. Surprisingly, we find that, for ∣z∣>1,|z|>1,∣z∣>1, there is an initial increase in entanglement followed by an extremely gradual decline to zero. Other aspects of classicality are also discussed over the transition in S, including the distinguishability, which decreases quickly and monotonically. We illustrate the distinguishability of spin-coherent states using the representation of Majorana.
Using the linear entropy as a measure of entanglement, we investigate the effect of a beam splitter on the Perelomov coherent states for the q-deformed Uq(su(2)) algebra. We distinguish two cases: in the classical q → 1 limit, we find that the states become Glauber coherent states as the spin tends to infinity; whereas for q ≠ 1, the states, contrary to the earlier case, become entangled as they pass through a beam splitter. The entanglement strongly depends on the q-deformation parameter and the amplitude Z of the state.
Recently, Markham and Vedral [Phys. Rev. A 67, 042113 (2003)] investigated the effect of beam splitting on the spin, or SU(2), coherent states for a single mode field. The spin coherent state is a binomial coherent state related to the Holstein-Primakoff realization of the su(2) Lie algebra given in terms of a set of single mode bose annihilation and creation operators. Upon beam splitting, the ordinary (or Glauber) coherent states merely split into products of ordinary coherent states with reduced amplitudes without becoming entangled, as one would expect for a classical-like field. The above authors expected the spin coherent states to go over to the ordinary coherent states in the limit of high spin, $j\ensuremath{\rightarrow}\ensuremath{\infty}$, and thus to become product states after beam splitting. But this expectation was not confirmed through numerical calculation of the entropy which, instead of going to zero, leveled off with increasing spin. In this paper we find similar behavior for SU(1,1) coherent states of the Perelomov type for large Bargman index kkk, but also find that the Barut-Girardello SU(1,1) coherent states appear to rapidly become product states after beam splitting for increasing kkk. We explain these results by showing that, in reality, neither the spin coherent states nor the Perelomov SU(1,1) coherent states go over to ordinary coherent states in the limits of large jjj or kkk, and that the Barut-Girardello coherent states merely go over to the vacuum in the large kkk limit. Finally, we examine the correct limiting procedure for obtaining separable states (i.e., products of coherent states) upon beam splitting by performing contractions of the su(2) and su(1,1) Lie algebras and of their associated coherent states.
Quantum-classical transitions have long attracted much attention. We study such transitions in quantum spin-(jjj,1/2) systems at thermal equilibrium. Unlike previous work [Phys. Rev. A 73, 064302 (2006)], we find that the threshold temperature of quantum entanglement decreases with increasing spin jjj and completely disappears in the limit $j\ensuremath{\rightarrow}\ensuremath{\infty}$. In the ground state of systems with highly symmetric interactions, the discord-type quantum correlations can exist even for arbitrarily large spin. Such correlations turn out to be unstable and are destroyed by small perturbations that violate the symmetry of the Hamiltonian. The stable quantum correlations gradually degrade as the spin jjj grows and eventually vanish when the classical limit is reached.
We prove that a beam splitter, one of the most common optical components, fulfills several classes of majorization relations, which govern the amount of quantum entanglement that it can generate. First, we show that the state resulting from kkk photons impinging on a beam splitter majorizes the corresponding state with any larger photon number ${k}^{\ensuremath{'}}>k$, implying that the entanglement monotonically grows with kkk. Then we examine parametric infinitesimal majorization relations as a function of the beam-splitter transmittance and find that there exists a parameter region where majorization is again fulfilled, implying a monotonic increase of entanglement by moving towards a balanced beam splitter. We also identify regions with a majorization default, where the output states become incomparable. In this latter situation, we find examples where catalysis may nevertheless be used in order to recover majorization. The catalyst states can be as simple as a path-entangled single-photon state or a two-mode squeezed vacuum state.
The study of new quantum states and methods for their generation is an important part of the current development of quantum technologies. One of the most important characteristics of such states is their quantum entanglement. Photon-added coherent states, which are defined as |α,n⟩=Nn(a^†)n|α⟩{\text{|}}\alpha ,n\rangle = {{\mathcal{N}}_{n}}{{({{\hat {a}}^{\dag }})}^{n}}{\text{|}}\alpha \rangle |α,n⟩=Nn​(a^†)n|α⟩ , where Nn{{\mathcal{N}}_{n}}Nn​ is the normalization constant, |α⟩{\text{|}}\alpha \rangle |α⟩ is the coherent state, a^†{{\hat {a}}^{\dag }}a^† is the creation operator, and n is the number of added photons, have been introduced relatively recently. The quantum entanglement of such states passing through a beam splitter has been considered for the first time in this work. A simple analytical representation of the wavefunction at the outputs of the beam splitter has been found and the quantum entanglement of the resulting two-qubit states has been considered. It has been shown that new quantum states have good prospects for their use in quantum technologies.
The nonclassicality of a macroscopic single-mode optical superposition state is potentially convertible into entanglement, when the state is mixed with the vacuum on a beam splitter. Considering light beams with polarization degree of freedom in Euclidean space as coherent product states in a bipartite Hilbert space, we propose a method to convert the two orthogonal polarizations into simultaneous entanglement and classical nonseparability through nonclassicality in the superpositions of coherent and displaced Fock states. Equivalent Bell state emerges from the resulted superpositions and the proportion of mixed entanglement and nonseparablity is determined by the displacement amplitudes along the polarization directions. We characterize the state nonclassicality via features in Wigner distributions and propose an experimental method for generating these states and measuring them via homodyne tomography.