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arXiv:quant-ph/0411040 · Physical Review A

Quantum entanglement as an interpretation of bosonic character in composite two-particle systems

C. K. Law

Published Mar 21, 2005• 74 citations
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Abstract

We consider a composite particle formed by two fermions or two bosons. We discover that composite behavior is closely related to the quantum entanglement between the constituent particles. By analyzing the properties of creation and annihilation operators, we show that bosonic character emerges if the constituent particles become strongly entangled. Such a connection is demonstrated explicitly in a class of two-particle wave functions.

DOI
10.1103/physreva.71.034306

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We consider a composite particle formed by two fermions or two bosons. We discover that composite behavior is closely related to the quantum entanglement between the constituent particles. By analyzing the properties of creation and annihilation operators, we show that bosonic character emerges if the constituent particles become strongly entangled. Such a connection is demonstrated explicitly in a class of two-particle wave functions.
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Composite particles made of elementary fermions can exhibit a wide range of behavior ranging from fermionic to bosonic depending on the quantum state of the fermions and the experimental situation considered. This behavior is captured by the fundamental operations of single-particle addition and subtraction and two-particle interference. We analyze the quantum channels that implement the physical operations of addition and subtraction of indistinguishable particles. In particular, we construct optimal Kraus operators to implement these probabilistic operations for systems of a finite number of particles. We then use these to measure the quality of bosonic and fermionic behavior in terms of single-particle addition and subtraction and two-particle interference. For the specific case of composite particles made of two distinguishable fermions, we find a transition from fermionic to bosonic behavior as a function of the entanglement between the two constituents. We also apply these considerations to composite particles of two distinguishable bosons and identify the relation between entanglement and bosonic behavior for these systems.
We build upon work by C. K. Law [Phys. Rev. A 71, 034306 (2005)10.1103/PhysRevA.71.034306] to show in general that the entanglement between two fermions largely determines the extent to which the pair behaves like an elementary boson. Specifically, we derive upper and lower bounds on a quantity chi{N+1}/chi{N} that governs the bosonic character of a pair of fermions when N such pairs approximately share the same wave function. Our bounds depend on the purity of the single-particle density matrix, an indicator of entanglement, and demonstrate that if the entanglement is sufficiently strong, the quantity chi{N+1}/chi{N} approaches its ideal bosonic value.
We consider composite bosons (cobosons) comprised two elementary particles, fermions or bosons, in an entangled state. First, we show that the effective number of cobosons implies the level of correlation between the two constituent particles. For the maximum level of correlation, the effective number of cobosons is the same as the total number of cobosons, which can exhibit the original Bose–Einstein condensation (BEC). In this context, we study a model of BEC for indistinguishable cobosons with a controllable parameter, i.e., entanglement between the two constituent particles. We find that bi-fermions behave in a predictable way, i.e., the effective number of the ground state coboson is an increasing function of entanglement between a pair of constituent fermions. Interestingly, bi-bosons exhibit the opposite behavior—the effective number of the ground state coboson is a decreasing function of entanglement between a pair of constituent bosons.
We study the necessary conditions for systems composed of two distinguishable fermions to exhibit bosonic behavior. We base our analysis on tools of quantum information theory such as entanglement, local operations and classical communication and the majorization criterion for probability distributions. Introduction of these tools yields new insights into the old and important problem of identifying the conditions for existence of composite bosons. We precisely identify the role entanglement plays in the formation and behavior of such composite particles.