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Register a new userEntanglement of Identical Particles and Coherence in the First Quantization Language
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We suggest a formalism to illustrate the entanglement of identical particles in the first quantization language (1QL). Our 1QL formalism enables one to exploit all the well-established quantum information tools to understand the indistinguishable ones, including the reduced density matrix and familiar entanglement measures. The rigorous quantitative relation between the amount of entanglement and the spatial coherence of particles is possible in this formalism. Our entanglement detection process is a generalization of the entanglement extraction protocol for identical particles with mode splitting proposed by Killoran et al. (2014).
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We reconsider the effect of indistinguishability on the reduced density operator of the internal degrees of freedom (tracing out the spatial degrees of freedom) for a quantum system composed of identical particles located in different spatial regions. We explicitly show that if the spin measurements are performed in disjoint spatial regions then there are no constraints on the structure of the reduced state of the system. This implies that the statistics of identical particles has no role from the point of view of separability and entanglement when the measurements are spatially separated. We extend the treatment to the case of n particles and show the connection with some recent criteria for separability based on subalgebras of observables.
Progress in the reliable preparation, coherent propagation and efficient detection of many-body states has recently brought collective quantum phenomena of many identical particles into the spotlight. This tutorial introduces the physics of many-boson and many-fermion interference required for the description of current experiments and for the understanding of novel approaches to quantum computing. The field is motivated via the two-particle case, for which the uncorrelated, classical dynamics of distinguishable particles is compared to the quantum behaviour of identical bosons and fermions. Bunching of bosons is opposed to anti-bunching of fermions, while both species constitute equivalent sources of bipartite two-level entanglement. The realms of indistinguishable and distinguishable particles are connected by a monotonic transition, on a scale defined by the coherence length of the interfering particles. As we move to larger systems, any attempt to understand many particles via the two-particle paradigm fails: In contrast to two-particle bunching and anti-bunching, the very same signatures can be exhibited by bosons and fermions, and coherent effects dominate over statistical behaviour. The simulation of many-boson interference, termed Boson-Sampling, entails a qualitatively superior computational complexity when compared to fermions. The hierarchy between bosons and fermions also characterises multipartite entanglement generation, for which bosons again clearly outmatch fermions. Finally, the quantum-to-classical transition between many indistinguishable and many distinguishable particles features non-monotonic structures. While the same physical principles govern small and large systems, the deployment of the intrinsic complexity of collective many-body interference makes more particles behave differently.
We identify the correlation in a state of two identical particles as the residual information beyond what is already contained in the 1-particle reduced density matrix, and propose a correlation measure based on the maximum entropy principle. We obtain the analytical results of the correlation measure, which make it computable for arbitrary two-particle states. We also show that the degrees of correlation in the same two-particle states with different particle types will decrease in the following order: bosons, fermions, and distinguishable particles.
Possible definitions for the relative momentum of identical particles are considered.
We probe the theoretical connection among three different approaches to analyze the entanglement of identical particles, i.e., the first quantization language (1QL), elementary-symmetric/exterior products (which has the mathematical equivalence to no-labeling approaches), and the algebraic approach based on the GNS construction. Among several methods to quantify the entanglement of identical particles, we focus on the computation of reduced density matrices, which can be achieved by the concept of emph{symmetrized partial trace} defined in 1QL. We show that the symmetrized partial trace corresponds to the interior product in symmetric and exterior algebra (SEA), which also corresponds to the subalgebra restriction in the algebraic approach based on GNS representation. Our research bridges different viewpoints for understanding the quantum correlation of identical particles in a consistent manner.
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