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arXiv:quant-ph/0612081 · Physical Review A

Detecting hidden differences via permutation symmetries

Robert B. A. Adamson, P. S. Turner, Morgan W. Mitchell, Aephraim M. Steinberg

Published Sep 22, 2008• 27 citations
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Abstract

We present a method for describing and characterizing the state of $N$ particles that may be distinguishable in principle but not in practice due to experimental limitations. The technique relies upon a careful treatment of the exchange symmetry of the state among experimentally accessible and experimentally inaccessible degrees of freedom. The approach we present allows a formalization of the notion of indistinguishability and can be implemented easily using currently available experimental techniques. Our work is of direct relevance to current experiments in quantum optics, for which we provide a specific implementation.

DOI
10.1103/physreva.78.033832

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We present a method for describing and characterizing the state of NNN particles that may be distinguishable in principle but not in practice due to experimental limitations. The technique relies upon a careful treatment of the exchange symmetry of the state among experimentally accessible and experimentally inaccessible degrees of freedom. The approach we present allows a formalization of the notion of indistinguishability and can be implemented easily using currently available experimental techniques. Our work is of direct relevance to current experiments in quantum optics, for which we provide a specific implementation.
Using tools from quantum information theory, we present a general theory of indistinguishability of identical bosons in experiments consisting of passive linear optics followed by particle number detection. Our results do neither rely on additional assumptions on the input state of the interferometer, such as, for instance, a fixed mode occupation, nor on any assumption on the degrees of freedom that potentially make the particles distinguishable. We identify the expectation value of the projector onto the N-particle symmetric subspace as an operationally meaningful measure of indistinguishability, and derive tight lower bounds on it that can be efficiently measured in experiments. Moreover, we present a consistent definition of perfect distinguishability and characterize the corresponding set of states. In particular, we show that these states are diagonal in the computational basis up to a permutationally invariant unitary. Moreover, we find that convex combinations of states that describe partially distinguishable and perfectly indistinguishable particles can lead to perfect distinguishability, which itself is not preserved under convex combinations.
Identical particles have dynamically invariant labels in all setups where their degrees of freedom can be partitioned into two parts, which we call internal and visible, with the visible part subject to unitary evolution, while the dynamically invariant internal part accounts for the partial distinguishability of particles. We give the explicit form of the visible state in terms of the indistinguishability function, generalizing the standard symmetrization/antisymmetrization postulate for bosons/fermions with internal modes. The indistinguishability function on the symmetric group accounts for the permutation symmetry of the dynamically invariant label state and gives a self-contained description of the symmetry spectrum of the visible state. It is shown that the indistinguishability of single particles emitted independently from a stable source is completely characterized by the projective measures on the symmetry spectrum. We reveal a hierarchy of successive upper bounds on the projective measure of indistinguishability in terms of marginal projective measures. We point out an experimentally feasible way to get the upper bounds on the indistinguishability of an arbitrarily large number of single photons by direct experimental readout of their marginal projective measures. We also analyze the indistinguishability of coherent superpositions and convex mixtures and resolve a recently posed problem.
Abstract We show how to characterize the indistinguishability of up to four identical, bosonic or fermionic particles, which are rendered partially distinguishable through their internal degrees of freedom prepared in mixed states. This is accomplished via their counting statistics when subjected to a highly symmetric unitary acting upon their external (i.e. dynamical) degrees of freedom. For pure internal states, we further extract information on the particles’ collective phases, which ultimately allows for an experimental reconstruction of the full many-particle density operator up to complex conjugation.
Abstract Quantum theory stipulates that if two particles are identical in all physical aspects, the allowed states of the system are either symmetric or antisymmetric with respect to permutations of the particle labels. Experimentally, the symmetry of the states can be inferred indirectly from the fact that neglecting the correct exchange symmetry in the theoretical analysis leads to dramatic discrepancies with the observations. The only way to directly unveil the symmetry of the states for, say, two identical particles is through the interference of the two-particle state and the physically permuted one, and measuring the phase associated with the permutation process, the so-called particle exchange phase. Following this idea, we have measured the exchange phase of indistinguishable photons, providing direct evidence of the bosonic character of photons.
If a system in atomic physics contains a number of particles of the same kind, e.g. a number of electrons, the particles are absolutely indistinguishable one from another. No observable change is made when two of them are interchanged … A satisfactory theory ought, of course, to count any two observationally indistinguishable states as the same state and to deny that any transition does occur when two similar particles exchange places. ( Dirac, 1958, p. 207 ) Introduction In our contribution to this volume we deal with discrete symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In physics we find that discrete symmetries frequently arise as ‘internal’, non-spacetime symmetries. Permutation symmetry is such a discrete symmetry, arising as the mathematical basis underlying the statistical behaviour of ensembles of certain types of indistinguishable quantum particle (e.g. fermions and bosons). Roughly speaking, if such an ensemble is invariant under a permutation of its constituent particles (i.e. permutation symmetric) then one doesn't ‘count’ those permutations which merely ‘exchange’ indistinguishable particles; rather, the exchanged state is identified with the original state. This principle of invariance is generally called the ‘indistinguishability postulate’ (IP), but we prefer to use the term ‘permutation invariance’ (PI).
We address the problem of completely characterizing multiparticle states including loss of information to unobserved degrees of freedom. In systems where nonclassical interference plays a role, such as linear-optics quantum gates, such information can degrade interference in two ways, by decoherence and by distinguishing the particles. Distinguishing information, often the limiting factor for quantum optical devices, is not correctly described by previous state-reconstruction techniques, which account only for decoherence. We extend these techniques and find that a single modified density matrix can completely describe partially coherent, partially distinguishable states. We use this observation to experimentally characterize two-photon polarization states in single-mode optical fiber.