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arXiv:0705.4177 · New Journal of Physics

Path–phase duality with intraparticle translational–internal entanglement

Michal Kolář, Tomáš Opatrný, Nir Bar‐Gill, Noam Erez A, Gershon Kurizki

Published May 17, 2007• 13 citations
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Abstract

The aim of this paper is to revisit the implications of complementarity when we inject into a Mach Zehnder interferometer particles with internal structure, prepared in special translational–internal entangled (TIE) intraparticle states. This correlation causes the path distinguishability to be interferometric phase-dependent in contrast to the standard case, where distinguishability depends on some external parameters (not interferometric phase). We show that such a TIE state permits us to detect small phase shifts along with almost perfect path distinguishability, beyond the constraints imposed by complementarity on simultaneous which-way and which-phase measurements for cases when distinguishability is uncoupled to interferometric phase.

DOI
10.1088/1367-2630/9/5/129

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The aim of this paper is to revisit the implications of complementarity when we inject into a Mach Zehnder interferometer particles with internal structure, prepared in special translational–internal entangled (TIE) intraparticle states. This correlation causes the path distinguishability to be interferometric phase-dependent in contrast to the standard case, where distinguishability depends on some external parameters (not interferometric phase). We show that such a TIE state permits us to detect small phase shifts along with almost perfect path distinguishability, beyond the constraints imposed by complementarity on simultaneous which-way and which-phase measurements for cases when distinguishability is uncoupled to interferometric phase.
Abstract. We examine two setups that reveal different operational implications of path-phase complementarity for single photons in a Mach-Zehnder interferometer (MZI). In both setups, the which-way (WW) information is recorded in the polarization state of the photon serving as a “flying which-way detector”. In the “predictive” variant, using a fixed initial state, one obtains duality relation between the probability to correctly predict the outcome of either a which-way (WW) or which-phase (WP) measurement (equivalent to the conventional path-distinguishibility-visibility). In this setup, only one or the other (WW or WP) prediction has operational meaning in a single experiment. In the second, “retrodictive ” protocol, the initial state is secretly selected for each photon by one party, Alice, among a set of initial states which may differ in the amplitudes and phases of the photon in each arm of the MZI. The goal of the other party, Bob, is to retrodict the initial state by measurements on the photon. Here, a similar duality relation between WP and WW probabilities, governs their simultaneous guesses in each experimental run.
We examine two setups that reveal different operational implications of path-phase complementarity for single photons in a Mach-Zehnder interferometer (MZI). In both setups, the which-way (WW) information is recorded in the polarization state of the photon serving as a "flying which-way detector". In the "predictive" variant, using a {\em fixed} initial state, one obtains duality relation between the probability to correctly predict the outcome of either a which-way (WW) or which-phase (WP) measurement (equivalent to the conventional path-distinguishibility-visibility). In this setup, only one or the other (WW or WP) prediction has operational meaning in a single experiment. In the second, "retrodictive" protocol, the initial state is secretly selected {\em for each photon} by one party, Alice, among a set of initial states which may differ in the amplitudes and phases of the photon in each arm of the MZI. The goal of the other party, Bob, is to retrodict the initial state by measurements on the photon. Here, a similar duality relation between WP and WW probabilities, governs their {\em simultaneous} guesses in {\em each experimental run}.
An operational way of measuring entanglement in a balanced two-path interferometers is presented, where path information is carried by some internal degree of freedom which, in turn, gets entangled with an ancilla system. The analysis is based on a tripartite description involving paths, an internal qubit degree of freedom, and some ancillary states entangled with the internal degree of freedom. It is then applied to two physically distinct experimental situations: a modified Stern-Gerlach interferometer with spin-1/2 particles and a Mach-Zehnder interferometer with photons carrying polarization. The ancilla degree of freedom may not be experimentally accessible. Tracing out the ancillary system, and employing a quantum erasing procedure based on the internal degree of freedom, it is demonstrated that a concurrence-based measure of the entanglement, between the internal degree of freedom and the ancilla, can be extracted directly from measurable asymmetry of the two output channels. These results show that loss of coherence, quantum erasure, and entanglement estimation in interferometric experiments arise from the same underlying correlation structure and provide a compact experimentally accessible framework for quantifying entanglement in qubit systems.
The complementarity relations impose the constraints on different aspects of quantum states. We study the complementarity relation within a multi-path interferometer that includes detectors and quantum memory. Here we consider the mixed states as the input states. We establish a duality relation between the visibility and the path distinguishability. Based on this duality, two triality relations, one is related with visibility, path distinguishability, and mixedness, the other is related with visibility, path distinguishability, and entanglement, were derived respectively. Therefore, the role of entanglement in multi-path interferometer is characterized quantitatively. These complementarity relations are all complete for the two-path interferometer.