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arXiv:1805.12023 · Physical Review A

Coincidence landscapes for polarized bosons

Jizhou Wu, Hubert de Guise, Barry C. Sanders

Published Jul 10, 2018• 5 citations
arXivHTMLPDF

Abstract

Passive optical interferometry with single photons injected into some input ports and vacuum into others is enriched by admitting polarization, thereby replacing the scalar electromagnetic description by a vector theory, with the recent triad phase being a celebrated example of this richness. On the other hand, incorporating polarization into interferometry is known to be equivalent to scalar theory if the number of channels is doubled. We show that passive multiphoton $m$-channel interferometry described by $\mathrm{SU}(m)$ transformations is replaced by $\mathrm{SU}(2m)$ interferometry if polarization is included and thus that the multiphoton coincidence landscape, whose domain corresponds to various relative delays between photon arrival times, is fully explained by the now-standard approach of using immanants to compute coincidence sampling probabilities. Consequently, we show that the triad phase is manifested simply as SU(6) interferometry with three input photons, with one photon in each of three different input ports. Our analysis incorporates passive polarization multichannel interferometry into the existing scalar-field approach to computing multiphoton coincidence probabilities in interferometry and demystifies the triad phase.

DOI
10.1103/physreva.98.013817

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Passive optical interferometry with single photons injected into some input ports and vacuum into others is enriched by admitting polarization, thereby replacing the scalar electromagnetic description by a vector theory, with the recent triad phase being a celebrated example of this richness. On the other hand, incorporating polarization into interferometry is known to be equivalent to scalar theory if the number of channels is doubled. We show that passive multiphoton mmm-channel interferometry described by SU(m)\mathrm{SU}(m)SU(m) transformations is replaced by SU(2m)\mathrm{SU}(2m)SU(2m) interferometry if polarization is included and thus that the multiphoton coincidence landscape, whose domain corresponds to various relative delays between photon arrival times, is fully explained by the now-standard approach of using immanants to compute coincidence sampling probabilities. Consequently, we show that the triad phase is manifested simply as SU(6) interferometry with three input photons, with one photon in each of three different input ports. Our analysis incorporates passive polarization multichannel interferometry into the existing scalar-field approach to computing multiphoton coincidence probabilities in interferometry and demystifies the triad phase.
We use permutation-group methods plus SU(3) group-theoretic methods to determine the action of a three-channel passive optical interferometer on controllably delayed single-photon pulse inputs to each channel. Permutation-group techniques allow us to relate directly expressions for rates and, in particular, investigate symmetries in the coincidence landscape. These techniques extend the traditional Hong-Ou-Mandel effect analysis for two-channel interferometry to valleys and plateaus in three-channel interferometry. Our group-theoretic approach is intuitively appealing because the calculus of Wigner D functions partially accounts for permutational symmetries and directly reveals the connections among D functions, partial distinguishability, and immanants.
We obtain coincidence rates for passive optical interferometry by exploiting the permutational symmetries of partially distinguishable input photons, and our approach elucidates qualitative features of multiphoton coincidence landscapes. We treat the interferometer input as a product state of any number of photons in each input mode with photons distinguished by their arrival time. Detectors at the output of the interferometer count photons from each output mode over a long integration time. We generalize and prove the claim of Tillmann et al. [Phys. Rev. X 5, 041015 (2015)] that coincidence rates can be elegantly expressed in terms of immanants. Immanants are functions of matrices that exhibit permutational symmetries and the immanants appearing in our coincidence-rate expressions share permutational symmetries with the input state. Our results are obtained by employing representation theory of the symmetric group to analyze systems of an arbitrary number of photons in arbitrarily sized interferometers.
Multiphoton interference is at the very heart of quantum foundations and applications in quantum sensing and information processing. In particular, boson-sampling experiments have the potential to demonstrate quantum computational supremacy while relying on only multiphoton interference in linear optical interferometers. However, even when photonic losses are negligible, scalable experiments are challenged by the rapid decrease of the probability of the success of current schemes with probabilistic sources for a large number of single photons in each experimental sample. Remarkably, we show a boson sampling scheme in which the probability of success increases instead of decreasing with the number of input photons, eventually approaching a unit value even with nondeterministic sources. It is achieved by sampling measurements not only at the interferometer output but also over all the possible numbers of photons at each input port and, different from scattershot boson sampling, allowing a random number of occupied input ports from one experimental run to another. Therefore, these results provide an exciting route toward future demonstrations of quantum computational supremacy with scalable experimental resources.
Quantum information holds enormous potential to achieve communication and computational tasks that are impossible using classical systems. Many quantum information processing tasks can be implemented by exploiting the passive interference of photons. In this thesis, we reports advances towards the analysis, simulation, characterization, and design of passive interferometry experiments for quantum information processing applications. First, we develop a theory of passive optical interferometry experiments that relates coincidence rates at the output of an interferometer to the permutational symmetries of partially distinguishable photons. In our formalism, coincidence rates are elegantly expressed in terms of immanants, which are matrix functions that exhibit permutational symmetries, and the immanants appearing in our coincidence-rate expressions share permutational symmetries with the input state. Our formalism thus provides an intuitive qualitative understanding of how the permutational symmetries of distinguishable photons determine their interference. Our approach can be used to better understand the effect of partial distinguishability in modeling passive interferometry experiments aiming to perform quantum information tasks such as linear optical quantum computing and BosonSampling. By exploiting symmetries, we can also reduce some calculational tasks and provide additional insights into relations between the coincidence rates in various situations. Second, we tackle the problem of characterizing passive interferometers, essential for modelling any passive interferometry experiment. We develop a characterization procedure that uses only one- and two-photon coincidence data, and makes a number of advancements over existing procedures. We estimate the errors in the characterization via bootstrapping, a statistical technique with the advantage of not assuming any particular error model on the experimental data. We also introduce a scattershot approach to reduce the experimental data collection time quadratically in the number of modes of the interferometer. Third, we study a relativistic quantum information task called quantum summoning. We cast quantum summoning as an interactive protocol between a verifier and a possibly dishonest prover. For the prover, we present a protocol and quantum codes to summon in any valid configuration of causal diamonds, and design the encoding and decoding circuits for our code. Our protocol decreases the space complexity for encoding by a factor of two compared to the previous best result and reduces the gate complexity from scaling as the cube to the square of the number of causal diamonds. Within our interactive protocol framework, we construct an operational definition of quantum summoning, and use this definition to present a verification test. We prove that our verification test is sound and complete, thereby showing that quantum summoning can be verified by a resource-limited verifier in an adversarial setting. Our protocol and verification procedures are amenable to implementation by passive interferometry, thereby extending the reach of passive interferometry in the field of relativistic quantum information processing.