We devise a multiphoton interferometry scheme for sampling a quadratic function of a specific immanant for any submatrix of a unitary matrix and its row permutations. The full unitary matrix describes a passive, linear interferometer, and its submatrix is used when photons enter in and are detected at subsets of possible input and output channels. Immanants are mathematical constructs that interpolate between the permanent and determinant; contrary to determinants and permanents, which have meaningful physical applications, immanants are devoid of physical meaning classically but here are shown to be meaningful in a quantum setting. Our quadratic form of immanants is sampled by injecting vacuum and single photons into interferometer input ports such that the photon arrival times are entangled, in contrast to previous methods that control arrival times without entangling. Our method works for any number of photons, and we solve explicitly the quadratic form for the two-, three-, and four-photon cases.
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.
This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary
nsnptimesnsnp unitary matrix on the state of light in
ns spatial and
np internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU
(n) basis states, which reduce the canonical subgroup chain, for arbitrary
n. The boson realizations are employed to construct
mathcalD-functions, which are the matrix elements of arbitrary irreducible representations, of SU
(n) in the canonical basis. I show that immanants of principal submatrices of a unitary matrix
T are a sum of the diagonal
mathcalD(Omega)-functions of group element
Omega over
t determined by the choice of submatrix and over the irrep
(lambda) determined by the immanant under consideration. The algorithm for
mathrmSU(n) mathcalD-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.
This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary
nsnp×nsnp unitary matrix on the state of light in
ns spatial and
np internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU
(n) basis states, which reduce the canonical subgroup chain, for arbitrary
n. The boson realizations are employed to construct
D-functions, which are the matrix elements of arbitrary irreducible representations, of SU
(n) in the canonical basis. I show that immanants of principal submatrices of a unitary matrix
T are a sum of the diagonal
D(Ω)-functions of group element
Ω over
t determined by the choice of submatrix and over the irrep
(λ) determined by the immanant under consideration. The algorithm for
SU(n) D-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.
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.