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arXiv:1901.06178 · Physical Review A

Method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe

Juan Carlos García-Escartín, Vicent Gimeno, Julio José Moyano‐Fernández

Published Aug 2, 2019• 10 citations
arXivHTMLPDF

Abstract

The evolution of quantum light through linear optical devices can be described by the scattering matrix $S$ of the system. For linear optical systems with $m$ possible modes, the evolution of $n$ input photons is given by a unitary matrix $U={\ensuremath{\varphi}}_{m,M}(S)$, derived from a known homomorphism, ${\ensuremath{\varphi}}_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states, $M$. We present a method to decide whether a given unitary evolution $U$ for $n$ photons in $m$ modes can be achieved with linear optics or not and the inverse transformation ${\ensuremath{\varphi}}_{m,M}^{\ensuremath{-}1}$ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

DOI
10.1103/physreva.100.022301

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The evolution of quantum light through linear optical devices can be described by the scattering matrix SSS of the system. For linear optical systems with mmm possible modes, the evolution of nnn input photons is given by a unitary matrix U=\ensuremathφm,M(S)U={\ensuremath{\varphi}}_{m,M}(S)U=\ensuremathφm,M​(S), derived from a known homomorphism, \ensuremathφm,M{\ensuremath{\varphi}}_{m,M}\ensuremathφm,M​, which depends on the size of the resulting Hilbert space of the possible photon states, MMM. We present a method to decide whether a given unitary evolution UUU for nnn photons in mmm modes can be achieved with linear optics or not and the inverse transformation \ensuremathφm,M\ensuremath−1{\ensuremath{\varphi}}_{m,M}^{\ensuremath{-}1}\ensuremathφm,M\ensuremath−1​ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.
Quantum linear optics without post-selection is not powerful enough to produce any quantum state from a given input state. This limits its utility since some applications require entangled resources that are difficult to prepare. Thus, a deeper understanding of linear optical state preparation is needed. In this work, we give a recipe to derive conserved quantities in the evolution of arbitrary states along any possible passive linear interferometer. One example of such an invariant is the spectrum of a density matrix mapped onto the Lie algebra of passive linear optical Hamiltonians. These invariants give necessary conditions for exact state preparation: if the input and output states have different invariants, it is impossible to design a passive linear interferometer that evolves one into the other. Moreover, we provide a lower bound to the distance between an output and target state based on the distance between their invariants. This gives a necessary condition for approximate or heralded state preparations. Therefore, the invariants allow us to narrow the search when trying to prepare useful entangled states, like NOON states, from easy-to-prepare states, like Fock states. We conclude that future exact and approximate state preparation methods will need to consider the necessary conditions given by our invariants to weed out impossible linear optical evolutions.
Linear optical networks play a paramount role in current implementations of quantum information protocols using photons. A paradigmatic example of their potential is given by boson sampling experiments, which recently have reached the quantum advantage regime [1], [2]. Such linear devices are essentiallyNNN-mode interferometers, whose operation is entirely captured by anN×NN\times NN×Nunitary matrixUUUwith complex entriesuji=∣uji∣eθjiu_{j i}=\left\vert u_{j i}\right\vert e^{\theta_{j i}}uji​=∣uji​∣eθji​. This matrix characterizes, in a classical framework, the transformation of the phasorsεj(in)\varepsilon_{j}^{(in)}εj(in)​of the input field across the different modes into the phasorsεj(out)\varepsilon_{j}^{(out)}εj(out)​of the output fields according to the linear relationεi(out)=∑iujiεi(in)\varepsilon_i^{({out })}=\sum_i u_{j i} \varepsilon_i^{({in })}εi(out)​=∑i​uji​εi(in)​).
Unitary operations are a specific class of linear transformations that have become an essential ingredient for the realization of classical and quantum information processing. The ability of implementing any n-dimensional unitary signal transformation by using a reconfigurable optical hardware has recently led to the pioneering concept of programmable linear optical processor, whose basic building block (BB) must be correctly designed to guarantee that the whole system is able to perform n × n universal (i.e., arbitrary) unitary matrix transformations. Here, it is demonstrated that the present architectures of the BB do not fulfil the universal unitary property (at least) in 2 × 2 optical processors, limiting the number of unitary matrix transformations that may be generated. Aiming to solve this fundamental constraint, the theoretical tools required to analyze and design 2 × 2 universal unitary optical circuits and their corresponding BBs are presented. The consequences of this mathematical framework are explored, obtaining a simple route to implement different BB architectures, all of them guaranteeing a true universal unitary functionality in the resulting 2 × 2 optical processors. These findings may pave the way to revisit the design of high-dimensional unitary optical processors, unleashing the potential of programmable integrated photonics technology.
Abstract We introduce procedures for decomposing N × N unitary matrices into smaller M × M unitary matrices. Our procedures enable designing modular and optimal architectures for implementing arbitrary discrete unitary transformations on light. Such architectures rely on systematically combining the M -mode linear optical interferometers together to implement a given N -mode transformation. Thus this work enables the implementation of large linear optical transformations using smaller modules that act on the spatial or the internal degrees of freedom of light such as polarization, time or orbital angular momentum. The architectures lead to a rectangular gate structure, which is optimal in the sense that realizing arbitrary transformations on these architectures needs a minimal number of optical elements and minimal circuit depth. Moreover, the rectangular structure ensures that each of the different optical modes incurs balanced optical losses, so the architectures promise substantially enhanced process fidelities as compared to existing schemes.