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Physical Review Letters

Experimental realization of any discrete unitary operator

Michael Reck, Anton Zeilinger, Herbert Joseph Bernstein, Philip Bertani

Published Jul 4, 1994• 2,153 citations
Publisher

Abstract

An algorithmic proof that any discrete finite-dimensional unitary operator can be constructed in the laboratory using optical devices is given. Our recursive algorithm factorizes any N\ifmmode\times\else\texttimes\fi{}N unitary matrix into a sequence of two-dimensional beam splitter transformations. The experiment is built from the corresponding devices. This also permits the measurement of the observable corresponding to any discrete Hermitian matrix. Thus optical experiments with any type of radiation (photons, atoms, etc.) exploring higher-dimensional discrete quantum systems become feasible.

DOI
10.1103/physrevlett.73.58

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An algorithmic proof that any discrete finite-dimensional unitary operator can be constructed in the laboratory using optical devices is given. Our recursive algorithm factorizes any N\ifmmode\times\else\texttimes\fi{}N unitary matrix into a sequence of two-dimensional beam splitter transformations. The experiment is built from the corresponding devices. This also permits the measurement of the observable corresponding to any discrete Hermitian matrix. Thus optical experiments with any type of radiation (photons, atoms, etc.) exploring higher-dimensional discrete quantum systems become feasible.
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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 $U={\ensuremath{\varphi}}_{m,M}(S)$, derived from a known homomorphism, ${\ensuremath{\varphi}}_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states, 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 ${\ensuremath{\varphi}}_{m,M}^{\ensuremath{-}1}$ 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.
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Nature provides us with a restricted set of microscopic interactions. The question is whether we can synthesize out of these fundamental interactions an arbitrary unitary operator. In this paper we present a constructive algorithm for realization of any unitary operator which acts on a (truncated) Hilbert space of a single bosonic mode. The algorithm itself is not unitary because it involves a conditional measurement. However, it does yield a constant probability of the conditional measurement which does not depend on the input state of the bosonic system. We consider a physical implementation of unitary transformations acting on one-dimensional vibrational states of a trapped ion. As an example we present an algorithm which realizes the discrete Fourier transform.
Dedicated to Prof. Iwo Białynicki-Birula on the occasion of his 60th birthdayUsing the correlated pairs produced in spontaneous parametric down--conversion, one can extract quantum states effectively defined in a Hilbert space of any dimension N. Furthermore, using just beam splitters and phase shifters one can build any unitary operator in the laboratory.We briefly discuss how this can be done, what kind of states could easily be produced in the laboratory, and we will discuss one explicit result pertaining to photon bunching in an N-dimensional Hilbert space.