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A General Linear-Optical Quantum State Generator

Dmitry B. Uskov, Nickolas M. VanMeter, Pavel Lougovski, Jonathan P. Dowling, Jens Eisert, Konrad Kieling

2007
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International Conference on Quantum Information
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unpublished

We introduce a notion of a linear-optical quantum state generator. This is a device that prepares a desired quantum state using product inputs from photon sources, linear-optical networks, and post-selection using photon counters. We show that this device can be concisely described in terms of polynomial equations and unitary constraints. We illustrate the power of this language by applying the Gröbner-basis technique along with the notion of vacuum extensions to solve the problem of how to
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... oblem of how to construct a quantum state generator analytically for any desired state, and use methods of convex optimization to identify success probabilities. In particular, we disprove a conjecture concerning the preparation of the maximally path-entangled NOON-state by providing a counterexample using these methods, and we derive a new upper bound on the resources required for NOON-state generation. There are many quantum states of light which are in great demand in various applications. Due to their high robustness to decoherence, and relatively simple manipulation techniques, photons are often exploited as primary carriers of quantum information. Recently, a number of schemes have been suggested enabling quantum information processing with photons using only beam splitters, phase shifters, and photodetectors [1, 2]. Moreover, exotic states of many photons, such as maximally path-entangled NOON states, have found their place in quantum metrology [3] and lithography [4] . A natural question arises: how can these complicated states of light be prepared? One approach to the problem is to make use of an optical nonlinearity. However, due to the relatively small average number of photons involved, the overall nonlinear effect is extremely weak and typically of little practical use [5] . An alternative way to enable an effective photonphoton interaction is to use ancilla modes and projective measurements. In this way a quantum state generator can be realized utilizing only linear optical elements (beam splitters and phase shifters) and photon counters, at the expense of the process becoming probabilistic [6] . Any linear-optical quantum state generator (LOQSG) can hence be thought of as consisting of two main blocks (see Fig. 1 ). The first block is a N -port linear-optical device, described by a unitary matrix U ∈ U (N ) combining N input into N output modes. The second block represents a projective measurement of some of the modes of the first block, in which a certain pattern of photons measured in some M < N of the modes is considered a "successful measurement", leading to a preparation of the desired state in the remaining modes (compare also Refs. [6, 7, 8] ). This measurement is probabilisticbut heralded -or "event-ready". Clearly, the output is determined by an interplay between the numbers of input photons, entries of the matrix U , and the numbers of photons detected. There are two types of problems that can be formulated around the concept of LOQSG. The first problem is the following: Given the matrix U and a known input state, which output states can be generated for different projective mea-0〉 0 L.O. m 2 m 1 ψ in 〉 ψ tar 〉 FIG. 1: The linear optical quantum state generator. surements? This is what could be referred to as the "forward problem". This question is equivalent to the problem of finding the effective nonlinearity generated by a given projective measurement and was addressed in Refs. [9, 10] . The second problem is that of state preparation: Given an input state, a projective measurement, and a target state, is it possible to determine a unitary matrix U , of appropriate dimension, involving potentially further auxiliary modes, describing the unitary block of the LOQSG? This important problem asks whether a device for the preparation of a certain quantum state can be identified, and if so, what elements it contains. This may be coined the "inverse problem". In this work, we provide a mathematical description of LOQSG and illustrate how methods of algebraic geometry can be used to solve this inverse problem. To be more specific, we start from a given input |ψ in in N optical modes obtained from photon sources. Then, we investigate all state vectors that can be reached from this using arbitrary networks of linear optical elements [11] . We hence investigate the orbit Ω = {|Ψ : |Ψ = U(U )|ψ in for some U ∈ U (N )} of an input state vector |ψ in . The unitary U(U ) acting in the Hilbert space of quantum states is the standard Fock-Bargmann irreducible representation of the unitary transformation U acting on modes. If we now denote with |ψ tar the desired output upon a particular projective measurement P = |ψ D ψ D |, then Θ = {|Ψ : |Ψ = α|ψ tar |ψ D + i β i |ψ any i |ψ D⊥ i , α = 0} is the set of all state vectors which can be converted into |ψ tar by the measurement of |ψ D in auxiliary modes. |α| 2 represents a probability of success, and each |ψ D⊥ i denotes a state vec-

doi:10.1364/icqi.2007.ithb5
fatcat:bh3jevaqurhy3f7fo4he4mlhha