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Universal Quantum Observables 22 July 2002 Physics Letters A 300 (2002) 1–6 www.elsevier.com/locate/pla Universal quantum observables Giacomo Mauro D’Ariano a,b a Quantum Optics and Information Group, Istituto Nazionale di Fisica della Materia, Unità di Pavia, Italy b Department of Electrical and Computer Engineering, Northwestern University, Evanston, IL 60208, USA Received 19 October 2001; received in revised form 24 May 2002; accepted 28 May 2002 Communicated by P.R. Holland Abstract It is shown how one can estimate the ensemble average of all operators of a quantum system by measuring only one fixed “universal” observable on an extended Hilbert space. This is equivalent to run a tomographic reconstruction in a kind of “quantum parallelism”, measuring all the quorum observables with a single universal observable. An experimental implementation in quantum optics is given, based on Kerr cross-phase modulated homodyning. 2002 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 03.67.-a Quantum tomography [1–5] is a method for es- alized to the estimation of an arbitrary observable of timating the ensemble average of all operators of a the field [14] and to arbitrary quantum system [15– quantum system—including its density matrix—from 18]. Finally, very recently, a method for tomographic a set of measurements of a quorum of observables, estimation of the unknown quantum operation [19] of i.e., a “complete” set of noncommuting observables. a quantum device has been presented [20], exploiting The name quantum tomography originated in quantum the “quantum parallelism” of an entangled input state optics, where the set of quadrature probability distri- which plays the role of a “superposition of all possible butions for varying phase was recognized [6] as the input states”. Radon transform of the Wigner function, the Radon In this Letter I will show how another kind of transform being the basic imaging tool in computer- quantum parallelism can be exploited in order to ized medical tomography. Such analogy gave the name run the whole tomographic process in parallel, by to a first qualitative technique for measuring the matrix measuring all the quorum observables in a single elements of the radiation density operator [7]. A first universal observable. In analogy with the tomographic quantitative method was given in Ref. [8] (for a re- Radon-transform reconstruction of a two-dimensional view, see Refs. [2,4]), and is currently used in quantum image [2–4] this method would resemble a kind optical labs [9–13]. The method has been then gener- of “quantum holography”, with the whole Radon- transform included in the single universal observable. After briefly reviewing the general tomographic E-mail address: [email protected] (G.M. D’Ariano). URL address: http://www.qubit.it. estimation approach, I will outline the construction of 0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII:S0375-9601(02)00752-1 2 G.M. D’Ariano / Physics Letters A 300 (2002) 1–6 universal observables, characterizing all of them, and erator expansion). Finally, it is easy to prove that any giving a way for deconvolving measurement noise. spanning set must be irreducible, i.e., it must satisfy I will also sketch an experimental implementation in the following properties: (i) there are no proper sub- quantum optics, based on Kerr cross-phase modulation spaces of H which are left invariant under the action and homodyning. of all c(l); (ii) it has a trivial centralizer, namely the Let us now recall the general quantum tomographic only operators that commute with all c(l) are multiple method. In the following, I will use a more convenient of the identity. definition of observable than the usual selfadjoint op- Given a quorum, i.e., a spanning set of observ- erator. I will call observable any complex operator O ables, the tomographic estimation of the ensemble that corresponds to a complete orthonormal resolution average A is simply obtained in form of double = || = of the identity I s s s ,wheres (s1,...,sn) average—over both the ensemble and the quorum—of denotes (a set of generally complex) “eigenvalues”. the unbiased estimator Tr[b†(l)A]c(l) with random l. In this way, all observables are in one-to-one corre- The most popular example of quantum tomography spondence with normal operators O, i.e., operators is the optical quantum homodyne tomography [2,4], commuting with their adjoint, as [O,O†]=0, and where the quorum (self-dual) is given by the operators = = 1 † iφ + −iφ any complex function F of an observable will be it- c(k,φ) exp(ikXφ), with Xφ 2 (a e ae ) self observable, F denoting the normal operator F = denoting the quadrature of the radiation field mode || s F(s) s s that resorts to the same physical mea- with annihilator a at phase φ with respect to the local surement, but with a different data processing [21]. oscillator. Notice that for estimating the density ma- As already mentioned, the idea of quantum tomog- trix also the maximum-likelihood strategy can be used raphy is the possibility of estimating the ensemble av- instead of the averaging procedure [16,24]. Moreover, erage of all operators of a quantum system from a set there is a general method [16] for deconvolving instru- of measurements of a quorum of observables, which in mental noise when measuring the quorum, which re- the following I will denote by {c(l)}. We call quorum sorts to finding the biorthogonal basis for the “noisy” any set of operators that span the linear space L(H) of quorum: this will be used in the following. For exam- operators on H. This means that any operator A can be ple, in homodyne tomography deconvolution of noise = [ † ] { } expanded as A l Tr b (l)A c(l) [23], where b(l) from non-unit quantum efficiency is possible [2,4]. † is the dual set of {c(l)} satisfying Tr[b (i)c(j)]=δij , For tomography of multipartite quantum systems, and we say that {c(l)} and {b(l)} form a biorthogonal when they are distinguishable—i.e., they can be mea- spanning couple for L(H). Which biorthogonal couples sured separately—a quorum is simply given by the correspond to spanning sets? They must satisfy the so- tensor product of single-system quorums, e.g., for two called orthogonality relation identical systems one has the quorum {c(k) ⊗ c(l)} with dual set {b(k) ⊗ b(l)} and expansion of any † n|b (l)|rs|c(l)|m=δnmδrs, (1) = [ † ⊗ † ] ⊗ joint operator J as J lk Tr b (k) b (l)J c(k) l c(l). An example of application is the experiment of where {|n} denotes any orthonormal basis on H.No- Ref. [12], which has been performed on a twin beam tice that the name “orthogonality relation” for iden- from parametric down-conversion of vacuum. If the tity (1) does not mean that the spanning set itself is or- systems are indistinguishable, a permutation-invariant thogonal, and most of the times the physical spanning global quorum on the tensor-product Hilbert space is sets of observables are overcomplete, and the index l needed: this is the case of multimode homodyne to- of the set has to be regarded as continuous, with the mography with a single local oscillator [25], where, in sum in Eq. (1) being replaced by an integral. Notice principle, the full joint density matrix of a multimode . that the continuous version ∆(i, j) = Tr[b†(i)c(j)] of radiation field can be recovered using a single homo- the “Kroneker delta” is generally not a “Dirac delta”: dyne detector with tunable mode-shape local oscilla- in fact, it can be bounded, and have support on the tor. whole index manifold (such “delta” is generally re- Let us now come back to the idea of “univer- ferred to as “self-reproducing kernel”, since it works sal observables”. How can I make a complete tomo- effectively as a Dirac delta under the integral of the op- graphic reconstruction by measuring only a single ob- G.M. D’Ariano / Physics Letters A 300 (2002) 1–6 3 servable? Consider the following observable H , with [H,H†]= 0, on the extended Hilbert space H ⊗ K H = c(l) ⊗|ll|≡c(K), (2) l Fig. 1. General scheme for a Universal Observable H . By changing where {|l} ∈ K denotes an orthonormal basis in the the data processing on the computer, one can measure the ensemble average of any operator of the quantum system with unknown ancillary Hilbert space K corresponding to an auxiliary density operator ρ by measuring the observable H on an extended = | | observable K l l l l (K is generally infinite- Hilbert space with the ancilla prepared essentially in any state σ dimensional, and in the continuous case |l denotes (see text). Notice that once a sample of measurement outcomes a Dirac-normalized orthogonal vector). Then, prepare is recorded, it can be reused for any other observable, by just the ancilla in a state σ with all nonvanishing diagonal re-evaluating a different data processing. matrix elements. In order to estimate the ensemble = † † average of the operator A on H measure the following via cross-Kerr interaction U exp(iκa ab b) as fol- function FA of the observable H lows = † Tr[b†(l)A] Xκb†b U X0U. (5) FA = c(l) ⊗|ll|, (3) l|σ|l Comparing this case with the general scheme for uni- l versal observables, here Xκb†b plays the role of the with the ancilla of the apparatus in the state σ .
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