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The Canonical Naimark Extension for the Pauli Quantum Roulette Wheel

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The Canonical Naimark Extension for the Pauli Quantum Roulette Wheel The canonical Naimark extension for generalized measurements involving sets of Pauli quantum observables chosen at random Carlo Sparaciari Dipartimento di Fisica, Universita` degli Studi di Milano, I-20133 Milano, Italy Matteo G. A. Paris∗ Dipartimento di Fisica, Universita` degli Studi di Milano, I-20133 Milano, Italy and CNISM, UdR Milano, I-20133 Milano, Italy (Dated: September 21, 2018) We address measurement schemes where certain observables Xk are chosen at random within a set of non- degenerate isospectral observables and then measured on repeated preparations of a physical system. Each P observable has a probability zk to be measured, with k zk = 1, and the statistics of this generalized mea- surement is described by a positive operator-valued measure (POVM). This kind of schemes are referred to as quantum roulettes since each observable Xk is chosen at random, e.g. according to the fluctuating value of an external parameter. Here we focus on quantum roulettes for qubits involving the measurements of Pauli matrices and we explicitly evaluate their canonical Naimark extensions, i.e. their implementation as indirect measurements involving an interaction scheme with a probe system. We thus provide a concrete model to real- ize the roulette without destroying the signal state, which can be measured again after the measurement, or can be transmitted. Finally, we apply our results to the description of Stern-Gerlach-like experiments on a two-level system. PACS numbers: 03.65.Ta, 03.67.-a I. INTRODUCTION involving an independent preparation of an ancillary (probe) system [6], an interaction of the probe with the system un- In this paper we deal with a specific class of general- der investigation, and a final step where only the probe is ized quantum measurements usually referred to as Quan- subjected to a (projective) measurement [7, 8]. A question tum Roulettes. These quantum measurements are achieved thus arises on the canonical implementation of the quantum through the following procedure. Consider K projective roulette’ POVM, and on the resources needed to realize the corresponding interaction scheme. This is the main point of measurements, described by the set fXkgk=1;:::;K of non- degenerate isospectral observables in a Hilbert space H. The this paper. In particular, we focus on quantum roulettes in- system is sent to a detector which, at random, performs the volving the measurements of Pauli matrices on a qubit system and explicitly evaluate their canonical Naimark extensions. measurement of the observable Xk. Each observable has a P probability zk to be measured, with k zk = 1. This scheme We remind that having the Naimark extension of a general- is referred to as quantum roulette since the measured observ- ized measurement is, in general, highly desirable, since it pro- able Xk is chosen at random, e.g. according to the fluctu- vides a concrete model to realize an apparatus which performs ating value of a physical parameter, as it happens for the out- the measurement without destroying the state of the system come of a roulette wheel. The generalized observable actually under investigation. Thereby, the state after the measurement measured by the detector is described by a positive operator- can be measured again, or can be transmitted, and the tradeoff valued measure (POVM), which provides the probability dis- between information gain and measurement disturbance may tribution of the outcomes and the post-measurement states [1– be evaluated [9–12]. Alternatively, the scheme may serve to 3]. perform indirect quantum control, [13]. As a matter of fact, any POVM on a given Hilbert space It should be emphasized that the concept of quantum arXiv:1211.0048v3 [quant-ph] 18 Jan 2013 may be implemented as a projective measurement in a larger roulette provides a natural framework to describe measure- one, e.g. see [4] for single-photon qudits. This measurement ment scheme where the measured observable depends on an scheme is usually referred to as a Naimark extension of the external parameter, which can not be fully controlled and fluc- POVM. Indeed, it is quite straightforward to find a Naimark tuates according to a given probability distribution. A promi- extension for the POVM of any quantum roulette in terms of a nent example is given by the Stern-Gerlach apparatus, which joint measurement performed on the system under investiga- allows one to measure a spin component of a particle in the tion and an ancillary one. direction individuated by an inhomogeneous magnetic field On the other hand, for any POVM the Naimark theorem [5] [14, 15]. Indeed, whenever the field is fluctuating, or the un- ensures the existence of a canonical Naimark extension, i.e. certainty in the splitting force is taken into account [16], the the implementation of the POVM as an indirect measurement measurement scheme is described by a quantum roulette. Also in this case, if we have the canonical Naimark extension for the roulette, then we have a concrete way to realize a mea- surement without destroying the state [17]. We remind that ∗Electronic address: matteo.paris@fisica.unimi.it in the continuous variable regime quantum roulettes involv- 2 ing homodyne detection with randomized phase of the local which states that a generalized measurement in a Hilbert space oscillator has been already studied theoretically [18] and real- HA may be always seen as an indirect measurement in a larger ized experimentally [19]. Hilbert space given by the tensor product HA ⊗ HB. This The paper is structured as follows. In the next Section indirect measure is known as canonical Naimark extension we introduce notation, briefly review the Naimark theorem, for the generalized measurement. Conversely, when we per- and gather all the necessary tools, e.g. the Cartan decompo- form a projective measure on the subsystem HB of a com- sition of SU(4) transformations, which allows us to greatly posite system HA ⊗ HB, the degrees of freedom of HB may reduce the number of parameters involved in the problem of be traced out and we obtain the same probability distribution finding the canonical Naimark extension. In Section III we of the outcomes of the projective measurement and the same introduce the concept of quantum roulette, derive the corre- post-measurement states performing a generalized measure- sponding POVM, and illustrate an example of non-canonical ment on the subsystem HA. Naimark extension. In Section IV we derive the canonical extension for one-parameter Pauli quantum roulettes and dis- |ρ ᐳ Πx |ρAᐳ Ax cuss details of their implementation, whereas Section V is de- |ρ ᐳ Πx |ρAᐳ Ax |ρAᐳ|ρAxᐳ U voted to analyze Stern-Gerlach-like experiments as quantum |ρ ᐳ|ρ|ω ᐳᐸωᐳ|ρ ᐳ| U P (a) A AxB B B x roulettes, i.e. taking into account the possibility that the mag- |ωBᐳᐸω|ρBᐳB| Px Px (a) (b) netic field is randomly fluctuating. Finally, Section VI closes Px (b) the paper with some concluding remarks. FIG. 1: The two measurement schemes linked by the Naimark the- orem. (a): a generalized measurement described by the POVM Π = M yM ; (b): its canonical Naimark extension, defined by II. NOTATION AND TOOLS x x x the triple fρB ; U; fPxgg, describing the probe state ρB , the evolu- tion operator U and the projective measurement fPxg on the probe A. POVMs and the Naimark theorem system, respectively. When we measure an observable on a quantum system, we The Naimark theorem gives a practical recipe to evaluate can not predict which outcome we will obtain in each run. the canonical extension for a generalized measurement in a What we know is the spectrum of possible outcomes and their Hilbert space HA: probability distribution. Given a system described by a state y ρ in the Hilbert space H, to obtain the probability distribution Πx = TrB I ⊗ ρB U I ⊗ Px U (1) of the outcomes x we use the Born Rule: where ρB 2 L(HB) describes the state of the probe system px = Tr [ρΠx] (or ancilla), the operators fPxg 2 L(HB) are a set of pro- jectors which describe the measurement on the ancilla and the In order to satisfy the properties of the distribution px, the op- unitary operator U 2 L(HA ⊗ HB) works on both the system erators Πx do not need to be projectors. The operators Πx and the ancilla. A canonical Naimark extension for the gener- have to be positive, Πx ≥ 0, since the probability distribu- alized measurement given by the operators fΠxg 2 L(HA) is tion px has to be positive for every j'i 2 H, and normalized, P thus individuated by the triple fρB; U; fPxgg. x Πx = I, since px is normalized. A decomposition of iden- Evaluating the canonical Naimark extension for a general- tity by positive operators Πx will be referred to as a positive ized measurement is desirable since it gives a concrete model operator-valued measure (POVM), and the operators Πx are to realize an apparatus which performs the measurement with- the elements of the POVM. out destroying the state. Then, the post-measurement state can We use Πx to get information about the probability dis- be transmitted, or measured again. tribution px, but if we are interested in post-measurement states we have to introduce the set of operators Mx, the de- tection operators. These operators should give the same prob- B. The Cartan decomposition of SU(4) transformations ability distribution given by Πx, thus they are obtained from p = Tr M ρM y = Tr [ρΠ ]. Therefore, detection op- x x x x p y erators that satisfy Πx = MxMx are Mx = Ux Πx, with y Ux a unitary operator such that UxUx = I, and this leaves a residual freedom on the post-measurement states.
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