The Canonical Naimark Extension for the Pauli Quantum Roulette Wheel

Home , POVM

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 measurement 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 realize 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 {Xk}k=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 measurement 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 theorem. (a): a generalized measurement described by the POVM Π = M †M ; (b): its canonical Naimark extension, defined by II. NOTATION AND TOOLS x x x the triple {ρB , U, {Px}}, describing the probe state ρB , the evolution operator U and the projective measurement {Px} 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 † 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 ∈ L(HB) describes the state of the probe system px = Tr [ρΠx] (or ancilla), the operators {Px} ∈ L(HB) are a set of projectors which describe the measurement on the ancilla and the In order to satisfy the properties of the distribution px, the op- unitary operator U ∈ 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 {Πx} ∈ L(HA) is tion px has to be positive for every |ϕi ∈ H, and normalized, P thus individuated by the triple {ρB, U, {Px}}. 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 detection 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 † = Tr [ρΠ ]. Therefore, detection op- x x x x √ † erators that satisfy Πx = MxMx are Mx = Ux Πx, with † Ux a unitary operator such that UxUx = I, and this leaves a residual freedom on the post-measurement states. The post- S1 R1 measurement states are then given by: V 1 † ρx = MxρMx px S0 R0

A measurement described by the operators Πx is referred to as generalized measurement. In order to link general measurements with physical FIG. 2: The Cartan decomposition of the operator X ∈ SU(4), schemes of measurement, we have the Naimark theorem, given by (R1 ⊗ R0)V (S1 ⊗ S0). 3

In the following we are going to deal with two-qubit inter- In terms of the α parameters the bounded region correspond- actions, i.e. unitary operators (with unit determinant) of the ing to the canonical class vectors is given by group SU(4), which are individuated by 15 parameters. In −π ≤ α ≤ 0 order to reduce the number of these parameters we will make 1 use of the Cartan decomposition, which allows us to factor 0 ≤ α2 ≤ −α1 a general operator in SU(4) into local operators working on α1 + α2 ≤ 2α3 ≤ 0 single qubits and a single two-qubit operator V individuated if α3 = 0 then α1 − α2 ≥ −π by 3 parameters [20–22], see (Fig. 2). According to the Cartan decomposition any X ∈ SU(4) As we will see in the next section, the Cartan decomposi- can be rewritten as X = (R1 ⊗ R0)V (S1 ⊗ S0), where tion simplifies the problem of finding the canonical Naimark P3 extension for the quantum roulette since we will be able to R1,R0,S1,S0 ∈ SU(2) and V = exp{i kjσj ⊗ σj}, j=1 neglect single-qubit operations and thus reducing the the 15- with k ≡ (k , k , k ) ∈ 3. The operators σ are the Pauli 1 2 3 R i parameters operator U to the 3-parameters operator V . Fur- matrices. If we introduce the following equivalence relation thermore, we will restrict the interval of the parameters of V in SU(4) to the bound region defined above.

A ∼ B if A = (R1 ⊗ R0)B(S1 ⊗ S0) (2) III. THE PAULI QUANTUM ROULETTE WHEEL we can split the whole space of unitary operators with unit determinant into equivalence classes. Then, since an operator X ∈ SU(4) is represented (by the equivalent relation given Let us consider K observables {Xk} in a Hilbert space HA here) by the matrix V , it is possible to establish a link be- with dimension d = dim(HA). All the observables are non- tween operators in SU(4) and real vectors: X ∼ k, where k degenerate and isospectral. Since the observables are non- is the class vector of X. The following operations are class- degenerate and the Hilbert space is ﬁnite-dimensional, each of preserving: them has d eigenvalues. We use a detector which chooses at random one of these observables and performs a measurement Shift: k can be shifted by ± π along one of its components. of that observable. Each observable has a probability zk of be- 2 P ing selected by the detector and k zk = 1. This scheme, de- Reverse: the sign of two components of k can be reversed. noted by the K-tuple {{X1, z1}, {X2, z2},..., {XK , zK }}, is referred to as Quantum Roulette. Swap: two components of k can be swapped. If we have a system represented by the state ρ ∈ L(HA) and we send it to the detector, the probability distribution of By the use of these operations it is always possible to reduce the outcomes is given by k K any into a bounded region given by: X (k) X h (k)i px = zkpx = zkTr ρPx (4) π 1. > k1 ≥ k2 ≥ k3 ≥ 0 k k 2 " # X π = Tr ρ z P (k) = Tr [ρ Π ] 2. k1 + k2 ≤ 2 k x x k π 3. If k3 = 0, then k1 ≤ 4 (k) where px is the probability distribution of the outcome x (k) (k)(k) The k ∈ K are referred to as canonical class vectors. for the observable Xk and Px = |xi hx| is the 1- The expression of the operator V may be simpliﬁed using a dimensional projector on the eigenspace of the eigenvalue x different set of parameters, e.g. for the observable Xk. In the last equality of the Eq. (4) we have introduced the POVM of the roulette, whose elements α1 − α2 are given by k1 = − 4 X (k) α + α Πx = zkPx (5) k = − 1 2 2 4 k α k = − 3 The Πx’s are positive operators, indeed, given any |ϕi ∈ 3 2 P (k) 2 HA, we have hϕ|Πx|ϕi = k zk|hϕ|xi | ≥ 0, and they represent a decomposition of identity, since 1 Besides, we introduce the operators Σi = 2 σi ⊗ σi, which X X X (k) X X (k) X are normalized in the space of 4 × 4 operators, with the in- Πx = zkPx = zk Px = zk I = I † ner product hA, Bi = Tr[B A]. Eventually, we obtain the x x k k x k following matrix V : On the other hand, Πx are not orthogonal projectors since Π Π 0 6= δ 0 Π . Indeed: 1 1 x x xx x V = exp −i (α1 − α2)Σ1 + (α1 + α2)Σ2 + α3Σ3 X (k) (k0) 2 2 ΠxΠx0 = zkzk0 Px Px0 6= 0 (3) k,k0 4

(k) 0 (k) In fact, while hx|x i has to be equal to δxx0 for a fixed overall state of the composite system. The operator U ∈ 0 value of k, the quantity (k)hx|x0i(k ) (with k 6= k0) could be SU(4), then it is defined by 15 parameters. Therefore, the to- different from zero also when x 6= x0. tal number of parameters that defines the Naimark extension is 19 (4 parameters from |ωBi and |xi plus 15 from U). As we will see this number can be greatly reduced by employing A. The non-canonical Naimark extension the Cartan decomposition.

The quantum roulette has a Naimark extension that is not the canonical one (i.e. the indirect measurement scheme de- A. Application of the Cartan decomposition to the Naimark scribed by the Naimark theorem) that can be obtained as fol- extension low: consider an additional Hilbert space HB, describing the ancilla, with dimension equal to the number K of observables The Naimark theorem provides a practical connection be- Xk ∈ L(HA), and a basis {|θki} in HB. Then we introduce tween the generalized measurement given by the quantum P (k) roulette and the indirect measurement described by the exten- projectors Qx = k Px ⊗ |θkihθk| in the larger Hilbert space H ⊗ H and we prepare the initial state of the ancilla sion. Indeed both the probability distribution of the outcomes A B√ p ρ in |ω i = P z |θ i, obtaining the probability distribution x and the post-measurement states Ax have to be equal for B k k k these two schemes. That is:

px = TrAB[ρ ⊗ |ωBihωB| Qx] † px = TrAB[UρA ⊗ ρBU I ⊗ Px] = TrA[ρAΠx] (6) that gives us the POVM’s elements

X (k) 1 † 1 † Πx = TrB[I ⊗ |ωBihωB| Qx] = zkPx ρAx = TrB[UρA ⊗ ρBU I ⊗ Px] = MxρAMx (7) px px k p ρ Moreover, the post-measurement states will be given by where the distribution x and the state Ax in the ﬁrst equality belong to the projective measurement while those in the last 1 equality belong to the generalized measurement. ρx = TrB[Qx ρ ⊗ |ωBihωB| Qx] px We focus now on the Born rule Eq. (6) in order to evaluate the operators Πx; after straightforward calculation, we obtain 1 X (k) (k) = zk Px ρPx the elements of the POVM: px k † † † † Πx = S1 TrB[(I ⊗ S0ρBS0)V (I ⊗ R0PxR0)V ] S1 This measurement scheme does not involve an evolution operator U ∈ L(H ⊗ H ) and the projective measure is per- † † A B Consider now S0ρBS and R PxR0; the operators R0,S0 formed on both system and ancilla, unlike the canonical ex- 0 0 ∈ L(HB) represent a rotation in the qubit Hilbert space HB. tension that involves a projective measurement on the sole an- Since both ρB and Px are not yet deﬁned and depend on some cilla. parameters, we can combine the rotation to them and we are left with a transformation from L(HB) to L(HB):

IV. THE CANONICAL NAIMARK EXTENSION OF THE 0 † ρB → ρ = S0ρBS PAULI QUANTUM ROULETTE WHEEL B 0

0 † We focus on quantum roulettes which work on qubit sys- Px → Px = R0PxR0 2 tems with Hilbert space HA ≡ C and address quantum i.e. we can neglect this transformation by a suitable roulettes involving the measurement of Pauli operators. In 0 0 order to obtain the canonical Naimark extension for these reparametrization of ρB and Px. Furthermore, we assume roulettes we have to add a probe system (the ancilla). We as- the operator S1 to be the identity (S1 = I). We make this sume a two-dimensional ancilla and show that this is enough assumption in order to simplify the research of the canonical to realize the canonical extension using Eq. (1), where the extension. This ansatz will be justiﬁed a posteriori: once we elements of the POVM are given by Eq. (5). ﬁnd the Naimark extension, if the probability distribution px obtained from the extension is equal to the one obtained from Since HB is a Hilbert space with dimension two, we choose the POVM, then the extension is correct and S1 = . the following representation for the state ρB = |ωBi hωB| and I θ Now we have POVM’s elements obtained by the canonical for the projector Px = |xi hx|, where |ωBi = cos 2 |0i + iϕ θ α iβ α extension: e sin 2 |1i and |xi = cos 2 |0i + e sin 2 |1i, with the parameters α, θ ∈ [0; π] and β, ϕ ∈ [0; 2π). Notice that |0i, |1i † Πx = TrB[(I ⊗ ρB)V (I ⊗ Px)V ] (8) are a basis in HB; we assume that |0i is the eigenvector of σ3 related to the eigenvalue 1, while |1i is the eigenvector related and these elements have to be equal to those evaluated for the to -1. quantum roulette in exam. Using the Cartan decomposition The last tool to individuate the canonical extension is the on the canonical Naimark extension reduces the number of evolution operator U ∈ L(HA ⊗ HB) which works on the parameter to 7 (4 from ρB and Px and 3 from V ). 5

B. Detection operators for the Quantum Roulette operator V into

0 V → (R1 ⊗ R0)V (S1 ⊗ S0) The operator R1 is not involved in the deﬁnition of the elements of the POVM, but it is necessary for the evaluation of and the operators R0,S0 ∈ SU(2) modify both the initial the post-measurement state ρAx. Since the operators R0 and state of HB and the orthogonal projector S0 were absorbed into, respectively, Px and ρB and S1 = I, then the decomposition of U is U = (R ⊗ )V and the left 1 I 0 0 † part of Eq. (7) becomes: ρB(θ , ϕ ) = S0 ρB(θ, ϕ) S0 P (α0, β0) = R† P (α, β) R 1 x 0 x 0 ρ = Tr [(R ⊗ )V ρ ⊗ ρ V †(R† ⊗ P )] Ax p B 1 I A B 1 x x Hence, to transform the αi and keep the correct extension is 1 † † necessary to modify ρB and Px. = R1TrB[V ρA ⊗ ρBV (I ⊗ Px)]R1 px

Therefore, the operator R1 describes a residual degree of E. The canonical extension freedom in the design of possible post-measurement states. This freedom was expected, since when we define a POVM We now focus on quantum roulettes given by Pauli opera- Πx, the post-measurement states can be evaluated using the tors σi and on their canonical Naimark extension. The most detection√ operators Mx. These operators are defined as Mx = general quantum roulette of this kind is {σi, zi}i=1,2,3 and Ux Πx, where Ux is a unitary operator. The operator Ux provides the same freedom given by R to the post-measurement its canonical extension depends on two undefined parameters 1 z z states. (e.g. 1 and 2). Finding the extension for the general roulette is analytically challenging and thus we focus to roulettes involving two Pauli operators. Let us consider the roulette {{σ1, z}, {σ3, 1 − z}}, where C. The general solution z gets values from the interval (0; 1). The POVM’s elements are give by The problem of finding the canonical Naimark extension for a given quantum roulette is now basically reduced to the 1 2 − z z 1 z −z Π = Π = (9) solution of four equations dependent on seven parameters. 1 2 z z −1 2 −z 2 − z Indeed, the considered roulettes are always in qubit spaces; hence, the elements of the POVM Π are self-adjoint 2 × 2 √ x and the detection operators by Mx = Ux Πx, x = ±1. operators on the field and the relation given by Eq. (8) pro- C Upon expanding them on the Pauli basis, i.e. Mx = a0 I + vides four equations: one from the element Π that is real, x11 a1 σ1 + a2 σ2 + a3 σ3, the coefficients ai are evaluated us- two from the element Πx12 (real part and imaginary part) and ing the inner product hX,Y i = Tr[XY †]. For the roulette one from the element Π . x22 {{σ1, z}, {σ3, 1 − z}}, we obtain ai = ai(z) (for i = 0, 1, 3) and a2 = 0. In other words, the detection operators of a roulette involving the Pauli operators σ1 and σ3 have no D. Exchange of the parameters component on the missing one, i.e. σ2. This result also holds for the other roulettes depending on two σ’s , e.g. for One may wonder if it is possible to look for the canonical {{σ2, z}, {σ3, 1 − z}}, the detection operators Mx have no component by σ . extension when the parameters αi get values from all R. The 1 Cartan decomposition does not impose restriction on the range The solution for the canonical extension corresponds to the of the components of the class vectors (that is, the parameters parameters α ), but we know that each operator in SU(4) is related (via i s the equivalence relation Eq. (2)) to a canonical class vector, 1 3 α = −π ; α = 0 ; α = arcsin(− − 1) whose components lie on the bounded region K ∈ R . On 1 2 3 1 − z α K 2 the other hand if we find an extension with i outside of , π it is possible to use the 3 class-preserving operations (shift, α = arccos(z − 1) ; β = π ; θ = ; ∀ϕ , reverse and swap) to bring back the parameters to K. 2 K May the parameters be brought back to after we have where α1, α2, θ, ϕ and β are in the correct range and we found the canonical extension? This is not possible, unless are left to check whether also α3 and α lies in the correct we also modify the other objects of the extension. Indeed, if range. First of all, cos α = z − 1, i.e. cos α ∈ (−1; 0); we have found the extension, then we have defined both αi π q 1 then α ∈ ( 2 ; π). Finally, sin α3 = − 1− z − 1, that is and θ, ϕ, α, β. But if the αi are modified by one of the three 2 π class-preserving operations, then also the other parameters are sin α3 ∈ (−1; 0); then α3 ∈ (− 2 ; 0). The parameter α3 α1+α2 modified and the state |ωBi and the operator Px change. In has to be in [ 2 ; 0], and since α1 = −π and α2 = 0, its π fact, the operations are class-preserving, so they transform the greatest range is [− 2 ; 0]. 6

The ingredients of the canonical extension are thus the state hand, we may assume that b B so that we can neglect the iϕ |ω i = √1 |0i + e√ |1i, the projectors other components of B. The interaction Hamiltonian is given B 2 2 by H = (B − bz)σ3 (neglecting the vacuum permittivity) and −iτ(B−bz)σ3 1 2 − z pz(2 − z) the corresponding evolution operator by U = e , P = (10) where τ is an effective interaction time. Starting from an ini- 1 2 pz(2 − z) z tial state which is factorized into a spin and a spatial part, i.e. P−1 = I − P1 |Ψii = (c0|0i + c1|1i) ⊗ |ψ(q)i, the evolved state is given by and the unitary U|Ψii = c0|0i ⊗ |ψ−(q)i + c1|1i ⊗ |ψ+(q)i

±iτ(B−bz) f(z) 0 0 0  where |ψ±(q)i = e |ψ(q)i. The evolution is thus 0 0 i f ∗(z) 0 coupling the spin and the spatial part. Moving to the momen- V =   (11)  0 i f ∗(z) 0 0  tum representation 0 0 0 f(z) Z 3 −iq·p |ψe±(p)i = d q e |ψ±(q)i rq with f(z) = 2−2z + √ i . 2−z 2 −1 z = |ψe(p ± τbe3)i In order to obtain the canonical Naimark extension for the roulettes {{σ1, z}, {σ2, 1 − z}} and {{σ2, z}, {σ3, 1 − z}}, and tracing out the spin part after the interaction, we have we have to remove our previous assumption S1 = I. In fact, that the motional degree of freedom after the interaction is to rotate a Pauli operator σi by an angle θ we have to use a described by the density operator rotation operator W = e−i(n·σ)θ, where n is the versor of the 2 2 direction around which the rotation is made. Then, to move %p =|c1| |ψe+(p)ihψe+(p)| + |c0| |ψe−(p)ihψe−(p)| from a two Pauli operators roulette to another, we need to apply the correct rotation in order to modify the σi. For example, As a consequence the beam is divided in two parts, and it to move from {{σ1, z}, {σ3, 1−z}} to {{σ2, z}, {σ3, 1−z}} is possible to perform measurements on a screen placed at a −i π σ we have to apply the operator W = e 4 3 , which changes given distance from the magnetic field, where we can see the σ1 into σ2 and leaves σ3 unchanged. beams as two different spots. Therefore, the extensions for the other roulettes depending If for some reason the direction of the magnetic field is on two Pauli operators are defined by the same parameters of tilted we have B = (B − bt) eα, where t is a cohordinate the extension for {{σ1, z}, {σ3, 1 − z}}, but the elements Πx along the new direction and eα = cos α e3 + sin α e⊥, e⊥ are rotated by the operator W , that is: denoting any direction perperdicular to the z-axis, say e1. The above analysis is still valid if we perform the substitu- † Πx → W ΠxW tion H → (B − bt)σθ where σθ is a Pauli matrix describing a sping component along a tilted axis. Assuming a rotation This means that, while for the first found extension the oper- along the x-axis we have that σθ corresponds to the rotated ator S1 can be assumed equal to I, for the extensions of the operator other roulettes the operator S1 has to be equal to the conju- † gate transpose of the rotation operator W . We find that, for σθ = Uθσ3U π θ i σ3 the roulette given by σ2 and σ3, S1 = e 4 while for the one π −i σ1 −iσ1θ given by σ1 and σ2, S1 = e 4 . where Uθ = e and θ = α/2.

V. THE STERN-GERLACH APPARATUS AS A QUANTUM A. The continuous quantum roulette ROULETTE Usually, the magnetic field of the apparatus is assumed to a The so called Stern-Gerlach apparatus allows one to mea- be a stable classical quantity. However, in any practical situ- sure a component (e.g.the component along the z-axis Sz) of ation the magnetic field unavoidably fluctuates. In particular, the quantum observable spin, i.e. the intrinsic angular momen- we focus on Stern-Gerlach apparatuses where the magnetic tum of a particle. The measurement is usually performed on field fluctuates in one dimension around a pre-established di- a collimated beam of particles (e.g. neutral atoms) sent with rection and provide a more detailed analysis of non-ideal se- thermal speed into a region of inhomogeneous magnetic field. tups [23, 24]. As mentioned above, a measurement of spin Here the particles are deflected by the field in some beams made with a tilted magnetic field corresponds to measure the which, after propagating into the vacuum, are collected by a operator σθ. If the magnetic field is fluctuating, then we may screen. The magnetic field is usually assumed of the form describe this situation using a continuous quantum roulette B = (B −bz) e3, where z is the cohordinate along the z-axis, where θ is randomly fluctuating around the z-axis according B is the field in the origin, and b is a constant. Actually, this is to a given probability distribution. an artificial model, since a field like this one does not respect In principle, the magnetic field may fluctuate in any direc- π the Maxwell equations, as ∇ · B = −b 6= 0. On the other tion on the zy-plane, i.e. the angle θ takes values between − 2 7

π and 2 . On the other hand, in a realistic situation, the mag- Let consider the parameters α1 and α2; the codomain of 1 netic field moves away from the pre-established direction (the the function f(∆) is (0; 2 ]. Therefore, if cos α1 = −2f(∆), π z-axis, in this case) just by a small angle. We thus introduce a then cos α1 ∈ (0; −1] and α1 ∈ (− 2 ; −π]. Instead, cos α2 = π Gaussian probability distribution z(θ) for the fluctuating val- 2f(∆), i.e. cos α2 ∈ (0; 1] and α2 ∈ ( 2 ; 0]. Both α1 and α2 ues of θ depend on the function f, so when we choose a value for f the two parameters are fixed. For example when f(∆) → 0, then 1 θ2 π π 1 α1 → − 2 and α2 → 2 ; on the other hand, if f(∆) = 2 then z(θ) = exp{− 2 } A 2∆ α1 = −π and α2 = 0. The ranges of these two parameters where the normalization A is: are correct and we have α2 ≤ −α1 ∀ f(∆). Finally, since we have fixed α3 = 0, we have to check whether α1 − α2 ≥ −π, π Z 2 θ2 √ π and this is case: as it can be easily checked α1 − α2 = −π √ A = exp{− 2 }dθ = 2π∆Erf( ) for all ∆ ∈ [0; +∞). The canonical extension is thus given π 2∆ 2 2∆ − 2 by the state |ωBi = |0i, the observable σ3 (measured on the In order to evaluate the elements of the POVM which de- ancilla), and the unitary V ∈ L(HA ⊗ HB) scribes this continuous quantum roulette, we need the projectors on the eigenspaces of σ , i.e. 3 θ 1 X V = vk σk ⊗ σk (15) 2 4 cos θ i cos θ sin θ k=0 P1(θ) = 2 (12) −i cos θ sin θ sin θ r1 r1 v0/3 = + f(∆) ± − f(∆) (16) P−1(θ) = I − P1(θ) 2 2 ∆→1 Therefore, the elements of the POVM are given by ' 1 ± ∆

π v1/2 = i v0/3 (17) Z 2 Πx = z(θ)Px(θ)dθ (13) − π 2 For a particle with spin up, represented by the pure state |0i, the probability distribution of the outcomes is given by that is the equation equivalent to Eq. (5) in the continuous p = 1 + f(∆), p = 1 − p and thus, when such a particle case. We can evaluate the elements of the POVM using the 1 2 −1 1 is measured, there is always a probability that the apparatus distribution z(θ) and the projectors P (θ) given above, and x measures the spin down |1i. we obtain:

1 2 + f(∆) 0 Π1 = 1 (14) 0 2 − f(∆) VI. CONCLUSIONS Π−1 = I − Π1

1 where the function f(∆) : [0; +∞) → [ 2 ; 0) is given by We have addressed Pauli quantum roulettes and found their canonical Naimark extensions. The extensions are minimal, π−i4∆2 π+i4∆2 Erf( √ ) + Erf( √ ) i.e they involve a single ancilla qubit, and provide a concrete f(∆) = 2 2∆ 2 2∆ . 4e2∆2 Erf( √π ) model to realize the roulettes without destroying the signal 2 2∆ state, which can be measured again after the measurement, or We have f(∆) ' 1/2 − ∆2 for vanishing ∆ and f(∆) ' can be transmitted. Our results provide a natural framework 1/8∆2 for ∆ → ∞. to describe measurement scheme where the measured observable depends on an external parameter, which can not be fully controlled and may fluctuate according to a given probability B. The canonical extension for the continuous roulette distribution. As an illustrative example we have applied our results to the description of Stern-Gerlach-like experiments on a two-level system, taking into account possible uncertainties We look for the canonical extension for this roulette in or- in the splitting force. der to obtain a practicable measurement scheme with the same behavior (same probability distribution and post-measurement states) of the Stern-Gerlach experiment with fluctuating magnetic field. A canonical extension may be found, corresponding to the parameters Acknowledgments

α = arccos(−2f(∆)) ; α = arccos(2f(∆)) ; α = 0 1 2 3 This work has been supported by MIUR through the project α = π ; β = 0 ; θ = 0 ; ϕ = 0 FIRB-RBFR10YQ3H-LiCHIS. 8

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