The von Neumann Model of Measurement in Quantum Mechanics

Transcription

1 The von Neumann Model o Measurement in Quantum Mechanics Pier A. Mello November 29, 2013 arxiv

2 Abstract We describe how to obtain inormation on a quantum-mechanical system by coupling it to a probe and detecting some property o the latter, using a model introduced by von Neumann, which describes the interaction o the system proper with the probe in a dynamical way. We irst discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain inormation on the system detecting the probe position. We ind the reduced density operator o the system, and show how Lüders rule emerges as the limiting case o strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we ind inormation on the system by detecting the position-position and momentum-position correlations o the two probes. The so-called Wigner?s ormula emerges in the strongcoupling limit, while Kirkwood?s quasi-probability distribution is ound as the weak-coupling limit o the above ormalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we ind a generalized transorm o the state and the observables based on the notion o successive measurements. 1 Introduction In a general quantum measurement one obtains inormation on the system o interest by coupling it to an auxiliary degree o reedom, or probe, and then detecting some property o the latter using a measuring device. This procedure, which was described by von Neumann in his classic book [1], will be reerred to as von Neumannn s model (vnm). Within the vnm, the combined system-system proper plus probe - is given a dynamical description. An example o the idea involved in the vnm is given by the Stern-Gerlach experiment, which is described in every textbook on QM (see, e.g., [2, 3]). In this experiment, presented schematically in Fig. 1, the observable we want to obtain inormation about is the z-component o the spin o a par-

3 ticle. The auxiliary degree o reedom, or probe, is its position r - still a microscopic quantity - ater it leaves the magnet, and this is what is recorded by a detecting device, which in this case is a position detector. e.g., or s = 1/2, we may wish to ind inormation on the two components ψ spin P 2 ± ψ spin = ± ψ spin 2 o the original state. Figure 1: The Stern-Gerlach experiment, designed to ind inormation on the z- component o the spin o a particle by detecting its position. Another example is provided by Cavity Quantum Electrodynamics (QED) experiments [4, 5, 6]. Here the observable is the number n o photons in a cavity. As shown schematically in Fig. 2, the probes are atoms successively sent through the cavity: ater they leave the cavity, they are detected by a measuring device. Figure 2: A Cavity QED experiment, designed to ind inormation on the number o photons inside a cavity, by sending atoms through the cavity and subsequently detecting them. In what ollows we irst discuss, in the next section, the Stern-Gerlach problem, which has become a paradigm or models o measurement in QM. 2

4 The discussion will pave the way or the analysis, in Sec. 3, o single measurements in QM, where the system proper is coupled to one probe with arbitrary coupling strength. The main goal is to study what inormation can we obtain on the system by detecting the probe position [7]. As a by-product o the analysis, we obtain the reduced density operator o the system proper ater its interaction with the probe, and derive the so-called Lüders rule [8] as the limiting case o strong coupling. We then generalize the vnm to two probes that interact successively with the system proper [7]. Again, we study what inormation can we obtain on the system by detecting the position-position and momentum-position correlations o the two probes. Indeed, we describe a state reconstruction scheme based on the procedure o successive measurements [7, 9]. We obtain the so-called Wigner?s ormula emerges in the strong-coupling limit, and Kirkwood?s quasi-probability distribution [] in the weak-coupling limit. We also ind a generalized transorm o the state and the observables based on the notion o successive measurements, and in terms o complex quasiprobabilities. 2 The Stern-Gerlach Experiment The Stern-Gerlach experiment, Fig. 1, has become a paradigm or models o measurement in QM. Although the experiment was irst perormed as early as 1922, it is ex- plained in every textbook on QM (see, e.g. Res. [2, 3]), and various reinements have been presented in the literature (as in Res. [12, 13, 14]), surprisingly, its complete (non- relativistic) solution has been given only recently [15]. By a complete solution we mean one that takes into account the translational and transverse motions o the atom, and a conined magnetic ield B(r) that satisies Maxwell s equations B = 0, B = 0 (1) Here we shall work with a model o the complete problem which has a simple exact solution and still shows the physical characteristic we want to exhibit, i.e., bending o the trajectory depending on the z-projection o the spin. We shall: i) assume B 0 outside the gaps; ii) assume that only B z is sig- 3

5 niicant; iii) assume B z (z) B (z)z inside the gaps; iv) simulate the translational motion in the y direction using a t-dependent interaction which lasts or the time the particle is inside the gap o the magnet. This could be achieved by adopting a rame o reerence moving with the particle; v) ignore the x degree o reedom. We then consider the model Hamiltonian Ĥ(t) = ˆp2 z 2m µ Bθ t 1,τ(t) = ˆp2 z 2m [µ BB z(0)τ] θ t 1,τ(t) ˆσ z ẑ τ = ˆp2 z 2m εg(t)ˆσ zẑ (2) where ˆσ z is one o the Pauli matrices, and ε = µ B B z(0)τ. The unction θ t1,τ(t) is nonzero and equal to unity only inside the time interval (t 1 τ/2, t 1 + τ/2), i.e., 1 t (t 1 τ/2, t 1 + τ/2) θ t1,τ(t) = (3) 0 t (t 1 τ/2, t 1 + τ/2) and g(t) = θ t 1,τ(t), τ g(t)dt = 1, (τ << t 1 ) (4) 0 We urther use the simpliication g(t) δ(t t 1 ) and write our model Hamiltonian as Ĥ(t) = ˆp2 z 2m εδ(t t 1)ˆσ z ẑ (5a) = Ĥ0 + ˆV (t) (5b) Ĥ 0 = ˆK z is the kinetic energy operator or the z variable. (5c) In the nomenclature introduced in the Introduction, ˆσ z is the observable or the system proper and ẑ, ˆp z are the probe canonical variables. From now on we adopt the nomenclature that we measure (perhaps a better word could be premeasure ) the observable ˆσ z, by detecting, with a suitable instrument, either ẑ or ˆp z. 4

6 We shall solve the Schödinger equation with the initial condition i h ψ(t) t = Ĥ(t) ψ(t) (6a) ψ(0) = ψ (0) spin χ(0) (6b) Here, ψ (0) spin is the initial state o the system proper and χ(0) the initial state o the probe. It will be advantageous to use the interaction picture (or a textbook presentation, see, e.g., Re. [16]), in which we have the ollowing relations ψ(t) I = Û 0 (t) ψ(t) (7a) = Û 0 (t)û(t) ψ(0) (7b) ÛI(t) ψ(0) Û I (t) = Û 0 (t)û(t) i h dûi(t) dt (7c) (7d) = ˆV I (t)ûi(t) ; Û(0) = Î (7e) Here, Û(0), Û(t), are the evolution operators in the Schrödinger picture associated with Ĥ0 and Ĥ, respectively, and ÛI(t) the evolution operator in the interaction picture; ˆV I (t) is the interaction in the interaction picture, i.e., The solution or ÛI(t) is ˆV (t) I (t) = e iĥ0t/ h ˆV te iĥ0t/ h (8a) = εδ(t t 1 )e iĥ0t/ hˆσ z ẑe iĥ0t/ h (8b) εδ(t t 1 )Ŵ Û I (t) = e iε h t 0 δ(t t 1 )dt Ŵ Û I, = e iĥ0t 1 / he iεˆσzẑ/ he iĥ0t 1 / h (8c) (9a) (9b) In the last line we have the inal evolution operator in the interaction picture, i.e., ater the interaction has ceased to act. From Eq. (7d) we ind the inal evolution operator in the Schrödinger picture as U U(t > t 1 ) = e i ˆK z(t t 1 )/ he iεˆσzẑ/ he i ˆK zt 1 / h (10) 5

7 whose physical interpretation is very clear: we irst have ree evolution rom t = 0 to t 1 ; the interaction acts at t = t 1, and we have ree evolution again thereater. The inal state vector, i.e., or t > t 1, is then Ψ(t) = P σ ψ (0) spin e i ˆK z(t t 1 )/ he iεˆσzẑ/ h χ(t 1 ) () σ=±1 We have introduced the projector P σ onto the state with eigenvalue σ = +1, 1 o ˆσ z. We observe that the component P σ ψ (0) spin o the original spin state gets entangled with the probe state e iεˆσẑ/ h χ(t 1 ) (we denote by χ (t 1) the probe state which has evolved reely rom t = 0 to t = t 1 ) which, in the z-representation, is e iεσz/ h χ(z, t 1 ) (12) meaning that it got a boost p σ = εσ in the z direction. This is precisely the essential physical eect occurring in the Stern-Gerlach experiment. 2.1 The probability o detecting a probe position z as a unction o time At t = 0, the probability density p(z, t) o a probe position z is just χ (0) (z) 2, where χ (0) (z) is the original z-wave unction. As time goes on, the wave packet spreads in time until just beore the interaction occurs, i.e., until t = t 1 (see the schematic illustration in Fig. 3). Ater the interaction has taken place at t = t 1, i.e., or t > t 1, that probability is given by p (z, t) = Ψ P z Ψ (13a) = ψ (0) spin P σ ψ (0) spin z e i ˆK z(t t 1 )/ he iεσẑ/ h χ(t 1 ) 2 σ=±1 (13b) = W σ (ˆσz) χ (z, σ; t) 2 (13c) σ=±1 where W σ (ˆσz) = ψ (0) spin P σ ψ (0) spin s the Born probability or the value σ o the spin projection in the original system state. We have also deined the σ-dependent probe wave unction or t > t 1 χ (z, σ; t) = z e i ˆK z(t t 1 )/ he iεσẑ/ h χ(t 1 ) U 0 (z, z ; t t 1 )e iεσz / hχ(z, t 1 )dz (14a) (14b) 6

8 which consists o the probe wave unction which evolves reely up to t = t 1, it gets multiplied by the plane wave e iεσz / h at t = t 1 (the boost reerred to above), and evolves reely thereater, through the ree evolution operator indicated as U 0. Figure 3: The probability p(z, t) o a probe position z as a unction o time in the Stern-Gerlach experiment, as explained in the text. Thus, because o the interaction occurring at t = t 1, the σ = 1 component o this state receives a positive boost in the z direction, and the σ = 1 component receives a negative boost, so that ater t = t 1 they travel in opposite directions. From Eq. (13c), the σ = 1 component occurs with a weight W (ˆσz) 1, and the σ = 1 component with a weight W (ˆσz) 1, as is also indicated in Fig Conditions generally required [4, 17] or the measurement o the observable o a system, when detecting a property o the probe What are called Âs and Âprobe in Res. [4, 17], are translated as  s = ˆσ z  probe = ẑ respectively, in our notation or the present Stern-Gerlach problem. requirements established in these reerences are: (15a) (15b) The 7

9 i) ˆV = (Âs) (16) Here,indeed: ˆV = εδ(t t1 )ˆσ z ẑ. ii) [ ˆV, Âprobe] 0 (17) so that Âprobe changes, enabling one to get inormation on Âs by detecting  probe. Although here [ ˆV, ẑ] = 0, we have [ ˆK z, ẑ] 0; the kinetic energy [ ˆK z causes a displacement o the two wave packets, as illustrated in Fig. 3: i we wait long enough, the 2 packets separate, and we can measure σ = ±1. In order to have a QM non-demolition measurement, these reerences also establish the additional conditions: iii) [ ˆV, Âs] = 0 ; here, indeed, [ ˆV, ˆσ z ] = 0 (18a) [Ĥs, Âs] = 0 ; here, H s = 0, so that [Ĥs, ˆσ z ] = 0 (18b) which, taken together, give [Ĥ, Âs] = 0 ; here, indeed, [Ĥ, ˆσ z] = 0 (18c) This has the consequence that starting with ψ (0) spin = σ = 1, say, the state will not get a component σ = The probability o detecting a probe momentum p z as a unction o time At t = 0, the probability density p(p z, t) o a probe momentum p z is χ (0) (p z ) 2, where χ (0) (p z ), the original wave unction in the momentum representation, is the Fourier transorm o the original wave unction in z-space. As time goes on, the wave packet conserves its shape until just beore the interaction occurs, i.e., until t = t 1 (see the schematic illustration in Fig. 4). Ater the interaction has taken place at t = t 1, i.e., or t > t 1, that probability 8

10 is given by p (p z, t) = Ψ P pz Ψ (19a) = ψ (0) spin P σ ψ (0) spin P z e i ˆK z(t t 1 )/ he iεσẑ/ h χ(t 1 ) 2 σ=±1 (19b) = W σ (ˆσz) χ (0) (p z εσ) 2 (19c) σ=±1 Just as in Eq. (13), W σ (ˆσz) is the Born probability or the value σ o the spin projection in the original system state; χ (0) (p z εσ) is the original wave unction in momentum space, evaluated at the displaced value p z εσ. Figure 4: The probability p(z, t) o a probe momentum p z as a unction o time in the Stern-Gerlach experiment, as explained in the text. Thus, at t = t + 1 the p z probability density splits into the two pieces indicated in Fig. 4, corresponding to the two values o σ, with weights W (ˆσz) 1 and, respectively, and remains unaltered thereater. W (ˆσz) Conditions generally required [4, 17] or the measurement o the observable o a system, when detecting a property o the probe In this case, the Âs and Âprobe o Res. [4, 17] are  s = ˆσ z  probe = ˆp z (20a) (20b) respectively, in our present notation. We recall that these reerences require: 9

11 i) ˆV = (Âs) (21) Here,indeed: ˆV = εδ(t t1 )ˆσ z ẑ. ii) [ ˆV, Âprobe] 0 (22) here indeed, [ ˆV, ˆp z ] [ẑ, ˆp z ] 0. For a QM non-demolition measurement, these reerences require: iii) [ ˆV, Âs] = 0 ; here, indeed, [ ˆV, ˆσ z ] = 0 (23a) [Ĥs, Âs] = 0 ; here, H s = 0, so that [Ĥs, ˆσ z ] = 0 (23b) so that [Ĥ, Âs] = 0 ; here, indeed, [Ĥ, ˆσ z] = 0 (23c) Just as in the previous case, the consequence is that starting with ψ (0) spin = σ = 1, say, the state will not get a component σ = 1. 3 Single Measurements in Quantum Mechanics We irst consider the measurement o an observable  using one probe and detecting a property o it. We are using the nomenclature introduced right ater Eqs. (5). For  we write the spectral representation  = a n P an (24) n are the eigen- where the eigenvalues a n are allowed to be degenerate and P an projectors. We assume the system to be coupled to a probe, considered, or simplicity, to 10

12 be one-dimensional, whose position and momentum are represented by the Hermitian operators ˆQ and ˆP. The system-probe interaction is taken to be [1] ˆV (t) = εg(t)â ˆP, t 1 > 0 (25) with an arbitrary interaction strength [18] ε. This interaction could be translated to the one or the Stern-Gerlach experiment discussed in Sec. 2.2 using the correspondence  ˆσ z, ˆP ẑ, ˆQ ˆp z, as illustrated in Fig. 5 below. Figure 5: The probability density p (Â) (Q) o the probe degree o reedom Q = p z as a unction o time in the Stern-Gerlach experiment, as explained in the text. This is basically Fig. 4, with the translation o variables  ˆσ z, ˆP ẑ, ˆQ ˆpz. The delta unction interaction o the previous section was generalized to g(t) (see Eq. (4)), a narrow unction with inite support, centered at t = t 1, so that 0 t g(t )dt G(t) (26a) G(0) = 0, G( ) = 1 (26b) We disregard the intrinsic evolution o the system and the probe, and assume that ˆV (t) o Eq. (25) represents the ull Hamiltonian, i.e., Ĥ(t) = εg(t)â ˆP (27)

13 The evolution operator is then given by Û(t) = e i t 0 Ĥ(t )dt / h = e iεg(t)â ˆP (28) I the density operator o the system plus the probe at t = 0 is the direct product ρ (0) = ρ (0) s ρ (0) π (π stands or probe ), ater the interaction has ceased to act, i.e., or t >> t 1, it is given by ρ (Â) = P an ρ s (0) P an (e iεan ˆP / hρ π (0) e iεa n ˆP / h) (29) nn From this expression we notice that, because o the interaction, the system and the probe are now correlated. Also notice the presence o the displacement operator exp ( (i/ h)εa n ˆP in this last equation. Now the idea is that at time t > t 1, i.e., ater the system-probe interaction is over, we detect the probe position ˆQ to obtain inormation on the system proper. This we study in what ollows. 3.1 The ˆQ probability density ater the interaction According to Born s rule, the Q probability density or t > t 1 is given by where p (Â) (Q) = Tr (ρ (Â) P Q ) = n W (Â) a n p 0 (Q εa n ) (30) W (Â) a n = Tr(ρ (0) s P an ) (31) is the Born probability or the result a n in the original system state, and p 0 (Q εa n ) = Q εa n ρ (0) π Q εa n (32) is the original Q probability density p 0 (Q) (which has a width = σ Q ), but displaced by εa n. In this problem we may take the point o view that knowing the system state ρ (0) s beore the process, and thus W a (Â) n, we can predict the detectable quantity (Q). We may also adopt the more interesting viewpoint that, detecting p (Â) p (Â) (Q), we can retrieve inormation on the system state. We examine this 12

14 latter attitude below. Beore doing that, we illustrate in Fig. 5(see above) the result (30) or the case o the Stern-Gerlach experiment studied in the last section, by means o the translation given right beore Eq. (26). An illustrative example o a more general case is presented in Fig. 6; or the values o the parameters indicated in the igure, Fig. 6a corresponds to the case o strong coupling, while Fig. 6b to that o weak coupling. Figure 6: Illustrative example o the probability density o the pointer position Q o Eq. (30). We have assumed seven eigenvalues a n or the system observable Â, with Born probabilities [Eq. (31)] given by: 0.1, 0.2, 0.2, 0.15, 0.2, 0.05 and 0.1. We chose the particular values: (a) ε = 1 or the interaction strength and σ Q = 0.05 or the width o the probe state prior to the measurement; this is a case o strong coupling ; (b) ε = 1 or the interaction strength and σ Q = 1 or the width o the probe state prior to the measurement; this is a case o weak coupling In the above scheme, what we detect is the probe-position probability p (Â) (Q). As it can be seen rom Fig. 6, it is only in the idealized limit o very strong coupling, ε/σ Q >> 1, that p (Â) (Q) mirrors the eigenvalues a n o the observable which we want to have inormation about. In this limit, p (Â) (Q) integrated around a n gives Born s probability W a (Â) n o a n. In this high-resolution limit this is thus the inormation we can retrieve about the system proper, by detecting Q. We shall study below what inormation can be extracted rom experiments with arbitrary resolution. Surprisingly, we shall ind cases where it is advantageous to use low resolution (see Sec. 5, last paragraph)! 13

15 3.2 The average o Q ater the interaction From Eqs. (30) and (31) one inds [20] that the average o the probe position ˆQ units o ε, ater the interaction is over, is given by 1 ε ˆQ 2 (Â) = a n W a (Â) n n = a n Tr(ρ s (0) P an ) = Tr(ρ (0) s Â) =  0 ε (33) n We have assumed the original Q distribution to be centered at Q = 0. As a result, detecting the average probe position ˆQ 2 (Â) ater the interaction is over allows extracting the Born average  0 o the observable  in the original state o the system. It is remarkable that this result is valid or arbitrary coupling strength ε. For example, in the two situations illustrated in Fig. 6 we would obtain the same result or ˆQ 2 (Â) /ε. Similar results can be ound or higher-order moments. e.g., or the second moment o ˆQ one inds 1 ε 2 [ ˆQ 2 (Â) σ 2 Q] = Tr(ρ (0) s  2 ) = Â2 0 ε (34) implying that detecting ˆQ 2 (Â) and knowing σq 2 allows extracting the second moment o the observable  in the original state o the system, i.e., Â2 0. More in general, i we detect the inal Q probability density (30), i.e., p (Â) (Q) = (ρ (0) n s ) nn p 0 (Q εa n ) (35) we obtain inormation on the diagonal elements (ρ s (0) ) nn o the original density operator, but not on the o-diagonal ones. Alternatively, we can write this result in terms o the characteristic unction p (Â) (k) = [ (ρ (0) n s ) nn e ikεan ] p 0 (k) = e ikεâ 0 p 0 (k) (36) implying that i we detect p (Â) (Q) and iner p (Â) (k), we can extract e ikεâ 0. Results (33) and (34) are particular cases o Eq. (36). In Sec. 5 we shall ind a procedure to extract all o the matrix elements o the original density operator, using the notion o successive measurements. 14

16 3.3 Measuring projectors A particular case o great interest is the measurement o a projector, like P aν, so that the Hamiltonian o Eq. (27) becomes Ĥ(t) = εg(t)ˆp aν ˆP t1 > 0 (37) We designate the eigenvalues o ˆP aν (ˆP aν ) τ. Then by τ = 1, 0, and its eigenprojectors by (ˆP aν ) 1 = ˆP aν ; ˆPaν (ˆP aν ) 1 = 1 (ˆP aν ) 1 (38a) (ˆP aν ) 0 = I ˆP aν ; ˆPaν (ˆP aν ) 0 = 0 (ˆP aν ) 0 (38b) For these eigenvalues and eigenprojectors, the probe-position probability density o Eq. (30) gives p (ˆP aν ) (Q) = Tr [ρ (0) s (ˆP aν ) 0 ] p 0 (Q) + Tr [ρ (0) s (ˆP aν ) 1 ] p 0 (Q ε) (39) This result is illustrated in Fig. 7 or the strong-coupling case, in which p (ˆP aν ) (Q) consists o two peaks centered at Q/ε = 0 and Q/ε = 1. Figure 7: Illustrative example (qualitative) o the probability density o the probe position Q o Eq. (39) or the strong-coupling case. The result consists o two peaks, centered at Q/ε = 0 and Q/ε = 1. Also shown is the average o the probe position which, in units o ε, is independent o ε and lies between 0 and 1. From Eq. (39), or rom Eq. (33), we ind 1 (Â) ˆQ = ε 1 τ=0 τtr[ρ (0) s (ˆP aν ) τ ] = Tr(ρ s (0) ˆPaν ) = W a (Â) ν Tr[ρ s (0) (ˆP aν ) τ=1 ] = W (ˆP) aν τ=1 (40) 15

17 a result independent o ε. Thus the inal average probe position, in units o ε gives directly the probability W a (Â) ν o a ν o the observable  in the original state (see the irst row o Eq. (40), where we used τ τ(ˆp aν ) τ = P aν ); this is illustrated by the arrow in Fig. 7). In contrast, we would not know how to extract the W a (Â) n s rom the sum in Eq. (33) i  is not a projector. We can also say that inerring the probability o the eigenvalue τ = 1 (second row in (40)), i.e., the probability o yes o the observable P aν, rom a (Â) detection o ˆQ /ε (LHS o (40)), is equivalent to retrieving the diagonal elements a ν ρ (0) s a ν o the original density operator o the system (irst row in (40)). As we shall see in Sec. 5, the extension o this last idea to successive measurements will allow retrieving the ull density operator. 3.4 The reduced density operator o the system proper ater the interaction We compute the reduced density operator o the system proper ater the interaction with the probe is over, by tracing ρ (Â) o Eq. (29) over the probe, with the result = (P an ρ s (0) P an ) Tr π [e iεan ˆP / hρ π (0) e iεa n ˆP / h] (41a) nn = g(ε(a n a n )P an ρ s (0) P an (41b) n,n where we have deined the characteristic unction o the probe momentum distribution as g(β) = e iβ ˆP / h (0) π = Tr [ρ (0) π e iβ ˆP / h] ; β = ε(a n a n (42) As an example, or the particular case o a Gaussian state or the probe we have χ(q) = e Q2 /4σ2 Q (2πσ 2 Q )1/4 g(ε(a n a n ) = χ (Q εa n )χ(q εa n )dq (43a) (43b) = e ε2 (a n a n ) 2 /8σ 2 Q = e (εσ p/ h) 2 (a n a n ) 2 /2 (43c) 16

18 The result (41) is valid or an arbitrary state o the probe and an arbitrary coupling strength ε/σ Q. In the strong-coupling limit ε/σ Q, ρ (Â) s, reduces to ρ (Â) s, = P an ρ s (0) P an (44) n This is called the von Neumann-Lüders rule, originally postulated by Lüders [8], and then given a dynamical derivation in Re. [20] using vnm. We have thus reproduced the result o a non-selective projective measurement [21] o the observable Â, as a limiting case o our general ormalism. Notice that ρ (Â) s, and ρ (0) s are not connected by a unitary transormation: indeed, their eigenvalues have changed. For example, in the particular case in which the initial system state is the pure state ρ s (0) = ψ s (0) ψ (0), the initial eigenvalues are 1, 0,, 0. On the other hand, one eigenstate o ρ (Â) s, P aν ψ s (0), ulilling the eigenvalue equation ρ (Â) s, = (P a ν ψ (0) s ) = ψ (0) s P aν ψ (0) s (P aν ψ (0) s ) (45) so that the corresponding eigenvalue is ψ s (0) P aν ψ s (0). This is not a contradiction, because it is the density operator or the whole system, i.e., the system proper plus the probe, that evolves unitarily (see Eqn. (29)), while here we are dealing with the reduced density operator, which is the ull density operator traced over the probe. We computed above the reduced density matrix or the system proper, ρ (Â) s,, ater the system-probe interaction; then the probe is detected. We now wish to make a model or the probe-detector (π D) interaction, taking place at some time t 2 > t 1, and investigate i the resulting reduced density matrix or the system proper is still the same as the one given above, in Eq. (41). Assume that this new interaction does not involve the system proper s. The inal density operator ater the interaction with the detector, to (ater inter. with detector be called ρ, will contain a new evolution operator U πd, that involves the probe and the detector, but not the system s. Tracing s is 17

19 (ater inter. with detector ρ over the probe and the detector, we obtain (ater inter. with detector ρ s, (ater inter. with detector = Tr πd (ρ ) (46a) = (P an ρ s (0) P an ) Tr πd [U πd e iεan ˆP / hρ (0) π e iεa n ˆP / h ˆρ (0) D U πd ] nn (46b) = g(ε(a n a n )P an ρ s (0) P an (46c) nn giving the same answer as beore, Eq. (41). We stress that this result holds when the probe-detector interaction does not involve s. 4 Successive Measurements in Quantum Mechanics We now generalize the problem o the previous section to describe the measurement o two observables in succession: Â, as deined in Eq. (24), at time t 1, and then ˆB = b m P bm (47) m (the b m s may also be degenerate), at some later time t 2. For this purpose, we assume that we employ two probes, which are the ones that we detect; their momentum and coordinate operators are ˆP i, ˆQ i, i = 1, 2. The interaction o the system with the probes deines the Hamiltonian Ĥ(t) = ε 1 g 1 (t)â ˆP 1 + ε 2 g 2 (t) ˆB ˆP 2 (48) Again, we have disregarded the intrinsic Hamiltonians o the system and o the two probes. The unctions g 1 (t) and g 2 (t) are narrow non-overlapping unctions, centered around t t 1 and t t 2, respectively (see Eqs. (26)), with 0 < t 1 < t 2. The unitary evolution operator is given by Û(t) = e iε 2G 2 (t) ˆB ˆP 2 / he iε 1G 1 (t)â ˆP 1 / h (49) I the density operator o the system plus the probes at t = 0 is assumed to be the direct product ρ (0) = ρ s ρ π1 ρ π2, or t >> t 2, i.e., ater the second 18

20 interaction has ceased to act, it is given by ρ ( ˆB Â) = (P bm P an ρ s (0) P an P bm ) nn mm (e iε 1a n ˆP1 / hρ (0) π 1 e iε 1a n ˆP 1 / h) (e iε 2b m ˆP2 / hρ (0) π 2 e iε 2b m ˆP 2 / h) (50) At t >> t 2 we detect the two probe positions and momenta in order to obtain inormation about the system. Two examples are considered below. We irst detect the two probe positions ˆQ 1 and ˆQ 2. Their correlation can be calculated as ˆQ 1 ˆQ2 ( ˆB Â) = Tr [ρ ( ˆB Â) ˆQ 1 ˆQ2 ] (51) with the result [7, 9] ˆQ 1 ˆQ2 ( ˆB Â) ε 1 ε 2 where R stands or the real part. We have deined = R a n b m W ( ˆB Â) (ε 1 ) (52) nm and W ( ˆB Â) (ε 1 ) = λ(ε 1 (a n a n ))Tr [ρ s (0) (P an P bm P an )] (53) n λ(β) = g(β) + 2h(β) g(β) = e iβ ˆP 1 / h (0) π 1 h(β) = 1 β e iβ ˆP 1 /(2 h) ˆQ1 e iβ ˆP 1 /(2 h) (0) π 1 (54a) (54b) (54c) Here, (0) π 1 indicates an average over the initial state o probe 1. Notice that ˆQ 1 ˆQ2 ( ˆB Â) may be a complicated unction o ε 1 ; however, it is linear in ε 2, the strength associated with the last measurement, just as or a single (Â) measurement we ound, in Eq. (33), that ˆQ ε. We also have the result (see Re. [9] or the relevant conditions) λ(0) = 1 (55) The ollowing comments are in order at this point. Knowing the original system state ρ (0) s, the auxiliary equation RW ( ˆB Â) (ε 1 ) appearing in Eq. (51) 19

21 allows predicting the detectable quantity ˆQ 1 ˆQ2 ( ˆB Â). It extends to two measurements the Born probability W a (Â) n = Tr(ρ (0) s P an ) o Eq. (31) which, (Â) or single measurements, allows predicting the detectable quantity ˆQ = ε n a n W a (Â) n o Eq. (33). More interestingly, detecting ˆQ 1 ˆQ2 ( ˆB Â), we investigate what inormation can we retrieve on the system state. This is the point o As the next view that will be taken in the next section. As the next example, we consider again the same Hamiltonian o Eq. (48) but, ater the second interaction has acted, i.e., or t >> t 2, we detect, on a second sub-ensemble, the momentum ˆP 1 o the irst probe instead o its position, and the position ˆQ 2 o the second probe. The resulting correlation between ˆP 1 and ˆQ 2 is [7, 9] where and ˆP 1 ˆQ2 ( ˆB Â) = 1 ( I ε 1 ε 2 2σQ 2 a n b m W ˆB Â) (ε 1 ) (56) nm W ( ˆB Â) (ε 1 ) = λ(ε1 (a n a n ))Tr [ρ s (0) (P an P bm P an )] (57) n λ(β) = λ(β) λ(0) λ(β) = 1 g(β) β β (58a) (58b) Just as in the irst example, the auxiliary unction I W ( ˆB Â) (ε 1 ) allows predicting the detectable quantity ˆP 1 ˆQ2 ( ˆB Â). Alternatively, detecting ˆP 1 ˆQ2 ( ˆB Â) we shall investigate what inormation can we retrieve on the system state. We illustrate in Fig. 8 the measurements o Eqs. (52) and (56) or the particular case o the Stern-Gerlach experiment studied in section 2, using the ollowing translation o observables: Â ˆσ z ˆB ˆσx ˆP 1 ẑ ˆP2 ˆx (59), ˆQ 1 ˆp z ˆQ2 ˆp x 20

22 Figure 8: Illustration o the measurement o Q 1 Q 2 and P 1 Q 2 or a Stern-Gerlach arrangement. The correspondence between the various operators is given in Eqs. (59). Only the axes Q 1 and Q 2 are shown. The procedure to measure Q 1 Q 2 would be: a) send 1 atom rom L to R; b) Measure Q 1 == p z, Q 2 = p x ([ ˆQ 1, ˆQ 2 ] = 0) and construct Q 1 Q 2 ; c) Within an ensemble o N elements, construct Q 1 Q 2 = 1 N N i=1 Q (i) 1 Q(i) 2. The procedure to measure P 1 Q 2 would be to construct P 1 Q 2 = 1 N N i=1 P (i) 1 Q(i) 2 in another ensemble o N elements. In what ollows we examine some properties o the auxiliary unctions W ( ˆB Â) (ε 1 ) ( ˆB Â) and W (ε 1 ) deined in Eqs. (53) and (57) (or more details, see Res. [7, 9]). 1) In the strong-coupling limit, ε 1, we ind W ( ˆB Â) (ε 1 ) W ( ˆB Â) (ε 1 ) W ( ˆB Â) Tr [ρ s (0) (P an P bm P an )] (60) This is the joint probability distribution given by the so-called Wigner s rule [10] (it is real and non-negative; notice its dependence on the order in which the two successive measurements are perormed), which is obtained or projective measurements as P (b m, a n ) = P (b m a n )P (a n ) = b m a n 2 a n ψ 2 = ψ P an P an P B m ψ (61) We have assumed a pure state and no degeneracy, and P (b m a n ) denotes a conditional probability. 2) In the weak-coupling limit, ε 1 0, we ind W ( ˆB Â) (ε 1 ) W ( ˆB Â) (ε 1 ) K ( ˆB Â) Tr [ρ (0) s (P bm P an )] (62) 21

23 This is the Kirkwood s-dirac joint quasi-probability, which is complex, in general [, 22, 23, 24] (see also Re. [25]). In this limit ε 1 0, the probes correlation can be written in various orms as ollows ˆQ 1 ˆQ2 ( ˆB Â) ε 1 ε 2 1 = a n b m Tr [ρ(0) s (P bm P an + P an P bm )] nm 2 (63a) = a n b m Wb MH ma n = a n b m Ŝmn 0 (63b) nm nm = 1 Tr [ρ(0) s ( 2 ˆB +  ˆB] (63c) where Wb MH ma n = 1 Tr [ρ(0) s (P bm P an + P an P bm )] = Ŝmn 0 (64a) 2 Ŝ mn 1 2 (P b m P an + P an P bm ) (64b) Wb MH ma n is the real part o the Kirkwood quasi-probability distribution [, 22, 23, 24], also called the Margenau-Hill (MH) distribution [26]: it may take negative values, and thus cannot be regarded as a joint probability in the classical sense. We remark that, i [P bm, P an ] 0, then the operator Ŝmn o Eq. (64b) has at least one negative e-value. For this pair o variables, i.e., a n, b m, and or the particular state o the system which is the eigenstate that gives rise to this negative eigenvalue, the Margenau-Hill distribution o Eq. (64a) is negative, i.e., W MH = Ŝmn 0 < 0, and the probes correlation ˆQ 1 ˆQ2 /ε 1 ε 2 may lie outside the range [(a n b m ) min, (a n b m ) max ] (see Eq. (63b)). We illustrate this point with an example. Consider a Hilbert space o dimensionality N = 2, and two operators  and ˆB having the ollowing eigenvalues and eigenvectors: a n = 1, 0; 1 = ( 1 0 ), 0 = (0 1 ) (65a) b m = 1, 0; 1 = 1 2 ( 1 1 ), 0 = 1 2 ( 1 1 ) (65b) 22

24 From Eq. (64b) we have ˆQ 1 ˆQ2 ( ˆB Â) 1 = a n b m Ŝmn 0 = Ŝ 0 (66) ε 1 ε 2 n,m=0 For the two bases o Eq. (65), we ind S and its eigenvalues as S = 1 4 ( ) λ + = 1 4 (1 + 2) > 0 ψ+ λ = 1 4 (1 2) < 0 ψ (67) For the state ψ, the position-position correlation o Eq. (66) becomes ˆQ 1 ˆQ2 ( ˆB Â) ε 1 ε 2 = ψ S ψ = 1 4 (1 2) < 0 (68) which lies outside the interval deined by the possible values 0,1 o the product ss. 3) For an intermediate, arbitrary ε 1, W ( ˆB Â) ( ˆB Â) (ε 1 ) and W (ε 1 ) can be regarded, as already noted above, as two auxiliary unctions, Rε 1, W ( ˆB Â) (ε 1 ) and I W ( ˆB Â) ( ˆB Â) W (ε 1 ) being taylored or predicting ˆQ 1 ˆQ2. and (ε 1 ) or ˆP 1 ˆQ2. In the next section we shall see that or some special  s and ˆB s we can realize the inverse case: rom the measurable quantities ˆQ 1 ˆQ2 and ˆP 1 ˆQ2 we can reconstruct ρ (0) S. 4) I the projectors P an, P bm appearing in Eqs.(53) and (57) commute, i.e., [P an, P bm ] = 0, n, m, then we ind, or arbitrary ε 1 W ( ˆB Â) ( ˆB Â) (ε 1 ) = W (ε 1 ) = Tr [ρ (0) s (P bm P an )], ε 1 (69) This result is the standard, real and non-negative, quantum-mechanical deinition o the joint probability o a n and b m or commuting observables. We also ind that the correlation o the two probe positions measured in units o ε 1 ε 2 coincides, or an arbitrary coupling strength ε 1, with the standard result or the correlation o the two observables  and ˆB, i.e., ˆQ 1 ˆQ2 = Tr [ρ s (0) ( ε 1 ε ˆB)], ε 1 (70) 2 23

25 5) For the particular case in which π 1 is described by a pure Gaussian state, we ind (see also Eqs. (43)) λ(β) = λ(β) = g(β) = e β2 /(8σ 2 Q 1 ) (71a) W ( ˆB Â) ( ˆB Â) (ε 1 ) = W (ε 1 ) (71b) This is the case studied in Re. [7]. 6) As a particular situation o property 4) above, suppose we measure successively the same observable Â. We thus set ˆB = Â. in the ormalism. From Eq. (53) we ind Eq. (52) then gives W ( ˆB Â) a na n (ε 1 ) = λ((a n a n ))Tr s [ρ (0) s (P an P a n P an ] n = Tr s (ρ (0) s P an ) δ ( a n, a n ) (72) ˆQ 1 ˆQ2 (Â Â) ε 1 ε 2 = a n a n RW ( ˆB Â) a na n n, n (ε 1 ) = n W (Â) a n a 2 n = Â2 0 (73) For simplicity, we restrict ourselves to the case in which, at t = 0, the system proper s and the two probes π 1, π 2 are described by pure states, and Ψ 0 = ψ (0) s χ (0) π 1 χ (0) π 2 (74) Then, or t >> t 2, i.e., ater the second interaction, the state vector is given by Ψ = e iε 2Â ˆP 2 / he iε 1Â ˆP 1 / h Ψ 0 = (P an ψ (0) s ) (e iε 1a n ˆP1 / h χ (0) π 1 ) (e iε 2a n ˆP2 / h χ (0) π 2 ) n (75a) (75b) The joint probability density (jpd) o the eigenvalues Q 1, Q 2 o the two position operators or times t > t 2, when the two interactions have 24

26 ceased to act, is then p (Q 1, Q 2 ) = Ψ ˆP Q1 ˆPQ2 Ψ (76a) = W a (Â) n χ (0) π 1 (Q 1 ε 1 a n ) 2 χ π (0) 2 (Q 2 ε 2 a n ) 2 n = n W (Â) a n e (Q 1 ε 1 an) 2 2σ Q 2 1 2πσ 2 q1 e (Q 2 ε 2 an) 2 2σ Q 2 2 2πσ 2 q2 (76b) (76c) In Eq. (76c) we have assumed the original pure states or the probes to be Gaussian. Also, W (Â) (0) (0) a n = ψ ˆP s an ψ s (77) is the Born probability or the value a n in the original system state, and we wrote and similarly or probe π 2. = Q 1 e iε 1a n ˆP1 / h χ (0) π 1 = χ (0) π 1 (Q 1 ε 1 a n ) (78) Clearly, result (73) can be veriied rom the jpd o Eq. (76c). From this jpd we also obtain ˆQ i = ε i  0, i = 1, 2 (79a) ˆQ 2 i = ε 2 i Â2 0 + σ 2 Q i, i = 1, 2 (79b) It is useul to consider the correlation coeicient between the two probe positions, which is deined as C(Q 1, Q 2 ) = Q 1 Q 2 Q 1 Q 2 [ Q 2 1 Q 1 2 ][ Q 2 2 Q 2 2 ] (80a) = ( σ Q 1 ε 1 ) 2 (varâ) 0 + ( σ Q 2 ε 2 ) 2 (80b) 1, as σ Qi /ε i 0 (80c) In the second line, (80b), we have used results (79) or a pure Gaussian state o the probes. As a result, in the strong-coupling limit σ Qi /ε 1 0, the outcomes or the probe position 1 and probe position 2 become completely correlated, which is the result we would have expected. This is illustrated schematically in Fig

27 Figure 9: Schematic illustration o the jpd o the two probe positions p (Q 1, Q 2 ) o Eq. (76c), when we measure subsequently the same observable Â. The illustration is or the strong-coupling limit σ Qi /ε i << 1, in which Q 1 and Q 2 are strongly correlated. 7) Relation o successive measurements to weak values. Â, with pre-selection ψ and post- The weak value o the observable selection φ, is deined as [19, 24] (Â) W = = φ Â ψ φ ψ ψ φ φ Â ψ ψ φ φ ψ = n Tr (ˆρ(0) s P φ Â) Tr (ˆρ (0) s P φ ) P φ P an P φ = ψ P φâ ψ ψ P φ ψ (81a) (81b) (81c) (81d) 26

28 We see that the weak value can be regarded as the correlation unction between the observable  and the projector P φ [27, 28]. Eq. (81c) gives the generalization o the deinition or a state described by a density operator. Eq. (81d) expresses the weak value as the sum over the states o a n times a complex probability o a n conditioned by φ. Consider now a successive measurement experiment in which the two observables are  and P φ. The Hamiltonian o Eq. (48) becomes Ĥ(t) = ε 1 g 1 (t)â ˆP 1 + ε 2 g 2 (t)â ˆP 2, 0 < t 1 < t 2 (82) One can show [29, 30] that the position-position and momentum-position correlation o the two probes are related to the real and imaginary parts o the weak value as = lim ˆQ 1 ˆQ2 (Pφ Â) ε 0 ε 1 ˆQ 2 = 1 ÂP φ + P φ  = (Pφ Â) R[(ÂW ] (83a) 2 P φ = lim ˆP 1 ˆQ2 (Pφ Â) ε 0 ε 1 ˆQ 2 = 1 P φ  ÂP φ = 2σ (Pφ Â) 2σQ 2 P 2 1 2i P φ 1 I[(ÂW ] (83b) 5 State Reconstruction Scheme Based on Successive Measurements We now use the above ormalism to describe a state tomography scheme. We build on previous work [21, 7] to identiy a set o observables which, when measured in succession, provide complete inormation about the state o a quantum system described in an N-dimensional Hilbert space. For this purpose we consider, in our Hilbert space, two orthonormal bases, whose vectors are denoted by k and µ, respectively, with k, µ = 1,..., N. Latin letters will be used to denote the irst basis while Greek letters will be used or the second basis. Given k, there is only one vector; given µ, there is also only one vector: i.e., we have no degeneracy. We assume the two bases are mutually non-orthogonal, i.e., k µ 0, k, µ (84) Then the two bases have no common eigenvectors. This is illustrated in Fig. 10 or dimensionality N = 3. 27

29 Figure 10: Illustration, or N = 2, o the act that i the two orthonormal bases have one common eigenvector, they cannot be mutually non-orthogonal. The basic vectors o the irst basis are î, ĵ, ˆk. Those o the second basis are û, ˆv, ŵ. Assume they have one common eigenvector: e.g., ˆk = ŵ. Then û, ˆv ˆk and î, ĵ ŵ, contradicting the assumption that the two bases have no mutually orthogonal eigenvectors. We now consider a successive-measurement experiment in which the two observables are the rank-one projectors P k and P µ onto the k- and µ-state o the irst and second basis respectively. The Hamiltonian o Eq. (48) becomes Ĥ(t) = ε 1 g 1 (t)p k ˆP1 + ε 2 g 2 (t)p µ ˆP2, 0 < t 1 < t 2 (85) The projectors P k and P µ possess two eigenvalues: 0 and 1. We denote by τ and σ the eigenvalues o P k and P µ, respectively, and the corresponding eigenprojectors by (P k ) τ and (P µ ) σ. They satisy relations o the type presented in Eqs. (38). In the present case, Eq. (52) or the probes position-position correlation 28

30 unction gives ˆQ 1 ˆQ2 (Pµ P k) ε 1 ε 2 = R 1 τ,σ=0 τσw (Pµ P k) στ (ε 1 ) (86a) = RW (Pµ P k) (ε 1 ) (86b) In Eq.(86a), W (Pµ P k) στ (ε 1 ) is a particular case o the quantity W ( ˆB Â) στ (ε 1 ) o Eq.(54) when Â, ˆB, a n, and b m are replaced by P k, P µ, τ and σ, respectively, i.e., W (Pµ P 1 k) στ (ε 1 ) = λ(ε 1 (τ τ ))Tr[ρ s (P k ) τ (P µ ) σ (P k ) τ ] (87) τ =0 and, in particular, W (Pµ P k) (ε 1 ) = Tr(ρ (0) s P k P µ P k ) + λ(ε 1 ) k ( k) Tr(ρ (0) s P k P µ P k ) (88a) = G kk (ε 1 )Tr(ρ s (0) P k P µ P k ) (88b) k where we have used Eq. (55) and we have deined G kk (ε 1 ) = δ k,k + λ(ε 1 )(1 δ k,k ) (89) An important result is that Eq. (88) can be inverted to obtain ρ s (0), giving [7, 9] k ρ (0) s k = µ W (Pµ P k) (ε 1 ) µ k G kk (ε 1 ) µ k (90) It is clear that Eq.(90) requires that µ k 0 µ, k, i.e., that the two bases must be mutually non-orthogonal. Eq. (80) is the main result o this chapter. As a consequence, we see that the set o complex quantities W (Pµ P k) (ε 1 ) µ, k, contains all the inormation about the state o the system ρ s (0). I we could iner W (Pµ P k) (ε 1 ) µ, k, rom measurement outcomes, we could reconstruct the ull ρ (0) s. Notice that both the real and imaginary parts o the complex quantities W (Pµ P k) (ε 1 ) are needed or tomography. However, rom the detected positionposition correlations ˆQ 1 ˆQ2 (Pµ P k) ε 1 ε 2, we directly extract only RW (Pµ P k) (ε 1 ), 29

31 as we see rom Eq.(86b). In order to ind IW (Pµ P k) (ε 1 ), we use the momentumposition correlation. Our aim is to show that the measured quantities, the position-position and momentum-position correlation unctions, are inormationally complete: that is, one can reconstruct W (Pµ P k) (ε 1 ) rom these quantities. We recall, rom Eq. (40), the analogous situation in the single-measurement case, where the measurable quantities ˆQ (ˆP aν ) allow the reconstruction o the diagonal matrix elements o ρ s (0), i.e., (1/ε) ˆQ (ˆP aν ) = W (ˆP aν ) τ=1 = a ν ρ s (0) a ν. The extension o this result to the two-probe case is Eq. (90), in conjunction with Eq. (86b) (that relates RW (Pµ P k) (ε 1 ) to measurable quantities), and Eq. (98) below (that relates IW (Pµ P k) (ε 1 ) to measurable quantities). In the present case, Eq. (56) or the probes momentum-position correlation unction gives ˆP 1 ˆQ2 (Pµ P k) = 2σP 2 ε 1 ε 1 I W (Pµ P k) (ε 1 ) (91) 2 where W (Pµ P k) (ε 1 ) = Tr(ρ (0) s P k P µ P k ) + λ(ε 1 ) k ( k) Tr(ρ (0) s P k P µ P k ) (92) Although the unction W is in general not equal to the unction W (except when the probes are described by pure Gaussian states, as in Re. [7]), we now prove that it contains the necessary inormation to ind the imaginary part o W (Pµ P k), and thereore enables a complete state reconstruction. I we write W (Pµ P k) (ε 1 ) = x µk + iy µk (93a) W (Pµ P k) (ε 1 ) = x µk + iỹ µk (93b) the correlation unctions, Eqs. (86b) and (91), become ˆQ 1 ˆQ2 (Pµ P k) ε 1 ε 2 = x µk (94a) ˆP 1 ˆQ2 (Pµ P k) ε 1 ε 2 = 2σ 2 P 1 ỹ µk (94b) 30

32 Our aim is to express y µk in terms o the measured quantities o Eqs. (94). We go back to the expressions (88) and (92) or W (Pµ P k) (ε 1 ) and W (Pµ P k) (ε 1 ). The quantities λ(ε 1 ) and λ(ε 1 ) are known i the state o the probe π 1 is known (see Eqs. (54) and (58)). We write them as λ(ε 1 ) = λ r (ε 1 ) + iλ i (ε 1 ) λ(ε 1 ) = λ r (ε 1 ) + i λ i (ε 1 ) (95a) (95b) On the other hand, the traces appearing in Eqs. (88) and (92) are unknown; we write them as Tr(ρ (0) s P k P µ P k ) = k µ 2 k ρ s k = k µ 2 x µ k (96a) µ Tr(ρ (0) s P k P µ P k ) = r µk + is µk (96b) k ( k) Using Eq. (90), we wrote Eq. (96a) in terms o measured quantities only. We introduce the deinitions (93), (95), (96) in Eqs. (88) and (92), which then give x µk = k µ 2 µ x µ k + λ r r µk λ i s µk (97a) y µk = λ i r µk + λ r s µk ỹ µk = λ i r µk + λ r s µk (97b) (97c) For every pair o indices µ, k we now have a system o three linear equations in the three unknowns r µk, s µk and y µk, which can thus be expressed in terms o the measured quantities ỹ µk and x µk o Eqs. (92). The result or y µk is y µk = I{λ(ε 1) λ (ε 1 )} R{λ(ε 1 ) λ (ε 1 )} (x µk k µ 2 λ(ε 1 ) x µ k) + 2 (98) µ R{λ(ε 1 ) λ (ε 1 )}ỹµk We have thus achieved our goal o expressing W (Pµ P k) (ε 1 ), and hence ρ s (0) o Eq. (90), in terms o the measured correlations o Eqs. (94). This completes our procedure. It is interesting to examine the strong- and weak-coupling limits o the above 31

33 procedure. In the strong-coupling limit, ε 1, rom Eq. (89) we see that G kk (ε 1 ) δ kk, and Eq. (90) gives k ρ (0) s k Tr(ρ (0) s P k P µ P k ) = W (Pµ P k) µk = Tr s (ρ s (0) P k ) (99) in terms o Wigner s joint probability deined in Eq. (61). Notice that in this limit only the diagonal elements ρ (0) s can be retrieved. This is the limit in which Wigner s ormula (60) arises. In the weak-coupling limit, ε 1 0, we have G kk (ε 1 ) 1, and Eq. (90) gives k ρ (0) s k Tr s (ρ s (0) P µ P k ) µ k µ µ k = µ µ K (Pµ P k) µ k µk µ k (100) in terms o Kirkwood s joint quasi-probability deined in Eq. (62). The result (100) was irst obtained in Re. [21]. At irst glance it seems that, in a N-dimensional Hilbert space, the present scheme or state reconstruction requires the measurement o the 2N 2 dierent correlations ˆQ 1 ˆQ2 (Pµ P k) and ˆP 1 ˆQ2 (Pµ P k). However, Hermiticity and the unit value o the trace o the density matrix ρ s (0) impose N restrictions among its matrix elements, so that ρ s (0) can be expressed in terms o N 2 1 independent parameters. These restrictions eventually imply that only N 2 1 o these correlations are actually independent and thus the measurement o only N 2 1 correlations is required. Let us take as an example N = 2. Labeling the states as k = 0, 1 and µ = +,, it is enough to measure the correlations ˆQ 1 ˆQ2 (P+ P 0), ˆQ 1 ˆQ2 (P P 0), ˆP 1 ˆQ2 (P P 0) (101) In terms o these correlations and using the notation o Eqs. (94), the density matrix elements o our system prior to the measurement are given by ρ (0) 00 = x +,0 + x,0 (102a) ρ (0) = 1 x +,0 x,0 (102b) ρ (0) 01 = x +,0 x,0 2iy,0 g(ε 1 ) ρ (0) 10 = x +,0 x,0 + 2iy,0 g(ε 1 ) 32 (102c) (102d)

34 We have used Gaussian states or the probes, or which y µk = ỹ µk (see Eqs. (71)). The relations appearing in Eqs. (88) between W (Pµ P k) (ε 1 ) and ρ s (0) shed light on the the strong- and weak-couping limits o the retrieval scheme described above. We consider again the case N = 2, or Gaussian states or the probes, and or the case in which the states k, k = 0, 1, are eigenstates o the Pauli matrix σ k and the states µ, µ = +,, are eigenstates o σ x. In the strong-coupling limit ε 1, which, or N = 2 gives W (Pµ P k) (ε 1 ) Tr(ρ (0) s P k P µ P k ) = k µ 2 ρ (0) kk (103) W (+ 0) 1 2 ρ(0) 00, W ( 0) 1 2 ρ(0) 00 (104a) W (+ 1) 1 2 ρ(0), W ( 1) 1 2 ρ(0) 00 (104b) Thus, W (Pµ P k) (ε 1 ) only contains inormation on the diagonal elements o ρ (0) s, so that only ρ (0) 00 can be retrieved. and ρ(0) In the weak-coupling limit ε 1 0, and or N = 2 W (Pµ P k) (ε 1 ) Tr(ρ (0) s P µ P k ) = K µk = k ρ (0) s µ µ k (105) W (+ 0) ρ(0) 00 + ρ(0) 01 2 W (+ 1) ρ(0) 10 + ρ(0) 2, W ( 0) ρ(0) 00 ρ(0) 01 2, W ( 1) ρ(0) 10 ρ(0) 2 We thus see that the our ρ s s matrix elements can be retrieved. (106a) (106b) In conclusion, to reconstruct a QM state using the successive-measurement scheme, it is better to perorm measurements with weak coupling ε 1, rather than with strong coupling. For the case o an arbitrary ε 1 we have the relations (102) giving the ρ (0) s 33

35 matrix elements in terms o the position-position and momentum-position correlations. Recall that g(ε 1 ) = exp ( ε 2 1 /(8σ2 Q 1 ). Even i g 0, but g << 1, a small experimental uncertainty in extracting ˆQ 1 ˆQ2 (µ k) and ˆP 1 ˆQ2 (µ k), which give W (µ k) (ε 1 ) = x µ,k + iy µ,k, is divided by a small number g << 1 when k k, and this makes the error in extracting ρ (0) kk large. Again, we see that, in general, it is advantageous to use a weak coupling rather than a strong coupling. 6 A Quasi-Distribution and a Generalized Transorm o Observables Conceptually, one attractive eature o the tomographic approach we have described is that the quantities W (Pµ P k) (ε 1 ) that enter the reconstruction ormula, Eq. (90), can be interpreted as a quasi-probability, as we now explain. Let Ô be an observable associated with an N-dimensional quantum system. Using Eq.(90), we can express its expectation value as Tr s (ˆρ (0) s ô) = k ρ (0) s k k Ô k = W (Pµ P k) (ε 1 )O(µ, k; ε 1 ) (107) kk kµ where we have deined the transorm o the operator Ô as O(µ, k; ε 1 ) = k µ k k Ô k µ k G k k(ε 1 ) (108) Eqs. (107-8) have a structure similar to that o a number o transorms ound in the literature, that express the quantum mechanical expectation value o an observable in terms o its transorm and a quasi-probability distribution. For example, the Wigner transorm o an observable and the Wigner unction o a state are deined in the phase space (q, p) o the system, q and p labeling the states o the coordinate and momentum bases, respectively. In the present case, the transorm (108) o the observable is deined or the pair o variables (µ, k), µ and k labeling the states o each o the two bases. 34

36 As Eq. (107) shows, the quantity W (Pµ P k) (ε 1 ) plays the role o a quasiprobability or the system state ˆρ s (0), and is also deined or the pair o variables (µ, k). It can be thought o as the joint quasi-probability (in general complex [25]) o two non-degenerate observables, the two bases being their respective eigenbases. Since any pair o mutually orthogonal bases can be used, we have a whole amily o transorms that can be employed to retrieve the state. In the literature it has been discussed how Wigner s unction can be considered as a representation o a quantum state (see, e.g., Re. [31], Chs. 3 and 4), in the sense that i) it allows retrieving the density operator, and ii) any quantum-mechanical expectation value can be evaluated rom it. Similarly, and or the same reasons, in the present context the quasi-probability W (Pµ P k) (ε 1 ) can also be considered as a representation o a quantum state. 7 Conclusions In this series o lectures we described how to obtain inormation on a quantummechanical system, by coupling it to an auxiliary degree o reedom, or probe, which is then detected by some measuring instrument. The model used in the analysis is the one introduced by von Neumann, in which the interaction o the system proper with the probe is described in a dynamical way. For single measurements we used the standard von Neumann model, which employs one probe coupled to the system with an arbitrary coupling strength; or successive measurements, we generalized von Neumann s model employing two probes. In the case o single measurements, we investigated the average, the variance and the ull distribution o the probe position ater the interaction with the system, and their relation with the properties o the latter. An interesting outcome o the analysis is the reduced density operator o the system which, in the limit o strong coupling between the system and the probe, was shown to reduce to the von-neumann-lüders rule, a result which is requently obtained in a non-dynamical way, as a result o non-selective projective measurements. 35