Classical and Quantum Mechanical State Reconstruction
aa r X i v : . [ qu a n t - ph ] D ec Classical and quantum-mechanical state reconstruction
F. C. Khanna, P. A. Mello, and M. Revzen Theoretical Physics Institute, Physics Department,University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Instituto de F´ısica, U.N.A.M., Apartado Postal 20-364, 01000 M´exico, D. F. Department of Physics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Abstract
We review the problem of state reconstruction in classical and in quantum physics, which israrely considered at the textbook level. We review a method for retrieving a classical state inphase space, similar to that used in medical imaging known as Computer Aided Tomography. Weexplain how this method can be taken over to quantum mechanics, where it leads to a descriptionof the quantum state in terms of the Wigner function which, although may take on negative values,plays the role of the probability density in phase space in classical physics. We explain anotherapproach to quantum state reconstruction based on the notion of Mutually Unbiased Bases, andindicate the relation between these two approaches. Both are for a continuous, infinite-dimensionalHilbert space. We then study the finite-dimensional case and show how the second method, basedon Mutually Unbiased Bases, can be used for state reconstruction.
PACS numbers: 03.65.Wj,03.67.Ac . INTRODUCTION The retrieval of the state of a physical system is an important problem in classical as wellas in quantum physics, and yet it is a subject which is seldom discussed in textbooks.The state of a system in classical physics is described by a density in phase space, ρ ( q, p ),which could be determined (i.e., reconstructed) by the directly measurable conditional prob-ability density of the position q for a given momentum p , P ( q | p ), and the measurable prob-ability density of p , P ( p ), through the relation ρ ( q, p ) = P ( q | p ) P ( p ) . (1.1)Thus we may envision the state as being specified by the set of measurable quantities P ( q | p )and P ( p ). An alternative approach for determining a classical state involves measuring alinear combination of position and momentum (the constants a and b are introduced for thepurpose of fixing dimensions) X θ = aCq + bSp, C = cos θ, S = sin θ, (1.2)sometimes termed, for electromagnetic-field state measurements, “rotated quadratures”(Ref. [1], p. 136). The probability for the new variable X θ for all values of θ can thenbe used to reconstruct the phase-space density ρ ( q, p ) [2]. This procedure is similar to thefamiliar one employed in medical imaging for the reconstruction of a two-dimensional (2D)configuration density ρ ( x, y ), known as the Computer Aided Tomography (CAT) scan [2–4]: one simply replaces the two-dimensional configuration-space variables ( x, y ) of the CATmethod by the two-dimensional phase-space variables ( q, p ).In quantum physics, a system may be prepared in a pure state described by a vector inHilbert space, or, more generally, in a mixed state described by a density operator ˆ ρ (Ref.[5], p. 204 and Ref. [6], p. 72). The problem of state retrieval involves the inverse inquiry,i.e., what are the measurable quantities whose values will suffice to determine the quantumstate. Historically, this question may be traced back to the Pauli query [7] whether onecan reconstruct the wave function, amplitude and phase, for a one-particle system, from theprobability of its position, i.e., | ψ ( x ) | , and that for its momentum, | ˜ ψ ( p ) | ; here ˜ ψ ( p ) is thewave function in the momentum representation, the tilde indicating the Fourier transform.We now know that, in general, this is not possible: we need more information than these2wo distributions. The literature on this subject, which is still of current interest, has grownenormously ever since. Here we have made a selection out of these approaches, with theidea of providing a link with the classical reconstruction scheme.The classical approach based on P ( q | p ), is, of course, untenable in quantum physics, wherea fixed momentum precludes a well-defined position probability. A similar observation isapplicable to the direct approach of measuring the joint probability of q and p . However,it is remarkable that the alternative method based on measuring X θ defined in phase spacecan be taken over to quantum mechanics (Ref. [1], p. 143, Ref. [8], p. 101). But thenthe question arises: how can that be, if there is no such thing as a joint probability density ρ ( q, p ) in quantum mechanics? It turns out that the answer one obtains by following thisprocedure is a function defined in phase space which, although is not a bona-fide probabilitydensity (it is real, but not-necessarily non-negative, and has sometimes been named a “quasi-probability”), contains all the information needed to compute any quantum mechanicalexpectation value we please, just as if we were given the complex wave function, or the densityoperator. This concept of quasi-probability was invented by Wigner [9] in the early daysof Quantum Mechanics, with the purpose of finding the quantum-mechanical correctionsto thermodynamic functions, and is known as the Wigner function . Thus retrieving theWigner function using this tomographic method is a true quantum-state reconstruction, andto explain how this is achieved, and its relation with the classical tomographic approach,constitutes the main goal of the present paper. The main results for this approach are to befound in Eq. (3.4) below for the classical case, and in Eq. (3.16) for the quantum-mechanicalone.There is another concept which has been very useful in the task of reconstructing a quan-tum state. To give a trivial example, consider the eigenvectors of position and momentum:if the state vector of a system is an eigenstate of momentum, the system is equally likely to be found in any of the eigenstates of position. Pairs of bases with a similar propertyhave been extensively studied [10, 11] and are known as
Mutually Unbiased Bases (MUB).It turns out that MUB constitute a powerful tool for state reconstruction, since it is possibleto express the density operator that defines the state of the system in terms of a completeorthonormal set of operators [12–14]. We will explain the MUB approach to the problemof state reconstruction and show that the result [see Eq. (4.23) below] is consistent withthat found with the method explained above, based on tomography in phase space and the3igner function. Even more important, we shall find that the two approaches correspond,essentially, to employing two ways of handling the same complete set of operators, thusproviding a unified description of both methods.The paper is organized as follows. In the next section we review the CAT scan method,as employed for the reconstruction of a classical 2D density. In Section III A we present ascheme for classical state reconstruction in phase space similar to the CAT method employedin configuration space. In Section III B we explain how the classical scheme can be takenover to quantum mechanics, and explain the role played by Wigner function. We thenpresent in Sec. IV the alternative method for quantum state reconstruction based on thenotion of MUB. So far, the discussion has been restricted to quantum systems described ina continuous, infinite-dimensional Hilbert space, because of our desire to make an analogywith classical physics. However, there have been many contributions to the problem ofstate reconstruction for quantum systems described in a finite-dimensional Hilbert spaceemploying the notion of MUB. Although these systems do not have a classical counterpart,still they allow us to draw an illuminating parallel with the various concepts that have beenintroduced for a continuous Hilbert space. This fact motivates the brief discussion on therole of MUB for a finite-dimensional Hilbert space presented in Sec. V . Finally, we give ourconclusions in Sec. VI. To avoid cluttering of the main text, we include some details of themathematical derivations in a number of appendices.We wish to emphasize that the main goal for writing this paper is to give a pedagogicalpresentation of a subject which has been studied for many years and is still of current interest.With this motivation, we use a language that can be followed by a physics graduate student.We do hope that the analysis is in a form that allows its incorporation in a graduate QuantumMechanics course.
II. THE CLASSICAL RECONSTRUCTION SCHEME
First we review briefly the method mentioned in the Introduction, the CAT scan, thatis used for the reconstruction of a two-dimensional (2D) configuration density ρ ( x, y ). Themathematical procedure can be translated directly to retrieve a classical 2D phase-spacedensity ρ ( q, p ) and, even more interesting for us, it can be taken over to quantum mechanics.In a 2D CAT-scan [2–4] a fine pencil beam of X-rays passes through a sample, shown as4he shaded area in Fig. 1, along the “line of sight” defined by r · n = x ′ ; r is the positionvector of a point on the “line of sight” and n a unit vector perpendicular to the line ofsight, forming an angle θ with the x -axis, so that r and n can be written as r = x i + y j , n = cos θ i + sin θ j , i and j being unit vectors along the x and y axes, respectively. Thenthe equation for the line of sight becomes x ′ = x cos θ + y sin θ. (2.1)The line of sight is offset by the amount x ′ from the rotated y ′ axis. The beam is attenuated r xline of sight θ n ρ (x,y) x’x’ y’ y FIG. 1:
In 2D Computer-Aided-Tomography (CAT), a beam of X-rays is passed through a sample,indicated by the shaded area, along the “line of sight” which is offset by the amount x ′ from the rotated y ′ axis. The unit vector n , perpendicular to the line of sight, forms an angle θ with the x -axis. Knowingthe response of the sample for all offsets x ′ and directions θ we can reconstruct the original sample density ρ ( x, y ). by scattering and absorption produced by the various parts of the sample encountered alongthe path. Assuming that the attenuation at ( x, y ) is proportional to the sample density ρ ( x, y ), the total attenuation will be proportional to ρ θ ( x ′ ) = Z Z dxdy δ ( x ′ − Cx − Sy ) ρ ( x, y ) , (2.2)where we have used an arbitrary offset value designated by x ′ , and C and S have been definedin Eq. (1.2). Now it is important to remark that knowing the response of the sample givenby ρ θ ( x ′ ) for all x ′ and directions θ , we can reconstruct the density ρ ( x, y ) of the sample.5he mathematics of this problem was actually developed by J. Radon at the beginning ofthe twentieth century [15] for the study of astronomical data. In fact, the function ρ θ ( x ) ofEq. (2.2) is known in the literature as the Radon transform of the density ρ ( x, y ). Thus thetask is to invert the Radon transform to find the sample density.It is shown in Appendix A that the sample density ρ ( x, y ) can be expressed in terms ofthe response of the sample, ρ θ ( x ′ ), for all x ′ and directions θ defined above (see Ref. [1], pp.144), as ρ ( x, y ) = − π Z π dθ P Z ∞−∞ dx ′ ∂ρ θ ( x ′ ) /∂x ′ x ′ − ( x cos θ + y sin θ ) , (2.3)where P stands for the Cauchy principal value of the integral. Indeed, Eq. (2.3) is theinverse Radon transform of ρ θ ( x ′ ).To gain some insight into the structure of the sample response ρ θ ( x ′ ), it is illustrative toconsider the particular case in which the sample density ρ ( x, y ) is isotropic, i.e., dependentonly on the distance r = p x + y from the origin and independent of the angle. If we write x and y in polar coordinates as x = r cos φ , y = r sin φ , Eq. (2.2) for the response ρ θ ( x ′ )takes the form ρ θ ( x ′ ) = Z π dφ Z ∞ dr r δ ( x ′ − r cos( φ − θ )) ρ ( r ) , = Z π dφ Z ∞ dr r δ ( x ′ − r cos φ ) ρ ( r ) , (2.4)showing that ρ θ ( x ′ ) is independent of θ for the isotropic case. By direct substitution, onemay also observe that in this case ρ θ ( x ′ ) is symmetric, i.e., ρ θ ( − x ′ ) = ρ θ ( x ′ ).It may also be pointed out that the sample density ρ ( x, y ) must be a non-negative quantity,although this fact is not explicitly manifest in Eq. (2.3). It is thus useful to verify thisproperty in some particular example. For this purpose, we choose the isotropic case studiedin the last paragraph. For example, at the origin of coordinates, x = y = 0, Eq. (2.3) gives ρ (0 ,
0) = − π P Z ∞−∞ dx ′ ∂ρ θ ( x ′ ) /∂x ′ x ′ , (2.5)Since ρ θ ( x ′ ) is a symmetric function of x ′ , ∂ρ θ ( x ′ ) /∂x ′ is antisymmetric. Since the quantity − x ′ appearing in Eq. (2.5) has precisely this same property, as illustrated in Fig. 2, theresulting density ρ (0 ,
0) at the origin of coordinates is positive.6 ρ θ (x’)−1/x’d (x’)/dx’ θρ ρθ d (x’)/dx’ FIG. 2: Schematic representation of the functions ρ θ ( x ′ ), dρ θ ( x ′ ) /dx ′ (solid lines) and − /x ′ (dashed line) as functions of x ′ , needed in the text to show that the sample density ρ (0 ,
0) atthe origin is positive.
III. CLASSICAL-QUANTUM PHYSICS STATE-RECONSTRUCTION ANAL-OGYA. Classical state reconstruction
A state in classical statistical physics is determined by a probability density in phasespace. In this paper we shall always consider, for simplicity, one-particle systems with onedegree of freedom. We write the probability density in phase space as ρ ( q, p ) which, forconvenience in our comparison with Quantum Mechanics, will be normalized as Z ρ ( q, p ) dqdp π = 1 . (3.1)After the discussion given in the previous section on CAT in 2D configuration space ( x, y ),it is clear that a similar method can be applied in 2D phase space ( q, p ): if we consider thelinear combination of position and momentum given in Eq. (1.2), the probabilities for thenew variable X θ for all values of θ can then be used to reconstruct ρ ( q, p ) [2].Before proceeding, we indicate our choice for the constants a and b which were introducedto fix dimensions. We choose a = 1 /q , b = 1 /p , where q and p represent any convenientscales for position and momentum. Subsequently renaming the dimensionless quantities7 /q and p/p again as q and p , respectively, the transformation of Eq. (1.2) reads X θ = Cq + Sp. (3.2)We go back to the probability density of the variable X θ . If we designate it as ρ θ ( x ′ ),where x ′ represents an arbitrary value of X θ , we have, just as in Eq. (2.2) ρ θ ( x ′ ) = Z Z δ ( x ′ − Cq − Sp ) ρ ( q, p ) dqdp π . (3.3)The goal is to find ρ ( q, p ) in terms of ρ θ ( x ′ ) by inverting Eq. (3.3). Proceeding as in theprevious section and Appendix A, we find the equivalent of Eq. (2.3) as ρ ( q, p ) = − π Z π dθ P Z ∞−∞ dx ′ ∂ρ θ ( x ′ ) /∂x ′ x ′ − ( q cos θ + p sin θ ) . (3.4) B. Quantum state reconstruction
As mentioned in the Introduction, the above method based on measuring X θ defined inphase space can be taken over to quantum mechanics (Ref. [1], p. 143). This leads to a quasi-probability density defined in phase space known as the Wigner function.In what follows we shall take units in which ~ = 1. Consider an arbitrary Hermiteanoperator ˆ A . We define its Wigner transform as [1, 8, 9] W ˆ A ( q, p ) = Z e − ipy D q + y (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) | q − y E dy . (3.5)For the case where the operator ˆ A is the density operator ˆ ρ defining the state of the system,we speak of the Wigner function of the state, which has the normalization property
Z Z W ˆ ρ ( q, p ) dqdp π = 1 , (3.6)similar to the normalization of Eq. (3.1) adopted for the classical distribution.It is well known [1] that Wigner function for a state may be negative in some parts ofphase space. Thus it does not qualify as a true probability density and is referred to as aquasi-probability density. An illustration of the fact that it plays in quantum mechanics arole analogous to that played by the classical probability density ρ ( q, p ) is the similarity ofEqs. (3.15) and (3.16) given below with Eqs. (3.3) and (3.4), respectively.An important property of Wigner function, obtained from the definition (3.5), is [1, 8] T r ( ˆ A ˆ B ) = Z Z W ˆ A ( q, p ) W ˆ B ( q, p ) dqdp π , (3.7)8or any two operators ˆ A and ˆ B . This implies that the trace of the product of two operatorsin Hilbert space can be evaluated as an integral in phase space of the corresponding Wignertransforms. The normalization of Eq. (3.6) is consistent with the property (3.7), takingˆ A = ˆ ρ and ˆ B = 1. The statistical expectation value of an observable ˆ A , obtained by usingEq. (3.7), can be expressed as h ˆ A i = T r ( ˆ ρ ˆ A ) = Z Z W ˆ ρ ( q, p ) W ˆ A ( q, p ) dqdp π , (3.8)i.e., as an integral in phase space of Wigner function for the state times Wigner transform ofthe observable. With these results, Wigner function of the state and the Wigner transformof observables can be employed to “do QM in phase space”.It is also a simple exercise to show that the above definition of the Wigner function ofthe state ˆ ρ is equivalent to the inverse Fourier transform of the characteristic function of thedensity operator (Ref. [8], Eqs. (3.12), (3.16)), i.e., W ˆ ρ ( q, p ) = 12 π Z Z ˜ W ( u, v ) e i ( uq + vp ) dudv , (3.9a)˜ W ( u, v ) = T r [ ˆ ρ e − i ( u ˆ q + v ˆ p ) ] . (3.9b)Now consider the observable ˆ X θ = C ˆ q + S ˆ p, (3.10)which is the QM counterpart of the classical quantity of Eq. (3.2). This observable satisfiesthe eigenvalue equation ˆ X θ | x ′ ; θ i = x ′ | x ′ ; θ i , (3.11)where x ′ denotes an eigenvalue and | x ′ ; θ i the corresponding eigenvector.Our program is as follows. If the system is prepared in the state defined by the densityoperator ˆ ρ , we first consider the probability density ρ QMθ ( x ′ ) that a measurement of theobservable ˆ X θ will give the value x ′ : this probability density will be initially expressed interms of ˆ ρ , Eq. (3.12) below, and then in terms of the Wigner function W ˆ ρ ( q, p ) in phasespace, Eq. (3.15) below. The final goal is to “invert” this relation and show that we canretrieve the Wigner function in terms of ρ QMθ ( x ′ ).The probability density ρ QMθ ( x ′ ) is given by the standard QM expression ρ QMθ ( x ′ ) = T r ( ˆ ρ P θx ′ ) , where P θx ′ = | x ′ ; θ i h x ′ ; θ | . (3.12)9aking use of Eq. (3.8), we write ρ QMθ ( x ′ ) = Z Z W ˆ ρ ( q, p ) W P θx ′ ( q, p ) dqdp π . (3.13)In this expression, W P θx ′ ( q, p ) is the Wigner transform of the projector P θx ′ , which is calculatedin Appendix B with the result W P θx ′ ( q, p ) = δ ( x ′ − ( Cq + Sp )) . (3.14)Then Eq. (3.13) takes the form ρ QMθ ( x ′ ) = Z Z W ˆ ρ ( q, p ) δ ( x ′ − ( Cq + Sp )) dqdp π . (3.15)This last equation is the QM counterpart of Eq. (3.3) for the classical probability density ρ θ ( x ′ ). It shows explicitly that what plays the role of the classical probability density ρ ( q, p )in phase space is now the quasi-probability density given by the Wigner function W ˆ ρ ( q, p ).Thus in order to invert Eq. (3.15) we just copy the result in Eq. (3.4) and write W ˆ ρ ( q, p ) interms of ρ QMθ ( x ′ ) as W ˆ ρ ( q, p ) = − π Z π dθ P Z ∞−∞ dx ′ ∂ρ QMθ ( x ′ ) /∂x ′ x ′ − ( q cos θ + p sin θ ) . (3.16)This equation allows reconstructing the QM state, in the sense that from the observableprobability density ρ QMθ ( x ′ ) the Wigner function of the density operator can be extracted;its knowldedge, in turn, is equivalent to that of the state itself.This completes our analysis that shows a close analogy between the classical and quantumstate reconstruction: both require the use of the inverse Radon transform. We now turn toan alternative quantum-state reconstruction scheme which does not require the use of theRadon transform. IV. MUTUALLY UNBIASED BASES AND STATE RECONSTRUCTION
Mutually unbiased bases (MUB) in concept were introduced by Schwinger [10] in hisstudies of vectorial bases for Hilbert spaces that exhibit “maximal degree of incompatibility”.The eigenvectors of ˆ x and ˆ p , | x i and | p i , respectively, are example of such bases. Theinformation-theoretical oriented appellation “mutual unbiased bases” was introduced byWootters [11]. 10onsider two complete and orthonormal vectorial bases, B , B , whose vectors will bedesignated by | u ; B i and | v ; B i , respectively. The two bases are said to be MUB if andonly if, for B = B , |h u ; B | v ; B i| = K , ∀ u, v, (4.1)where K is a constant independent of u and v (see Ref. [12]). This property means thatthe absolute value of the scalar product of vectors from different bases is independent of thevectorial label within either basis. This implies that if a system is measured to be in oneof the states, say | u ; B i , of B , it is equally likely to be found in any of the states | v ; B i of any other basis B , when B and B are MUB. The value of K may depend on the bases B , B , which indeed is the case for a continuous Hilbert space. For a Hilbert space with afinite dimensionality d , K = 1 /d .The concept of MUB is found to be of interest in several fields. For instance, the ideasare useful in a variety of cryptographic protocols [16] and signal analysis [17].In what follows we outline a scheme for state reconstruction based on MUB [18] which isan alternative to the one presented in the previous section. A. Some properties of the operator ˆ X θ and its eigenstates We begin with a review of the properties of the operator ˆ X θ and its eigenstates | x ′ , θ i ,Eqs. (3.10), (3.11), and show that the bases {| x ; θ i} , {| x ; θ i} ( θ = θ , fixed) are MUB.We repeat the definition (3.10) of the operator ˆ X θ and introduce the new operator ˆ P θ asˆ X θ = C ˆ x + S ˆ p, ˆ P θ = − S ˆ x + C ˆ p ; (4.2)ˆ X θ and ˆ P θ are canonically conjugate , i.e., [ ˆ X θ , ˆ P θ ] = i , just as the original operators ˆ x , ˆ p .As a first step we solve the eigenvalue equation (3.11) in the coordinate representation.In this representation we define the wave function ψ x ′ ,θ ( x ) = h x | x ′ ; θ i , (4.3)which satisfies the equation (cid:18) x cos θ − i sin θ ∂∂x (cid:19) ψ x ′ ,θ ( x ) = x ′ ψ x ′ ,θ ( x ) . (4.4)11he solution of this equation is ψ x ′ ,θ ( x ) = F ( x ′ , θ ) e − i θ ( x cos θ − xx ′ ) , (4.5)where F ( x ′ , θ ) is an arbitrary function of x ′ and θ . It is shown in Appendix C that F ( x ′ , θ )can be completely determined, up to an arbitrary overall phase, by imposing on the states | x ′ ; θ i the requirements [20] h x , θ | x , θ i = δ ( x − x ) , (4.6a) h x , θ | ˆ X θ | x , θ i = x δ ( x − x ); h x , θ | ˆ P θ | x , θ i = − i δ ′ ( x − x ) , (4.6b) ψ x ′ ,θ ( x ) = h x | x ′ , θ i → δ ( x − x ′ ) , as θ → ψ x ′ ,θ = π/ ( x ) = e ix ′ x √ π . (4.6c)Here, Eq. (4.6a) expresses the ortho-normalization (in the sense of the Dirac delta func-tion) of the states | x ′ ; θ i . Equation (4.6b) requires that the matrix elements of the newcanonically conjugate operators ˆ X θ and ˆ P θ with respect to the new states | x ′ ; θ i be equalto the matrix elements of the old canonically conjugate operators ˆ x and ˆ p with respect tothe old states | x i , as demanded by a canonical transformation . The first Eq. (4.6c) requiresthat in the limit θ → | x ′ , θ i and the old one | x i be aDirac delta function. The second Eq. (4.6c) requires that for θ = π/
2, i.e., when ˆ X θ = π/ isthe momentum ˆ p , the wave function ψ x ′ ,θ = π/ ( x ) be a plane wave with no extra phases.The final result for the wave function ψ x ′ ,θ ( x ), up to an overall constant phase, is ψ x ′ ,θ ( x ) = e i [ π sgn(sin θ ) − θ ] p π | sin θ | e − i θ [( x ′ + x ) cos θ − xx ′ ] . (4.7)Notice the symmetry of this expression under the interchange x ↔ x ′ . Since | − x ′ , θ + π i = | x ′ , θ i , it suffices to consider state vectors in the range −∞ < x ′ < ∞ and − π/ ≤ θ ≤ π/ θ repeating the eigenvectors in this range (see Ref. [1], p. 144).We relate the new state | x ′ , θ i to the old one | x ′ i through a unitary transformation as | x ′ , θ i = ˆ U † ( θ ) | x ′ i . (4.8)For the reader’s convenience, we mention that the operator ˆ U used in this article coincideswith the one designated by ˆ V in Ref. [19], and that called ˆ U † in Ref. [20]. Using Eqs. (4.8)and (4.7) we find, for the matrix elements of the unitary operator ˆ U † ( θ ) in the old basis, h x | ˆ U † ( θ ) | x ′ i = e i [ π sgn(sin θ ) − θ ] p π | sin θ | e − i θ [( x ′ + x ) cos θ − xx ′ ] . (4.9)12sing the unitary operator ˆ U ( θ ) we write the eigenvalue equation (3.11) asˆ U ( θ ) ˆ X θ ˆ U † ( θ ) | x ′ i = x ′ | x ′ i , implying ˆ x = ˆ U ( θ ) ˆ X θ ˆ U † ( θ ). Thus the operator ˆ x and, similarly,its canonically conjugate ˆ p transform asˆ X θ = ˆ U † ( θ ) ˆ x ˆ U ( θ ) , and ˆ P θ = ˆ U † ( θ ) ˆ p ˆ U ( θ ) . (4.10)The above unitary transformation is given by the operatorˆ U ( θ ) = e − iθ ˆ n , (4.11)where ˆ n = a † a is the number operator, and a , a † are the annihilation and creation operators,respectively, given by a = √ (ˆ x + i ˆ p ), a † = √ (ˆ x − i ˆ p ) . Indeed, using the operator identity(Ref. [5], p. 339) e ˆ A ˆ Be − ˆ A = ˆ B + [ ˆ A, ˆ B ] + 12! [ ˆ A, [ ˆ A, ˆ B ]] + · · · , (4.12)we readily find that the operator (4.11) gives the transformation properties of a and a † ˆ U † ( θ )ˆ a ˆ U ( θ ) = e − iθ ˆ a, ˆ U † ( θ )ˆ a † ˆ U ( θ ) = e iθ ˆ a † , (4.13)that lead to the transformation properties of ˆ x and ˆ p , Eq. (4.10) [with Eq. (4.2)]. It isalso shown in Appendix D that the matrix elements of the operator exp ( iθ ˆ n ) (the adjointof (4.11)) are identical to those of Eq. (4.9) found earlier.Finally, we verify that the bases {| x ; θ i} and {| x ; θ i} (with fixed θ = θ ) that we havebeen studying above are MUB. Using Eq. (4.9) and the relation U ( θ ) U † ( θ ) = U † ( θ − θ ),which follows from (4.11), we find |h x ; θ | x ; θ i| = |h x | U † ( θ − θ ) | x i| = 12 π | S ( θ , θ ) | , (4.14)where S ( θ , θ ) = sin( θ − θ ). The number |h x ; θ | x ; θ i| is thus independent of x and x , so that, according to definition (4.1), the two bases are MUB. As an example, for θ = π/ | x ′ ; θ = π i is an eigenfunction of ˆ p with eigenvalue x ′ , whose projection in the x representation is e ix ′ x / √ π [see the second Eq. (4.6c)], its absolute value squared beingconsistent with Eq. (4.14).In Appendix E we present a simple way to derive the result of Eq. (4.14), which is anapplication of the idea of doing QM in phase space using Wigner transforms, mentionedright below Eq. (3.8). 13 . State reconstruction based on MUB Now we show that the MUB introduced above can be used to perform a quantum-mechanical state reconstruction. We first introduce the set of operatorsˆ Z ( a, b ) = e ia ˆ x e ib ˆ p = e − i ab e i ( a ˆ x + b ˆ p ) , −∞ < a, b < + ∞ , (4.15)where we have used the BCH identity, Eq. (B4). These operators form a complete andorthogonal operator basis [1, 8]. They satisfy the orthogonality property Z Tr h ˆ Z † ( a ′ , b ′ ) ˆ Z ( a, b ) i da db π = δ ( a ′ − a ) δ ( b ′ − b ) . (4.16)Thus we express the density operator as a linear combination of the operators ˆ Z ( a, b ) asˆ ρ = Z c ( a, b ) ˆ Z ( a, b ) da db π , c ( a, b ) = Tr h ˆ ρ ˆ Z † ( a, b ) i . (4.17)In the above equations, a and b play the role of Cartesian coordinates. We go over to polarcoordinates, defining a = r cos θ , b = r sin θ , so that Eq. (4.17) takes the formˆ ρ = 12 π Z ∞ dr r Z π dθ Tr (cid:2) ˆ ρ e − ir ( C ˆ x + S ˆ p ) (cid:3) e ir ( C ˆ x + S ˆ p ) . (4.18)[We use the abbeviations C and S from Eq. (1.2)]. Using similar arguments to those thatled from Eq. (A5) to (A6), we rewrite the above equation asˆ ρ = 12 π Z ∞−∞ dt | t | Z π dθ Tr (cid:2) ˆ ρ e − it ( C ˆ x + S ˆ p ) (cid:3) e it ( C ˆ x + S ˆ p ) . (4.19)Since in the exponent of this last equation we have the operator ˆ X θ [see Eq. (4.2)], theexponential can be written in its spectral representation [see Eq. (3.11)] as e it ( C ˆ x + S ˆ p ) = e it ˆ X θ = Z e itx ′ ˆ P x ′ ,θ dx ′ , (4.20)where the projection operator ˆ P x ′ ,θ is defined in Eq. (3.12). Similarly,Tr (cid:2) ˆ ρ e − it ( C ˆ x + S ˆ p ) (cid:3) = Z e − itx ′ Tr (cid:16) ˆ ρ ˆ P x ′ ,θ (cid:17) dx ′ (4.21a)= Z e − itx ′ ρ QMθ ( x ′ ) dx ′ , (4.21b)where we have used the definition of the QM probability density ρ QMθ ( x ′ ), Eq. (3.12). ThenEq. (4.19) for ˆ ρ , using Eqs. (4.20) and (4.21b), becomesˆ ρ = 12 π Z ∞−∞ dt | t | Z π dθ Z Z ∞−∞ dx ′ dx ′′ e − it ( x ′ − x ′′ ) ρ QMθ ( x ′ )ˆ P x ′′ ,θ (4.22a)= 12 π ‘ lim ǫ → + Z π dθ Z Z ∞−∞ dx ′ dx ′′ f ǫ ( x ′ − x ′′ ) ρ QMθ ( x ′ )ˆ P x ′′ ,θ . (4.22b)14n the last line we have performed the radial integral and used the definition (A8) thatwas introduced in our earlier analysis, in the course of inverting the Radon transform. Itis important to note that in the present context we have been able to express the densityoperator ˆ ρ directly in terms of the probability ρ QMθ ( x ′ ), thanks to the expansion of ˆ ρ , Eq.(4.19), in terms of MUB, together with Eq. (4.21b) which relates the trace on its left-handside with ρ QMθ ( x ′ ).The Wigner function for the state ˆ ρ of Eq. (4.22b) is identical to the result found abovein Eq. (3.16), which we reproduce here for completeness W ˆ ρ ( q, p ) = − π Z π dθ P Z ∞−∞ dx ′ ∂ρ QMθ ( x ′ ) /∂x ′ x ′ − ( q cos θ + p sin θ ) . (4.23)This result can be proved as follows. Application of Eq. (3.5) –defining the Wigner function–to the density operator ˆ ρ , Eq. (4.22b), gives W ˆ ρ ( q, p ) = 12 π lim ǫ → + Z π dθ Z Z Z ∞−∞ dydx ′ dx ′′ f ǫ ( x ′ − x ′′ ) ρ QMθ ( x ′ ) × e − ipy D q + y (cid:12)(cid:12)(cid:12) ˆ P x ′′ ,θ (cid:12)(cid:12)(cid:12) q − y E . (4.24)We evaluate the matrix element of the projector ˆ P x ′′ ,θ by using its definition in Eq. (3.12),the unitary transformation, Eq. (4.8), and its explicit expression, Eq. (4.9), to find D q + y (cid:12)(cid:12)(cid:12) ˆ P x ′′ ,θ (cid:12)(cid:12)(cid:12) q − y E = e i sin θ y ( x ′′ − q cos θ ) π sin θ . (4.25)Substituting this result in Eq. (4.24) and performing the integration over y we have W ˆ ρ ( q, p ) = lim ǫ → + π Z π dθ Z Z ∞−∞ dx ′ dx ′′ f ǫ ( x ′ − x ′′ ) ρ QMθ ( x ′ ) × δ ( x ′′ − ( q cos θ + p sin θ ))= lim ǫ → + π Z π dθ Z ∞−∞ dx ′ f ǫ ( x ′ − ( q cos θ + p sin θ )) ρ QMθ ( x ′ ) . (4.26)The last line is 2 π times the right-hand side of Eq. (A9), with x replaced by q and y by p .We thus take over the result of Eq. (2.3), making these replacements and multiplying by2 π , and find Eq. (4.23).Finally, we calculate the matrix elements of the density operator (4.22b) in the coordinaterepresentation, h x | ˆ ρ | x i , which is the counterpart in Hilbert space of Eq. (4.23). For thematrix elements of the projector ˆ P x ′′ ,θ we find, just as in Eq. (4.25), D x (cid:12)(cid:12)(cid:12) ˆ P x ′′ ,θ (cid:12)(cid:12)(cid:12) x E = e i sin θ ( x − x )( x ′′ − x x cos θ ) π sin θ , (4.27)15o that h x | ˆ ρ | x i = 12 π lim ǫ → + Z π dθ Z Z ∞−∞ dx ′ dx ′′ f ǫ ( x ′ − x ′′ ) ρ QMθ ( x ′ ) e i sin θ ( x − x )( x ′′ − x x cos θ ) π sin θ . (4.28)We compare this last equation with Eq. (A9) and use the result of Eq. (2.3) to obtain h x | ˆ ρ | x i = − π Z π dθ P Z Z ∞−∞ dx ′ dx ′′ x ′ − x ′′ ∂ρ QMθ ( x ′ ) ∂x ′ e i sin θ ( x − x )( x ′′ − x x cos θ ) π sin θ , (4.29)which shows explicitly how ρ QMθ ( x ′ ), which is a probability density, and hence a measurablequantity, can be used to find the matrix elements of the density operator.This completes our demonstration of the consistency of the two approaches to the prob-lem of quantum-mechanical state reconstruction that we have considered in this paper, forsystems described in a continuous Hilbert space. On the one hand, the approach presentedin the previous section based on tomography in phase space and the Wigner function and,on the other, the one given in the present section based on the expansion of the densityoperator in terms of operators defined via MUB. The difference in the strategies of thesetwo approaches involves, essentially, two ways of handling the complete orthonormal oper-ators e ia ˆ x e ib ˆ p : the Wigner function approach that led us to the results in Sec. III B can beregarded as using these operators to construct the Fourier transform of the density operator,as in Eq. (3.9). If, on the other hand, we consider their spectral representation, Eq. (4.20),we are led to the MUB approach of the present section. V. MUTUALLY UNBIASED BASES AND STATE RECONSTRUCTION IN AFINITE-DIMENSIONAL HILBERT SPACE
Considerable work has been devoted to the study of MUB in a finite, d -dimensionalHilbert space [21–24]. In this paper we restrict our study to the case in which the dimen-sionality d is a prime number : for this case the number of MUB is exactly d + 1 [11, 14, 22].The finite-dimensional theory is intriguingly connected with sophisticated mathematical no-tions [23, 25] that we do not consider here.In the finite, d -dimensional Hilbert space problem, where a Radon-like transform is notavailable, we shall follow a procedure which is analogous to that presented in the last sectionfor a continuous, infinite-dimensional Hilbert space.16e first consider the d -dimensional Hilbert space to be spanned by d distinct states | n i , with n = 0 , , · · · , ( d − | n + d i = | n i .These states are designated as the “computational basis” of the space. We shall followSchwinger [10] and introduce the unitary operators ˆ X and ˆ Z , which play a role analogous tothat of the position operator ˆ x and the momentum operator ˆ p of the continuous case. TheSchwinger operators are defined by their action on the states of the computational basis bythe equations ˆ Z | n i = ω n | n i , ω = e πi/d , (5.1a)ˆ X | n i = | n + 1 i . (5.1b)These definitions lead to the commutation relationˆ Z ˆ X = ω ˆ X ˆ Z. (5.2)The two operators ˆ Z and ˆ X form a complete algebraic set, in that only a multiple ofthe identity commutes with both [10]. As a consequence, any operator defined in our d -dimensional Hilbert space can be written as a function of ˆ Z and ˆ X .The d -dimensional matrix space is spanned by the complete orthonormal d operatorsˆ X m ˆ Z l , with m, l = 0 , , .. ( d − d × d matrix can be written as a linearcombination of these d operators. A familiar example is a 2-dimensional Hilbert space,where any 2 × σ x , σ z , σ x σ z and I .The operators ˆ X m ˆ Z l are orthonormal under the trace operation,Tr (cid:20) ˆ X m ˆ Z l (cid:16) ˆ X m ′ ˆ Z l ′ (cid:17) † (cid:21) = d δ m,m ′ δ l,l ′ , (5.3)a relation which can be proved directly using the defining Eqs. (5.1). Completeness followsfrom the set consisting of d linearly independent operators.We shall need to do arithmetic operations on the numbers n = 0 , , · · · , ( d −
1) that labelour states, assuming the periodic condition d = 0[mod d ]. When d is a prime number, theoperations of multiplication and division, modulo d , can be defined consistently [26]. As asimple example, we find, for d = 3, that 1 / · ω , all the operators of the formˆ X m ˆ Z l , with m = 0, by ( ˆ X ˆ Z b ) m , with b = lm − = 0 , , ... ( d − X m ˆ Z l = ω − m ( m − b ( ˆ X ˆ Z b ) m , (5.4)17ith l = mb [mod d ]. ¿From Eq. (5.3) we find, for the new quantities ( XZ b ) m , the orthogo-nality relation Tr (cid:20) ( ˆ X ˆ Z b ) m (cid:16) ( ˆ X ˆ Z b ′ ) m ′ (cid:17) † (cid:21) = d δ b,b ′ δ m,m ′ , m, m ′ = 0 . (5.5)The operators ( ˆ X ˆ Z b ) m are d ( d −
1) in number; these, together with the d operators ˆ Z l ,with l = 0 , , ... ( d −
1) (this last set evidently contains the identity: ˆ Z = ˆ Z d = I ), forma complete orthonormal set of d operators which is equivalent to the set ˆ X m ˆ Z l consideredabove. Thus we may express an arbitrary density operator as a linear combination of theseoperators as ˆ ρ = 1 d ( d − X b =0 d − X m =1 Tr h ˆ ρ (( ˆ X ˆ Z b ) m ) † i ( ˆ X ˆ Z b ) m + d − X l =0 Tr h ˆ ρ ( ˆ Z l ) † i ˆ Z l ) . (5.6)For a given b , the operator ˆ X ˆ Z b possesses d eigenvectors, which we denote by | c ; b i , c = 0 , , · · · , d −
1. In terms of the computational basis these eigenvectors are given by [22] | c ; b i = 1 √ d d − X n =0 ω b n ( n − − cn | n i , ˆ X ˆ Z b | c ; b i = ω c | c ; b i . (5.7)This equation defines d distinct bases ( b = 0 , , · · · , d −
1) which, when supplemented withthe computational basis, which is an eigenbasis of the operator ˆ Z [see Eq. (5.1a)], forms aset of d + 1 MUB bases, i.e., h c ; b | c ′ ; b i = δ c,c ′ , |h c ; b | c ′ ; b ′ i| = 1 d , b = b ′ , (5.8a) h n | n ′ i = δ n,n ′ , |h n | c ; b i| = 1 d . (5.8b)These equations can be proved straightforwardly by direct evaluation.We rewrite Eq. (5.6) by adding and subtracting the m = 0 terms asˆ ρ = 1 d ( d − X b =0 d − X m =0 Tr h ˆ ρ (( ˆ X ˆ Z b ) m ) † i ( ˆ X ˆ Z b ) m − d I + d − X l =0 Tr h ˆ ρ ( ˆ Z l ) † i ˆ Z l ) . (5.9)The spectral representation of the operator ˆ X ˆ Z b is given byˆ X ˆ Z b = d − X c =0 | c ; b i ω c h c ; b | . (5.10)Note that the eigenvalues ω c are non-degenerate. We obtainTr[ ˆ ρ (( ˆ X ˆ Z b ) m ) † ] = d − X c =0 h c ; b | ˆ ρ | c ; b i ω − cm . (5.11)18 ABLE I:
Analogy between quantities for a discrete and a continuous Hilbert space
Discrete case Continuous caseˆ X m ˆ Z l e ia ˆ x e ib ˆ p ˆ X m ˆ Z l = ω − m ( m − b ( ˆ X ˆ Z b ) m e ia ˆ x e ib ˆ p = e − i r · CS (cid:2) e i ( C ˆ x + S ˆ p ) (cid:3) r | c ; b i | x ′ ; θ i ˆ X ˆ Z b | c ; b i = ω c | c ; b i e i ( C ˆ x + S ˆ p ) | x ′ ; θ i = e ix ′ | x ′ ; θ i ( ˆ X ˆ Z b ) m | c ; b i = ω mc | c ; b i (cid:2) e i ( C ˆ x + S ˆ p ) (cid:3) r | x ′ ; θ i = e irx ′ | x ′ ; θ i Hence the density operator of Eq. (5.9) takes the formˆ ρ = d − X b,c =0 | c ; b ih c ; b | ˆ ρ | c ; b ih c ; b | + d − X n =0 | n ih n | ˆ ρ | n ih n | − I . (5.12)The matrix elements of ˆ ρ in the computational basis are then given by h n ′ | ˆ ρ | n ′′ i = d − X b,c =0 h n ′ | c ; b ih c ; b | ˆ ρ | c ; b ih c ; b | n ′′ i + h n ′ | ˆ ρ | n ′ i δ n ′ ,n ′′ − δ n ′ ,n ′′ . (5.13)The density operator ˆ ρ is given in terms of probabilities, Eq. (5.12), which are observablequantities; e.g., h c ; b | ˆ ρ | c ; b i is the probability to find the state | c ; b i when the system isdescribed by the density operator ˆ ρ . We thus find that ˆ ρ is reconstructed by using d +1 measurements [21]. Each of these measurements yields d − d + 1)( d −
1) = d − d -dimensional density matrix.Finally, we wish to call the reader’s attention to the analogy between several quantitiesused in the present section and those introduced in the previous one, where a continuous,infinite-dimensional Hilbert space was used. This correspondence is indicated in Table I. VI. CONCLUSIONS AND REMARKS
We have reviewed the approach to the quantum-state reconstruction problem based onthe Wigner function and the Radon transform, pointing out its close analogy with classicaltomography. We put emphasis on the role played by the Wigner function, which was shownto be analogous to that of the probability density in phase space for the classical problem.19he analysis underscores the intriguing fact that to reconstruct a quantum state werequire the probabilities of all the phase-space plane, and not merely the probabilities alongthe position and momentum axes as might be implied by a positive reply to Pauli’s queryposed in the Introduction.Then we reviewed an alternative route for the state reconstruction which is based onthe notion of mutually unbiased bases and does not make use of the Radon transform. Wedescribed its connection with the method based on the Wigner function.In addition, we showed that the concept of mutually unbiased bases can be applied tothe problem of state reconstruction for a finite-dimensional Hilbert space, which is quiterelevant for all applications to quantum computing. Finally, a parallel with the case of acontinuous, infinite-dimensional Hilbert space is drawn.
Acknowledgments
F. C. K. acknowledges financial support from NSERCC. P. A. M. and M. R. expresstheir gratitude to the Physics Department of the University of Alberta, Canada, where partof this work was carried out, for its hospitality. P. A. M. acknowledges financial supportfrom CONACyT, M´exico, through grant No. 79501, as well as from the Sistema Nacionalde Investigdores, M´exico. Informative discussions with Professors J. Zak, A. Mann and O.Kenneth are gratefully acknowledged.
Appendix A: Inverting the Radon Transform: proof of Eq. (2.3)
Multiplying both sides of Eq. (2.2) by e − ikx ′ and integrating over x ′ we find Z ∞∞ e − ikx ′ ρ θ ( x ′ ) dx ′ = Z e − ik ( Cx + Sy ) ρ ( x, y ) dxdy. (A1)We identify the two sides of this equation with the Fourier transform ˜ ρ θ ( k ) of ρ θ ( x ′ ), andthe Fourier transform ˜ ρ ( k x , k y ) of ρ ( x, y ), respectively, so that˜ ρ θ ( k ) = ˜ ρ ( k x = Ck, k y = Sk ) , k ∈ ( −∞ , ∞ ) . (A2)We recover ρ ( x, y ) as the inverse Fourier transform of ˜ ρ ( k x , k y ), ρ ( x, y ) = 1(2 π ) Z ∞∞ dk x dk y e i ( k x x + k y y ) ˜ ρ ( k x , k y ) , (A3)20here k x and k y are the Cartesian components of a wave number vector k ; in polar coordi-nates we have k x = K cos φ, k y = K sin φ, (A4a) K = | k | > . (A4b)The density ρ ( x, y ) becomes ρ ( x, y ) = 1(2 π ) Z ∞ dK K Z π dφ e iK ( x cos φ + y sin φ ) ˜ ρ ( K cos φ, K sin φ ) . (A5)While the variable k in Eq. (A2) is defined in the interval ( −∞ , ∞ ), the radial variable K inEq. (A5) is defined to be non-negative and in the interval (0 , ∞ ). The range of integrationof the variable K can be extended to the full real axis by first splitting the interval ofintegration of φ into the intervals (0 , π ) and ( π, π ) and then making the change of variables φ = φ ′ + π , K = − K ′ in the integral over the second interval, to obtain ρ ( x, y ) = 1(2 π ) Z ∞−∞ dk | k | Z π dθ e ik ( x cos θ + y sin θ ) ˜ ρ ( k cos θ, k sin θ ) . (A6)We identify the last factor with the quantity ˜ ρ θ ( k ), Eq. (A2), and substitute ˜ ρ θ ( k ) from theleft-hand side of Eq. (A1) to write ρ ( x, y ) = 1(2 π ) Z ∞−∞ dk | k | Z π dθ e ik ( x cos θ + y sin θ ) Z ∞−∞ dx ′ e − ikx ′ ρ θ ( x ′ ) . (A7)Defining the integral f ǫ ( ξ ) ≡ Z ∞−∞ | k | e − ikξ −| k | ǫ dk, ǫ > , (A8)and identifying ξ = x ′ − ( x cos θ + y sin θ ) , we write Eq. (A7) as ρ ( x, y ) = 1(2 π ) lim ǫ → + Z π dθ Z ∞−∞ dx ′ f ǫ ( x ′ − ( x cos θ + y sin θ )) ρ θ ( x ′ ) . (A9)Thus our task is to study the function f ǫ ( ξ ), which we write as f ǫ ( ξ ) = Z −∞ ( − k ) e − ik ( ξ + iǫ ) dk + Z ∞ ke − ik ( ξ − iǫ ) dk = ∂∂ξ (cid:18) ξ + iǫ + 1 ξ − iǫ (cid:19) ≡ ∂g ǫ ( ξ ) ∂ξ , (A10)where g ǫ ( ξ ) = 2 ξξ + ǫ . (A11)21sing the abbreviation α = x cos θ + y sin θ , we write the last integral in Eq. (A9) as I ǫ ≡ Z ∞−∞ ρ θ ( x ′ ) f ǫ ( x ′ − α ) dx ′ = − Z ∞−∞ ∂ρ θ ( x ′ ) ∂x ′ g ǫ ( x ′ − α ) dx ′ (A12)where we have used the definition (A10) and we have integrated by parts, assuming theintegrated term to vanish for sufficiently large values of the argument.The function g ǫ ( ξ ) is shown schematically in Fig. 3. As ǫ →
0, the integral of Eq. (A12) ξ−ε ε g ε (ξ) FIG. 3:
Schematic plot of the function g ǫ ( ξ ) defined in Eq. (A11). tends to the principal-value integrallim ǫ → I ǫ = − P Z ∞−∞ ∂ρ θ ( x ′ ) /∂x ′ x ′ − α dx ′ . (A13)Substituting this result in Eq. (A9), we then find Eq. (2.3) in the text. Appendix B: Proof of Eq. (3.14)
We first remark that it is easy to prove the operator identity P θx ′ = δ ( x ′ − ˆ X θ ) . (B1)Therefore, we compute the required Wigner transform of the projection operator (B1) as W P θx ′ ( q, p ) = W δ ( x ′ − ˆ X θ ) ( q, p ) (B2a)= Z D q + y (cid:12)(cid:12)(cid:12) δ [ x ′ − ( C ˆ q + S ˆ p )] (cid:12)(cid:12)(cid:12) q − y E e − ipy dy (B2b)22t is convenient to work with the Fourier transform of this last expression with respect tothe variable x ′ ; i.e., Z e ikx ′ W P θx ′ ( q, p ) dx ′ (B3a)= Z D q + y (cid:12)(cid:12)(cid:12) e ik ( C ˆ q + S ˆ p ) (cid:12)(cid:12)(cid:12) q − y E e − ipy dy (B3b)= e i k SC Z D q + y (cid:12)(cid:12)(cid:12) e ikC ˆ q e ikS ˆ p (cid:12)(cid:12)(cid:12) q − y E e − ipy dy , (B3c)where use was made of the Baker-Campbell-Hausdorff (BCH) identity (Ref. [5], p. 442) e ˆ A + ˆ B = e ˆ A e ˆ B e − [ ˆ A, ˆ B ] , (B4)valid for any two Hermitean operators ˆ A , ˆ B , whose commutator commutes with each ofthem, i.e., [ ˆ A, [ ˆ A, ˆ B ]] = [ ˆ B, [ ˆ A, ˆ B ]] = 0 . Introducing inside the matrix element of Eq. (B3c)a complete set of eigenstates of position and of momentum right after the first and secondexponentials, respectively, we find Z e ikx ′ W P θx ′ ( q, p ) dx ′ = e ik ( Cq + Sp ) . (B5)The inverse Fourier transform of this last expression gives the result of Eq. (3.14). Appendix C: Proof of Eq. (4.7)
To prove Eq. (4.7) we impose the requirements of Eq. (4.6) on the solution, Eq. (4.5).1) The ortho-normalization condition, Eq. (4.6a), imposed on the wave function ψ x ′ ,θ ( x )of Eq. (4.5) gives, for the function F ( x ′ , θ ), F ( x ′ , θ ) = e iφ θ ( x ′ ) p π | sin θ | ; (C1) φ θ ( x ′ ) is an arbitrary phase, dependent on x ′ and θ . The wave function ψ x ′ ,θ ( x ) becomes ψ x ′ ,θ ( x ) = 1 p π | sin θ | e − i θ [ ( x cos θ − xx ′ )+ iφ θ ( x ′ ) ] . (C2)2) The first requirement in Eq. (4.6b) for the matrix elements of ˆ X θ with the wave functionof Eq. (C2) is automatically fulfilled, since i) our starting point has been the eigenvalueequation, Eq. (3.11), and ii) the wave function of Eq. (C2) satisfies the orthonormalizationcondition, Eq. (4.6a). 23) We first compute the matrix element of ˆ P θ which appears on the left-hand side of thesecond requirement in Eq. (4.6b) . Using the definition of ˆ P θ given in Eq. (4.2) and Eq.(C2) we have, in the coordinate representation h x , θ | ˆ P θ | x , θ i = Z ψ ∗ x ,θ ( x ) (cid:20) − sin θ x − i cos θ ∂∂x (cid:21) ψ x ,θ ( x ) dx, = Z ψ ∗ x ,θ ( x ) (cid:20) − θ x + cos θ sin θ x (cid:21) ψ x ,θ ( x ) dx, = − i e i [ φ θ ( x ) − φ θ ( x )] δ ′ ( x − x ) + cos θ sin θ x δ ( x − x ) . (C3)We write, for the above exponential, the Taylor expansion e i [ φ θ ( x ) − φ θ ( x )] = 1 + i (cid:20) ( x − x ) φ ′ θ ( x ) + ( x − x ) φ ′′ θ ( x ) + · · · (cid:21) − (cid:20) ( x − x ) ( φ ′ θ ( x )) + 2 ( x − x ) φ ′ θ ( x ) φ ′′ θ ( x ) + · · · (cid:21) + · · · , (C4)where the primes mean derivatives with respect to the argument. We use the δ -functionidentities xδ ′ ( x ) = − δ ( x ), x n δ ′ ( x ) = 0 , n ≥
2, to write the matrix element, Eq. (C3), as h x , θ | ˆ P θ | x , θ i = [ − iδ ′ ( x − x ) + φ ′ θ ( x ) δ ( x − x )] + cos θ sin θ x δ ( x − x ) . (C5)In order to satisfy the second requirement in Eq. (4.6b) we thus need φ ′ θ ( x ′ ) = − cos θ sin θ x ′ , (C6)with the solution φ θ ( x ′ ) = − cos θ sin θ x ′ ϕ ( θ ) , (C7)where ϕ ( θ ) is an arbitrary function of θ .The wave function ψ x ′ ,θ ( x ) of Eq. (C2) then becomes ψ x ′ ,θ ( x ) = e iϕ ( θ ) p π | sin θ | e − i θ [( x + x ′ ) cos θ − xx ′ ] . (C8)4) Choosing, for the phase ϕ ( θ ), ϕ ( θ ) = π θ ) − θ , (C9)we satisfy the requirements of Eq. (4.6c).We finally find the wave function of Eq. (4.7).24 ppendix D: The matrix elements of the operator exp( iθ ˆ n ) In this Appendix we compute the matrix elements of the operator exp ( iθ ˆ n ) in the originalbasis | x i . We have h x | e iθ ˆ n | x ′ i = X n ψ ∗ n ( x ) ψ n ( x ′ ) e inθ (D1a)= 1 √ π e − x x ′ X n H n ( x ) H n ( x ′ ) ( e iθ ) n n n ! , (D1b)where ψ n ( x ) are the one-dimensional harmonic oscillator wave functions (see Ref. [27], p.61, Eq. (5.24), where the variable x has been replaced by the dimensionless x/ q ~ mω , as usedin this paper) ψ n ( x ) = 1 π / √ n n ! e − x H n ( x ) , (D2) H n ( x ) being Hermite polynomials. We compute the sum in Eq. (D1b) using the identity(Ref. [28], p. 781, Problem 6.12) X n H n ( x ) H n ( x ′ ) t n n n ! = 1 √ − t exp (cid:20) xx ′ t − t ( x + x ′ )1 − t (cid:21) , (D3)with the result h x | e iθ ˆ n | x ′ i = e − i θ e − i θ [( x + x ′ ) cos θ − xx ′ ] p π ( − i ) sin θ . (D4)This result is identical to that of Eq. (4.9) if we choose, for the square root, the branch √− i sin θ = e − i π p | sin θ | , for sin θ > e + i π p | sin θ | , for sin θ < , (D5)= e − i π sgn(sin θ ) p | sin θ | . (D6) Appendix E: A simple way to derive the result (4.14)
It will suffice to evaluate the quantity |h x | x ′ ; θ i| ; this is the probability to find x in aunit interval around the value x when the system has been prepared in the state | x ′ ; θ i . Wefind |h x | x ′ ; θ i| = h x | ˆ P x ′ ; θ | x i = T r (ˆ P x ′ ; θ ˆ P x ) (E1a)= Z Z W ˆ P x ′ ; θ ( q, p ) W ˆ P x ( q, p ) dqdp π , (E1b)25here in the last line we have used Eq. (3.7) to express our probability in terms of Wignertransforms. The Wigner transform of the projector ˆ P x ′ ; θ is found from Eq. (3.14) and thatfor the projector ˆ P x is simply δ ( x − q ). We thus write the last equation as |h x | x ′ ; θ i| = Z Z δ ( x ′ − ( qC + pS )) δ ( x − q ) dqdp π (E2a)= 12 π Z δ ( x ′ − ( xC + pS )) dp . (E2b)On the one hand, this integral can be evaluated directly, giving |h x | x ′ ; θ i| = 12 π Z | S | δ (cid:18) p − x ′ − xCS (cid:19) dp = 12 π | S | , (E3)just as in Eq. (4.14) with θ = θ and θ = 0. On the other hand, the appearance of thefactor | S | in the denominator of the result (E3) can be understood by using an intuitivegeometrical argument starting from (E2b), as follows. We approximate the delta functionoccurring in Eq. (E2b) by the step δ ( x ′ − ( xC + pS )) ≈ u δx ′ ( x ′ ) ≡ δx ′ , if x ′ ∈ ( x ′ − δx ′ , x ′ + δx ′ )0 , if x ′ / ∈ ( x ′ − δx ′ , x ′ + δx ′ ) , (E4)the delta function being attained in the limit δx ′ →
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