1 Introduction

Inverse spin-s portrait and representation of qudit states by single probability vectors

Abstract

Using the tomographic probability representation of qudit states and the inverse spin-portrait method, we suggest a bijective map of the qudit density operator onto a single probability distribution. Within the framework of the approach proposed, any quantum spin- state is associated with the -dimensional probability vector whose components are labeled by spin projections and points on the sphere . Such a vector has a clear physical meaning and can be relatively easily measured. Quantum states form a convex subset of the simplex, with the boundary being illustrated for qubits () and qutrits (). A relation to the - and -dimensional probability vectors is established in terms of spin- portraits. We also address an auxiliary problem of the optimum reconstruction of qudit states, where the optimality implies a minimum relative error of the density matrix due to the errors in measured probabilities.

INVERSE SPIN- PORTRAIT
AND REPRESENTATION OF QUDIT STATES
BY SINGLE PROBABILITY VECTORS

Sergey N. Filippov and Vladimir I. Man’ko

Moscow Institute of Physics and Technology (State University)

Institutskii per. 9, Dolgoprudnyi, Moscow Region 141700, Russia

P. N. Lebedev Physical Institute, Russian Academy of Sciences

Leninskii Prospect 53, Moscow 119991, Russia

Keywords: spin tomography, spin portrait, qubit, qudit, probability representation.

1 Introduction

In the early years of quantum mechanics, it was proposed by Landau [1] and von Neumann [2] to represent the quantum states by the density matrices. This approach turned out to be applicable to spin states as well [3]. The density matrix formalism proved to be very useful and all physical laws such as the time evolution and energy spectrum were formulated in terms of this notion. However, many alternative ways to describe a spin- state were proposed; for instance, with the help of different discrete Wigner functions (these and other analytical representations were reviewed in [4]) and a fair probability-distribution function called spin tomogram [5, 6]. The latter function depends on the spin projection along all possible unit vectors . All these proposals are, in fact, merely different mappings of the density operator. Inverse mappings are also developed thoroughly and are based on the fact that, if a (quasi)probability distribution is given, the density matrix can be uniquely determined. On the other hand, there is a redundancy of information contained in spin tomogram . An attempt to avoid such a redundancy was made in [7, 12, 11, 14, 10, 9, 8, 13, 15]. According to [7], the density matrix can be determined by measuring probabilities to obtain spin projection if a Stern-Gerlach apparatus is oriented along specifically chosen directions in space. In [12] it was shown that the density matrix can also be reconstructed if the probabilities to get the highest spin projection are known for appropriately chosen directions in space. In other words, one deals with the values of function , and solves a system of linear equations to express the density matrix elements in terms of the probabilities . The conditions on vectors and an inverse method were also presented [12].

In this paper, we address the problem of identification of a qudit- state with a single probability distribution vector. Moreover, such a vector must have a clear physical interpretation. These arguments make this problem interesting from both theoretical and practical points of view. An attempt to construct a bijective map of the density operator onto a probability vector with some interpretation of vector components was made in the series of papers [20, 18, 19, 17, 16] (see also the recent review [21]). The problem was shown to have an explicit solution for the lower values of spin .

In [22], a unitary spin tomography was suggested for describing spin states by probability distribution functions , where is a unitary matrix. Information contained in this probability distribution function (called unitary spin tomogram) is even more redundant than that contained in spin tomogram . Nevertheless, these redundancies can be used to solve explicitly the problem under consideration, i.e., to find an invertible map of the spin density operator onto a probability vector with a clear physical interpretation for an arbitrary spin . The aim of our work is to present a construction of the following invertible map. It provides the possibility to identify any spin state with the probability vector with components . Here, random variables and are the spin projection and unitary matrix, respectively. The spin projection takes values and a finite set of unitary matrices describes the unitary rotation operations in the finite-dimensional Hilbert space of spin states.

Thus, the function is the joint probability distribution of two random variables and . The function is related to the spin unitary tomogram by the formula

 w(m,uk)=P(m,uk)∑jm=−jP(m,uk).

This fact makes it possible to determine the density operator . On the other hand, the probability distribution contains extra information in comparison with that contained in the density matrix. The matter is that, if the density matrix is given, a formula for the probability distribution in terms of does not exist. To obtain such a formula, one needs to take into account some additional assumptions. For example, we can assume the uniformity of the probability distribution of unitary rotations , i.e., each matrix (or direction in the case ) is taken with the same probability , where is an appropriate number of unitary rotations. If this is the case, the density matrix provides an explicit formula of the probability distribution . One can choose another nonuniform distribution of unitary rotations, but knowledge of this distribution is necessary information to be added to information contained in the density matrix.

In the context of probability theory, the vector (or the point on a simplex determined by this vector) is defined by the set of spin projections and unitary rotations , which can be chosen with some probability. In view of this, the probability under consideration is a fair joint probability distribution function of two discrete random variables.

This paper is organized as follows.

In Sec. 2, a short review of spin and unitary spin tomograms is presented and a spin- portrait method is introduced. In Sec. 3, we review a density matrix reconstruction procedure proposed in [12]. In Sec. 4, a representation of quantum states by probability vectors of special form [21] is given. In Sec. 5, an inverse spin-portrait method is presented. This method allows to construct a map of the spin density operator onto the probability vector . Two cases of used unitary rotations are given: and , . In Sec. 6, the inverse map is presented in explicit form for rotations. In Sec. 7, particular properties of symbols are analyzed: star product kernel is presented and relation to symbols is considered. In Sec. 8, examples of qubits () and qutrits () are presented. In Sec. 9, the probability vector is considered on the corresponding simplex and a boundary of quantum states is presented for qubits and qutrits. In Sec. 10, conclusions are presented.

2 Unitary Spin Tomography and Spin-s Portrait of Tomograms

We begin with some notation. Unless stated otherwise, qudit states with spin are considered. Any state vector of such a system is uniquely determined through the basis vectors , which are eigenvectors of both angular momentum operator and square of total angular momentum, i.e., . The spin projection takes the values .

A unitary spin tomogram of the state given by the density operator is defined as follows:

 w(j)(m,u)=⟨jm|^u†^ρ^u|jm⟩=Tr(^ρ ^u|jm⟩⟨jm|^u†)=Tr(^ρ ^U(j)(m,u)), (1)

where, in general, is a unitary transform of the group . For the sake of convenience, starting from now we will identify and its matrix representation in the basis of states , assuming that the matrix defines, in fact, the unitary transform . The tomogram is a function of the discrete variable and the continuous variable . The operator is called the dequantizer operator because it maps an arbitrary density operator onto the real probability distribution function . The dequantizer satisfies a sum rule of the form for all . In view of this fact, the tomogram is normalized, i.e., .

The particular case leads to the so-called spin tomogram , where the direction determines the dequantizer operator (some properties of spin tomogram were discussed in [23, 24, 25]). Here, we introduced a rotation operator defined through

 ^R(n)=e−i(n⊥⋅ ^J)θ,n⊥=(−sinϕ,cosϕ,0). (2)

The inverse mapping of spin tomogram onto the density operator is relatively easily expressed through the quantizer operator as follows:

 ^ρ=j∑m=−j14π2π∫0dϕπ∫0sinθdθ w(j)(m,n(θ,ϕ))^D(j)(m,n(θ,ϕ)). (3)

Different explicit formulas of both dequantizer and quantizer operators are known [5, 6, 26, 27, 28], with the ambiguity being allowed for the quantizer operator. In this paper, we preferably use the orthogonal expansion of the form [29]

 ^U(j)(m,n)=2j∑L=0f(j)L(m)^R(n)^S(j)L^R†(n)=2j∑L=0f(j)L(m)^S(j)L(n), (4) ^D(j)(m,n)=2j∑L=0(2L+1)f(j)L(m)^R(n)^S(j)L^R†(n)=2j∑L=0(2L+1)f(j)L(m)^S(j)L(n), (5)

where the coefficient is an -degree polynomial of the discrete variable , and the operator is the same polynomial of the operator variable .

For instance, in the case of qubits () and qutrits (), we have

 f(1/2)0(m)=1√2,f(1/2)1(m)=√2m,^S(1/2)0(n)=1√2^I,^S(1/2)1(n)=√2(^J⋅n), (6) f(1)0(m)=1√3,f(1)1(m)=m√2,f(1)2(m)=3m2−2√6, (7) ^S(1)0(n)=1√3^I,^S(1)1(n)=1√2(^J⋅n),^S(1)2(n)=1√6(3(^J⋅n)2−2^I). (8)

In the case of an arbitrary spin , , , and all other coefficients are expressed via the recurrence relation that was presented in [29] and relates , , and . Expansions (4) and (5) are orthogonal in the following sense:

 Tr(^S(j)L(n)^S(j)L′(n))=j∑m=−jf(j)L(m)f(j)L′(m)=δLL′. (9)

This means that functions are orthogonal polynomials of a discrete variable, with the weight function being identically equal to unity. Using the theory of classical orthogonal polynomials of a discrete variable [30, 31], it is not hard to prove that the function is expressed through the discrete Chebyshev polynomial or Hahn polynomial as follows:

 f(j)L(m)=1dLtL(j+m,2j+1)=1dLh(0,0)L(j+m,2j+1),dL=√(2j+L+1)!(2L+1)(2j−L)!. (10)

2.1 Spin-s Portrait

The tomogram of a system with spin is a function of the discrete spin projection . This means that the tomogram can be represented in the form of the following -dimensional probability vector:

 w(j)j(u)=⎛⎜ ⎜ ⎜ ⎜⎝w(j)(j,u)w(j)(j−1,u)⋯w(j)(−j,u)⎞⎟ ⎟ ⎟ ⎟⎠. (11)

We will refer to such a vector as the spin- portrait (in analogy with the qubit-portrait concept [32]). Note, that vector (11) is a fair probability distribution vector since and for all unitary matrices . Since the components of vector (11) are fair probabilities, the sum , where , can also be treated as a probability and has a clear physical meaning. Summing some components of vector (11), one can construct a probability vector of less dimension , where plays the role of pseudospin and can take values . To be precise, reads

 w(j)s(u)=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝∑m∈A1w(j)(m,u)∑m∈A2w(j)(m,u)⋯∑m∈A2s+1w(j)(m,u)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠, (12)

where for all , , and for all . Vector (12) is referred to as the spin- portrait of a qudit- state. In the case , we obtain the so-called qubit portrait of the form

 w(j)1/2(u)=(∑m∈A1w(j)(m,u)∑m∈A2w(j)(m,u))=(∑m∈A1w(j)(m,u)1−∑m∈A1w(j)(m,u))=(1−∑m∈A2w(j)(m,u)∑m∈A2w(j)(m,u)),

with , . In the particular case of and , the qubit portrait (2.1) reads

 w(j)1/2(u)=(w(j)(j,u)∑j−1m=−jw(j)(m,u))=(w(j)(j,u)1−w(j)(j,u)). (14)

Qubit portraits of this kind are implicitly used in several reconstruction procedures considered in subsequent sections. The qubit-portrait method is introduced in [32] and successfully applied not only to spin systems [33] but also to the light states [34].

The inverse spin-portrait method is to construct a single probability distribution vector with the help of several spin- portraits. Indeed, one can stack spin- portraits into the following single probability vector of final dimension :

 1Ns⎛⎜ ⎜ ⎜ ⎜ ⎜⎝w(j)s(u1)w(j)s(u2)⋯w(j)s(uNs)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (15)

This idea is elaborated for the case in Sec. 5. Here, we restrict ourselves to the discussion of a particular case (qubit portrait). Only one component of two-vector (2.1) contains information on the system. The density operator is determined by real numbers without regard to the normalization condition. This fact shows that, if a single probability distribution (15) contains complete information on the system, then it should comprise at least different qubit portraits. Then the dimension of vector (15) is , but only components are relevant. In the following section, we review the reconstruction procedure that was suggested in [12] and that employed qubit portraits of the form (14) with , . The symmetric informationally complete POVM (positive operator-valued measure) to be outlined in Sec. 4 can also be treated as an implicit use of qubit portraits (14) with unitary matrices , , which satisfy additional requirements and for all .

3 Amiet–Weigert Reconstruction of the Density Matrix

In this section, we review the approach [12] to reconstruct the density matrix of a spin through the Stern–Gerlach measurements, where one measures the probabilities to obtain the maximum spin projection for appropriately chosen directions in space.

According to (1), for each fixed direction , , the probability to obtain the spin projection is given by the formula

 w(j)(j,nk)=⟨jj|^R†(nk)^ρ^R(nk)|jj⟩. (16)

We denote by W a vector comprising all these probabilities

 W=(w(j)(j,n1)w(j)(j,n2)…w(j)(j,n(2j+1)2))T. (17)

Moreover, the density matrix can be written in the form of a -component vector

 ρ=(ρ11ρ21…ρ2j+1,1ρ12ρ22…ρ2j+1,2…ρ1,2j+1ρ2,2j+1…ρ2j+1,2j+1)T,

which implies that the density-matrix columns are merely stacked up in a single column. Let be a vector constructed from the operator using the same rule. Then the probability is nothing else but the scalar product of two -component vectors

 w(j)(m,n)=(U(j)(m,n)⋅ρ)=Tr[U(j)(m,n)T ρ]. (19)

From this, it is not hard to see that the map reads

 W=∥M∥ρ, (20)

where is a matrix of the form

 ∥M∥=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝U(j)(j,n1)TU(j)(j,n2)T⋯U(j)(j,n(2j+1)2)T⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (21)

Whenever , there exists an inverse map and

 ρ=∥M−1∥W. (22)

In [12], it is shown that a particular choice of directions , ensures that the condition is fulfilled. Moreover, such a choice of directions simplifies significantly the inversion procedure because it relies on the Fourier transform. Namely, the following directions were proposed:

 nk≡nqr≡(sinθqcosφqr,sinθqsinφqr,cosθq),0≤q,r≤2j, (23)

with , if and

 φqr=2π2j+1(r+qΔ),0<Δ≤12j+1. (24)

Examples of such a choice of the directions for spins and are shown in Fig. 1. We see that the directions are divided into groups that form nested cones. Some modifications to free cones and spirals were presented in [15]. An arbitrary choice of the directions was discussed in [14].

4 Quantum States as Probability Distributions

To begin, we showed in Sec. 2 that any quantum state can be interpreted as a probability distribution function. Since tomogram (1) is a function depending on the unitary matrix (direction in the case of ), there is a redundancy of information contained in the tomogram. It is tempting to reduce such a redundancy and associate a quantum state with the single probability distribution. The proposal to associate quantum states with single probability vectors was made in [21, 20, 18, 19, 17, 16]. To avoid any redundancy, it was suggested to use the minimum informationally complete POVM (positive operator-valued measure). This suggestion is related to constructing the minimum tomographic set discussed in [36]. Using the language of spin states, any spin- state is associated with the following -component probability vector:

 p=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝tr(^ρ^E1)tr(^ρ^E2)⋯tr(^ρ^E(2j+1)2)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠, (25)

where , are the corresponding POVM effects. There are many ways to introduce the minimum informationally complete POVM effects, and a possible choice that is valid in any dimension was presented in [19]. A relatively new tendency is to use the symmetric informationally complete (SIC) POVMs. If this is the case, the effects are one-dimensional projectors satisfying the following condition:

 tr(^Ek^El)=1(2j+1)2(2j+2)ifk≠l. (26)

As stated in [20], the effects that meet the above requirements were found numerically in dimensions and analytically in dimensions .

It is worth noting that the Amiet–Weigert construction considered in the previous section can be treated as a single probability distribution. For this, one needs to normalize vector (17). In this case, the mappings (20) and (22) are slightly modified

 W′=∥M∥ρ∑(2j+1)2i,k=1∥M∥ikρk,ρ=∥M∥−1W′∑2j+1i=1∑(2j+1)2k=1∥M∥−1(i−1)(2j+1)+i,kW′k. (27)

5 Inverse Spin-Portrait Method

In this section, starting from spin tomograms, we associate each quantum state with the corresponding probability distribution vector, which has a clear physical meaning and can be measured experimentally.

The unitary spin tomogram is a function of the unitary matrix and depends on a discrete parameter . Fixing the unitary rotation , we obtain a -component probability vector , called the spin- portrait of the system (see Sec. 2.1). Given only one spin- portrait of the system, it is impossible, in general, to define without doubt a state of the system. Nevertheless, the state is determined if one has an adequate number of different spin- portraits. Then we can introduce a joint probability distribution function of two random variables and

 P(m,uk),m=−j,−j+1,…,j,k=1,2,…,Nu. (28)

The physical meaning of this joint probability distribution is that, if one randomly chooses a unitary rotation from the set and a spin projection within the interval , the value of gives the probability of the detector’s click. Function (28) can also be written in the form of the following -component probability distribution vector:

 P=(P(j,u1)…P(−j,u1)P(j,u2)…P(−j,u2)…P(j,uNu)…P(−j,uNu))T

with the normalization condition of the form

 j∑m=−jNu∑k=1P(m,uk)=1. (30)

Since the tomogram is nothing else but the probability to obtain the spin projection if the rotation is fixed, the relation between the spin tomogram and the joint probability distribution function reads

 w(j)(m,uk)=P(m,uk)∑jm=−jP(m,uk), (31)

where the denominator has the sense of the probability to choose the unitary rotation . If the probabilities are known a priori, then the vector is easily expressed via spin- portraits (11), namely,

 P=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝p1w(j)j(u1)p2w(j)j(u2)⋯pNuw(j)j(uNu)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (32)

If the unitary rotations , are equiprobable, then

 Peq=1Nu⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝w(j)j(u1)w(j)j(u2)⋯w(j)j(uNu)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (33)

It is worth mentioning that, even if a priori the probabilities are not known, formula (31) provides a direct way of mapping onto the vector .

Let us now consider an open problem of the minimum number of spin portraits. In other words, is the number of unitary rotations that is needed to identify any quantum state with a single probability vector of the form (32) and to minimize the redundancy of information contained in this vector. Subsequently, two main cases are presented, namely, the use of rotations and rotations with . These particular problems can be of interest for experimentalists because rotations can be relatively easily realized in some modifications of the Stern–Gerlach experiment, while rotations may require more difficult apparatus. On the other hand, it will be shown that, to extract information on the system, one can use a smaller number of rotations than in the case of matrices.

5.1 Su(2) Rotations

Like the Amiet–Weigert scanning procedure (20), the map of the density operator onto the probability vector (32) can be written as follows:

 P=∥Q∥ρ, (34)

where is an rectangular matrix of the form

 ∥Q∥=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝p1U(j)(j,n1)Tp1U(j)(j−1,n1)T⋯p1U(j)(−j,n1)T⋮pNuU(j)(j,nNu)TpNuU(j)(j−1,nNu)T⋯pNuU(j)(−j,nNu)T⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (35)

The map (34) is invertible iff . Since the rank of a matrix is equal to the number of linearly independent rows, the number can be defined as the minimum natural number such that the set , , contains linearly independent vectors. According to [29], each vector can be resolved to the sum of orthogonal vectors , ; namely,

 U(j)(m,n)=2j∑L=0f(j)L(m)S(j)L(n), (36)

where . The vector corresponds to the operator acting on the Hilbert space of spin- states (see Sec. 2). The operator is shown to be the same polynomial of degree , with the argument being replaced: . Suppose , then there exists only one linear independent vector of the form which corresponds to the identity operator . If , no more than three linear independent vectors , , which correspond to the operators , , and , respectively, can exist. Note that the three vectors , are independent iff vectors are not coplanar, i.e., their triple product . Taking into account the normalization conditions and , in the case , we obtain five linear independent vectors . Using the matrix representation of the operator , we see that it is composed of independent -diagonal operators ( for a diagonal one, for a super-diagonal one, for a sub-diagonal one, and so on for all ). Increasing by unity, two more diagonals are filled. We draw the conclusion that for a fixed the maximum number of linearly independent vectors is equal to . Since vectors and with different and are orthogonal, the total number of linear independent rows equals . On the other hand, it must be equal to . From this, it is readily seen that .

The directions , cannot be chosen arbitrarily because of the condition . As was shown above, the directions are divided into sets of one, three, five, and so on directions. Without loss of generality, it can be assumed that these sets are , , , , , respectively. If this is the case, the requirement is equivalent to the condition

 Δ1Δ2⋅…⋅Δ2j≠0, (37)

where , are expressed through and associated Legendre polynomials as follows:

 Δq=det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝P(0)q(cosθ1)⋯P(q)q(cosθ1)cosqφ1P(q)q(cosθ1)sinqφ1P(0)q(cosθ2)⋯P(q)q(cosθ2)cosqφ2P(q)q(cosθ2)sinqφ2P(0)q(cosθ3)⋯P(q)q(cosθ3)cosqφ3P(q)q(cosθ3)sinqφ3⋯⋯⋯⋯P(0)q(cosθ2q)⋯P(q)q(cosθ2q)cosqφ2qP(q)q(cosθ2q)sinqφ2qP(0)q(cosθ2q+1)⋯P(q)q(cosθ2q+1)cosqφ2q+1P(q)q(cosθ2q+1)sinqφ2q+1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (38)

In the particular case of , we have

 Δ1=det⎛⎜ ⎜ ⎜⎝P(0)1(cosθ1)P(1)1(cosθ1)cosφ1P(1)1(cosθ1)sinφ1P(0)1(cosθ2)P(1)1(cosθ2)cosφ2P(1)1(cosθ2)sinφ2P(0)1(cosθ3)P(1)1(cosθ3)cosφ3P(1)1(cosθ3)sinφ3⎞⎟ ⎟ ⎟⎠=(n1⋅[n2×n3]). (39)

It is worth mentioning that there exists an optimum choice of the directions