Extended Cahill-Glauber formalism for finite-dimensional spaces. II.
Applications in quantum tomography and quantum teleportation
Marcelo A. Marchiolli*
Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970, São Carlos, SP, Brazil
Maurizio Ruzzi† and Diógenes Galetti‡
Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, São Paulo, SP, Brazil
共Received 13 April 2005; published 5 October 2005兲
By means of a mod共N兲-invariant operator basis, s-parametrized phase-space functions associated with bounded operators in a finite-dimensional Hilbert space are introduced in the context of the extended Cahill-Glauber formalism, and their properties are discussed in details. The discrete Cahill-Glauber-Sudarshan, Wigner, and Husimi functions emerge from this formalism as specific cases ofs-parametrized phase-space functions where, in particular, a hierarchical process among them is promptly established. In addition, a phase-space description of quantum tomography and quantum teleportation is presented and new results are obtained.
DOI:10.1103/PhysRevA.72.042308 PACS number共s兲: 03.67.Lx, 03.67.Hk, 03.65.Wj
I. INTRODUCTION
The first proposal of a unified formalism for quasiprob-ability distribution functions in continuous phase space has its origin in the seminal works produced by Cahill and Glauber 关1兴. Since then a huge number of papers have ap-peared in the literature covering a wide range of practical applications in different physical systems modeled by means of infinite-dimensional Hilbert spaces关2,3兴. In particular, the phase-space description of some important effects in quan-tum mechanics, such as interference, entanglement, and de-coherence, has opened up astounding possibilities for the comprehension of intriguing aspects of the microscopic world 关4兴. However, if physical systems with a finite-dimensional space of states are considered, then the qua-siprobability distribution functions are described by a set of discrete variables defined over a finite lattice关5–15兴. In this sense, Opatrnýet al.关9兴were the first researchers to propose a unified approach to the problem of discrete quasiprobabil-ity distribution functions in the literature. Basically, they used a discrete displacement-operator expansion to introduce
s-parametrized phase-space functions associated with opera-tors defined over a finite-dimensional Hilbert space. Further-more, the authors showed that the discrete Glauber-Sudarshan, Wigner, and Husimi functions are particular cases of s-parametrized phase-space functions and depend on the arbitrary reference state whose characteristic function cannot have zero values. It is worth mentioning that the dependence on the right choice of reference state and the associated prob-lems with the mod共N兲 invariance of the discrete displace-ment operators represent two important restrictions inherent to their approach which deserve to be carefully investigated. Nowadays, beyond these fundamental features, discrete qua-siprobability distribution functions in finite-dimensional
phase spaces have potential applications for quantum-state tomography 关16,17兴, quantum teleportation关18–21兴, phase-space representation of quantum computers关22兴, open quan-tum systems 关23兴, quantum information theory 关24兴, and quantum computation关25兴.
The main aim of this paper is to present a consistent for-malism for the quasiprobability distribution functions de-fined over a discrete N2-dimensional phase space, which is based upon the mathematical fundamentals developed in 关26兴. First, we review important topics and introduce new properties concerning the mod共N兲-invariant operator basis, which leads us not only to define a parametrized phase-space function in terms of the discrete s-ordered characteristic function, but also to discuss some characteristics inherent to the extended Cahill-Glauber formalism for finite-dimensional spaces. The restriction on the right choice of reference state is overcome in this approach through the vacuum state established by Galetti and de Toledo Piza关8兴, whose analytical properties were extensively explored in 关10兴. Consequently, the discrete Glauber-Sudarshan, Wigner, and Husimi functions are well defined in the present context and represent specific cases of s-parametrized phase-space functions describing density operators associated with physi-cal systems whose space of states is finite. In addition, we also establish a hierarchical order among them through a smoothing process characterized by a discrete phase-space function that closely resembles the role of a Gaussian func-tion in the continuous phase space. In this point, it is worth emphasizing that our ab initio construction inherently em-bodies the discrete analogs of the desired properties of the Cahill-Glauber approach. Next, we apply such a discrete ex-tension to the context of quantum information processing, quantum tomography, and quantum teleportation in order to obtain a phase-space description of some topics related to unitary depolarizers, discrete Radon transforms, and general-ized Bell states. In particular, we attain results within which some of them deserve to be mentioned:共i兲we show that the symmetrized Schwinger operator basis introduced in关6兴can be considered a unitary depolarizer;共ii兲we establish a link *Electronic address: marceloគmarch@bol.com.br
†Electronic address: mruzzi@ift.unesp.br ‡
between measurable quantities and s-ordered characteristic functions by means of discrete Radon transforms, which can be used to construct any quasiprobability distribution func-tions defined over aN2-dimensional phase space; and finally, 共iii兲we present a quantum teleportation protocol that leads us to reach a generalized phase-space description of the physi-cal process discussed by Bennettet al.关18兴.
This paper is organized as follows. In Sec. II we present some basic properties inherent to our discrete mapping ker-nel which allow us to define a parametrized phase-space function in terms of a discretes-ordered characteristic func-tion. Following, in Sec. III we show that the extended Cahill-Glauber formalism not only introduces mathematical tools for the analysis of finite quantum systems, but also can be applied in the context of quantum information processing, quantum tomography, and quantum teleportation. Moreover, we also employ a slightly modified version of the scattering circuit to measure any discrete Wigner function in the phase-space representation. Finally, Sec. IV contains our summary and conclusions.
II. MAPPING KERNEL
There is a huge variety of probability distribution func-tions defined in continuous quantum phase spaces whose range of practical applications in physics covers different areas and scenarios 关2,3兴. For example, the well-known Cahill-Glauber formalism 关1兴 provides a general mapping technique of bounded operators which permits, in particular, to define a generalized probability distribution function
F共s兲共q,p兲= Tr关T共s兲共q,p兲兴associated with an arbitrary
physi-cal system described by the density operator. In this
ap-proach, the mapping kernel共hereafterប= 1兲
T共s兲共q,p兲=
冕
dq⬘
dp⬘
2 exp关i共q
⬘
p−p⬘
q兲兴D共s兲共q
⬘
,p⬘
兲 共1兲is defined as a Fourier transform of the parametrized operator
D共s兲共q
⬘
,p⬘
兲= exp关共s/4兲共q⬘
2+p⬘
2兲兴D共q⬘
,p⬘
兲, 共2兲 whereD共q⬘
,p⬘
兲= exp关i共p⬘
Q−q⬘
P兲兴is the usual displacement operator written in terms of the coordinate and momentum operators satisfying the Weyl-Heisenberg commutation rela-tion 关Q,P兴= i1 and s is a complex parameter. Thus, fors= −1, 0, and +1 the generalized probability distribution function leads to the so-called Husimi, Wigner, and Glauber-Sudarshan functions, respectively. Besides, these functions present specific properties and correspond to different or-dered power-series expansions in the annihilation and cre-ation operators of the density operator: the Husimi function H共q,p兲 is infinitely differentiable and it is associated with the normally ordered form; the Wigner functionW共q,p兲is a continuous and uniformly bounded function; it can take negative values and corresponds to the symmetrically or-dered form; and finally, the Glauber-Sudarshan function P共q,p兲is highly singular; it does not exist as a regular func-tion for pure states and it corresponds to the antinormally ordered form. After this condensed review of the Cahill-Glauber formalism for the quasiprobability distribution
func-tions, we will establish the discrete representatives of these functions in anN2-dimensional phase space.
A. mod„N…-invariant operator basis
Let us introduce the symmetrized version of the unitary operator basis proposed by Schwinger关27兴as
S共,兲=
冑
1Nexp
冉
i
N
冊
UV, 共3兲
where the labels and are associated with the dual coor-dinate and momentum variables of a discreteN2-dimensional phase space. Consequently, these labels assume integer val-ues in the symmetrical interval关−ᐉ,ᐉ兴, withᐉ=共N− 1兲/ 2. A comprehensive and useful compilation of results and proper-ties of the unitary operators Uand Vcan be found in Ref. 关10兴, since the initial focus of our attention is the essential features exhibited by Eq.共3兲. Note that the set ofN2 opera-tors 兵S共,兲其,=−ᐉ,. . .,ᐉ constitutes a complete orthonormal
operator basis which allows us, in principle, to construct all possible dynamical quantities belonging to the system关27兴. Thus, the decomposition of any linear operator O in this basis is written as
O=
兺
,=−ᐉᐉ
O共,兲S共,兲, 共4兲
with the coefficientsO共,兲given by Tr关S†共,兲O兴. It must be stressed that this decomposition is unique since the rela-tions S†共,兲=S共−, −兲 and Tr关S†共,兲S共
⬘
,⬘
兲兴 =␦⬘,
关N兴
␦
⬘,
关N兴
are promptly verified. The superscript关N兴on the Kronecker delta denotes that this function is different from zero when its labels are mod共N兲 congruent.
The mod共N兲-invariant operator basis recently proposed in 关26兴,
T共s兲共,兲=
冑
1N,
兺
=−ᐉ ᐉexp
冋
i⌽共,;N兲−2iN 共+兲
册
⫻S共s兲共,兲, 共5兲
is defined by means of a discrete Fourier transform of the extended mapping kernel
S共s兲共,兲=关K共,兲兴−sS共,兲,
where the extra termK共,兲 can be expressed as a sum of products of Jacobi theta functions evaluated at integer argu-ments关28兴,
K共,兲=兵2关3共0兩ia兲3共0兩4ia兲 +4共0兩ia兲2共0兩4ia兲兴其−1兵
3共a兩ia兲3共a兩ia兲 +3共a兩ia兲4共a兩ia兲exp共i兲
+4共a兩ia兲3共a兩ia兲exp共i兲
bell-shaped function in the discrete variables共,兲and equals 1 for == 0; in addition, the complex parameter s obeys 兩s兩艋1. The phase⌽共,;N兲=NINI
N−I
N−I
Nis responsible for the mod共N兲 invariance of the operator basis 共5兲, IN=关/N兴being the integral part ofwith respect toN. This definition stands for the discrete version of the continuous mapping kernel 共1兲 and represents the cornerstone of the present approach.
By analogy with decomposition共4兲, the expansion
O= 1
N,
兺
=−ᐉ ᐉO共−s兲共,兲T共s兲共,兲 共7兲
can also be verified for any linear operator. Here, the coeffi-cientsO共−s兲共,兲= Tr关T共−s兲共,兲O兴 correspond to a one-to-one mapping between operators and functions belonging to anN2-dimensional phase space characterized by the discrete labels and . In particular, if one considers s= −1 and
O= in Eq.共7兲, we obtain the diagonal representation
=
1
N,
兺
=−ᐉ ᐉP共,兲兩,典具,兩, 共8兲
whereP共,兲= Tr关T共1兲共,兲
兴is the discrete version of the
Glauber-Sudarshan function for finite Hilbert spaces and
T共−1兲共,兲 is the projector of discrete coherent states 关10兴. Fors= 0, we verify that
=
1
N,
兺
=−ᐉ ᐉW共,兲G共,兲 共9兲
recovers the well-established results in 关8兴, W共,兲 = Tr关G†共,兲
兴 being the discrete Wigner function and
G共,兲 the mod共N兲-invariant operator basis whose math-ematical properties were studied in 关10兴. Furthermore, we note that the Husimi function in the discrete coherent-state representation, H共,兲= Tr关T共−1兲共,兲兴, can be promptly
obtained from Eq.共8兲or 共9兲 by means of a trace operation. Next, we will discuss some properties inherent to the set of
N2 operators 兵T共s兲共,兲其,=−ᐉ,. . .,ᐉ with emphasis on
estab-lishing a hierarchical process among the quasiprobability dis-tribution functions in finite-dimensional spaces.
B. Basic properties
The discrete mapping kernelT共s兲共,兲presents some in-herent mathematical features that lead us to derive a set of properties which characterize its algebraic structure. For in-stance, it is straightforward to show that the equalities
共i兲 1
N,
兺
=−ᐉ ᐉT共s兲共,兲= 1,
共ii兲Tr关T共s兲共,兲兴= 1,
共iii兲Tr关T共−s兲共,兲T共s兲共
⬘
,⬘
兲兴=N␦⬘,
关N兴 ␦
⬘,
关N兴
are promptly verified where, in particular, the third property has been reached with the help of the auxiliary relation
Tr关T共t兲共,兲T共s兲共
⬘
,⬘
兲兴= 1
N,
兺
=−ᐉ ᐉexp
再
2iN 关共
⬘
−兲+共⬘
−兲兴冎
⫻关K共,兲兴−共t+s兲.
Note that for s= −1, the first property coincides with the completeness relation of the discrete coherent states 共the proof of this relation was given in关10兴兲; the second property simply states thatT共s兲共,兲has a unit trace. Finally, the third property is the counterpart to the orthogonality rule estab-lished for the operatorsS共,兲. Furthermore, we also verify the condition T共s*兲共,兲=关T共s兲共,兲兴†, which implies that for real values of the parameters, the discrete mapping ker-nel is Hermitian; consequently, the mappings of Hermitian operators in theN2-dimensional phase space lead us to obtain real functions. Now, let us establish a hierarchical process among the discrete Glauber-Sudarshan, Wigner, and Husimi functions.
The connection between the discrete Glauber-Sudarshan and Wigner functions is reached with the help of Eq. 共8兲 through a smoothing process ofP共,兲—i.e.,
W共,兲= 1
N
兺
⬘,⬘=−ᐉ ᐉ
E共
⬘
−,⬘
−兲P共⬘
,⬘
兲, 共10兲where E共
⬘
−,⬘
−兲 ⬅Tr关T共0兲共,兲T共−1兲共⬘
,⬘
兲兴 is ex-pressed by means of a discrete Fourier transform of the func-tionK共,兲—note that E共⬘
−,⬘
−兲can be interpreted as a Wigner function evaluated for the discrete coherent states labeled by ⬘
and ⬘
. Similarly, the link between discrete Wigner and Husimi functions can also be established through Eq.共9兲as follows:H共,兲= 1
N
兺
⬘,⬘=−ᐉ ᐉ
E共
⬘
−,⬘
−兲W共⬘
,⬘
兲. 共11兲Therefore, Eqs.共10兲and共11兲exhibit a sequential smoothing which characterizes a hierarchical process among the qua-siprobability distribution functions in finite-dimensional spaces,P共,兲→ W共,兲→ H共,兲. It is worth mentioning that
H共,兲= 1
N
兺
⬘,⬘=−ᐉ ᐉ
兩具,兩
⬘
,⬘
典兩2P共⬘
,⬘
兲 共12兲have zero values. Here, we have established a suitable math-ematical procedure that allows us to overcome some intrinsic problems encountered in关9兴, the vacuum state being defined in关8,10兴as our reference state.
Next, we present two important properties associated with the trace of the product of two bounded operators and the matrix elements 具m兩T共s兲共,兲兩n典 in the finite number basis 兵兩n典其n=0,. . .,N−1. The first one corresponds to the overlap
共iv兲Tr共AB兲= 1
N,
兺
=−ᐉ ᐉA共−s兲共,兲B共s兲共,兲,
where, in particular, fors= 0, the trace of the product of two density operators coincides with the overlap of the discrete Wigner functions of each density operator,
Tr共12兲= 1
N,
兺
=−ᐉ ᐉW1共,兲W2共,兲.
In addition, the mean value of any bounded operator can also be obtained from this property,
具O典 ⬅Tr共O兲=
1
N,
兺
=−ᐉ ᐉO共−s兲共,兲F共s兲共,兲, 共13兲
the parametrized functionF共s兲共,兲being defined as the ex-pectation value of the discrete mapping kernel共5兲—i.e.,
F共s兲共,兲 ⬅Tr关T共s兲共,兲兴
=
冑
1N,
兺
=−ᐉ ᐉexp
冋
i⌽共,;N兲−2iN 共+兲
册
⫻⌶共s兲共,兲, 共14兲
while ⌶共s兲共,兲 ⬅Tr关S共s兲共,兲兴 represents the discrete s-ordered characteristic function 关1兴. Note that ⌽共,;N兲 can be discarded in Eq.共14兲since the discrete labelsand
are confined into the closed interval 关−ᐉ,ᐉ兴. In fact, this phase will be important only in the mapping of the product of M quantum operators关10兴. Besides, fors= −1 , 0 , + 1 the parametrized function is directly related to the discrete Hu-simi, Wigner, and Glauber-Sudarshan functions, respectively. Hence, the characteristic function can now be promptly cal-culated for each situation through the inverse discrete Fou-rier transform of the generalized probability distribution functionF共s兲共,兲.
The second one refers to the nondiagonal matrix elements in the finite number basis
共v兲 具m兩T共s兲共,兲兩n典= 1
N,
兺
=−ᐉ ᐉexp
冋
i⌽共,;N兲−2i
N 共+兲
册
关K共,兲兴−s⌫ mn共,兲,
with
⌫mn共,兲= exp
冉
− iN
冊
兺
=−ᐉ ᐉexp
冉
2iN
冊
F,nF−,m*
共15兲 written in terms of the coefficients关8兴
F,n= Nn 共− i兲n
冑
N 兺
=−⬁ ⬁exp
冉
−N
2+2i
N 
冊
Hn冉
冑
2
N 
冊
,where Nn is the normalization constant and Hn共z兲 is a Her-mite polynomial. It is easy to show that ⌫mn共,兲 satisfies the relations ⌫mn共0 , 0兲=␦m关N兴,n and ⌫00共,兲=K共,兲, which are associated with the orthogonality rule for the finite num-ber states and the diagonal matrix element 具0兩T共s兲共,兲兩0典 for the vacuum state. Moreover, adopting the mathematical procedure established in 关29兴 for the continuum limit, we obtain
⌫mn共q
⬘
,p⬘
兲=冑
m!n!
冉
q
⬘
+ ip⬘
冑
2冊
n−mLm共n−m兲
冉
兩q⬘
+ ip⬘
兩 22
冊
⫻exp
冋
−1 4共q⬘
2+p
⬘
2兲册
共n艌m兲,with Ln共m兲共z兲being the associated Laguerre polynomial. Con-sequently, the nondiagonal matrix elements for 兩s兩⬍1 take the analytical form
具m兩T共s兲共q,p兲兩n典= 2 1 −s
冑
m!
n!
冉
− 1 +s1 −s
冊
m
⫻
冋
冑2
共q− ip兲1 −s
册
n−m
Lm共n−m兲
冉
2共q 2+p2兲 1 −s2冊
⫻exp
冉
− q 2+p2 1 −s冊
.This result coincides exactly with that obtained by Cahill and Glauber关1兴for the mapping kernel共1兲, sinceT共s兲共,兲goes toT共s兲共q,p兲 in the limitN→⬁. Following, we will discuss some applications for the generalized probability distribution functionF共s兲共,兲with emphasis on the discrete phase-space representation of quantum tomography and quantum telepor-tation.
III. APPLICATIONS
other formalisms for finite-dimensional Hilbert spaces with convenient inherent mathematical properties which can also be applied in the description of similar quantum systems 关6–15,26兴. In this section, we will show that the present for-malism not only introduces mathematical tools for the analy-sis of finite quantum systems but also can be applied, for example, to the context of quantum information processing, quantum tomography, and quantum teleportation.
A. Quantum information processing
Within the most important quantum operations in quan-tum information processing, unitary operations have a prominent position 关30兴. Besides, in the scope of quantum information theory, the unitary depolarizers play an impor-tant role in quantum teleportation and quantum dense coding 关21,31兴. With respect to N-dimensional Hilbert spaces, uni-tary depolarizers are defined on a domain⍀as elements of the set
D共N兲=兵X⑀兩X⑀X⑀†=X⑀†X⑀=1,⑀苸⍀其, 共16兲 which satisfy the relation
1
N⑀
兺
苸⍀X⑀OX⑀†
= Tr共O兲1 共17兲
for any linear operatorOacting on finite-dimensional vector spaces, where 1 is an identity operator. Recently, Ban 关32兴 has shown that the Pegg-Barnett phase operator formalism is useful for quantum information processing as well as in in-vestigating quantum optical systems. In this sense, it is worth mentioning that the symmetrized version of the Schwinger operator basisS共,兲can also be considered a unitary depo-larizer, since the elements of the set
D共N兲=兵
冑
NS共,兲兩关冑
NS共,兲兴关冑
NS共,兲兴† =关冑
NS共,兲兴†关冑
NS共,兲兴=1,−ᐉ艋,艋ᐉ其共18兲
obey the property
1
N,
兺
=−ᐉ ᐉ关
冑
NS共,兲兴O关冑
NS共,兲兴†= Tr共O兲1. 共19兲This result shows that the average over all possible discrete dual coordinate and momentum shifts on theN2-dimensional phase space completely randomizes any quantum state defined on the finite-dimensional vector space. Furthermore, for s= i and 苸R, the elements of the set 兵冑NS共i兲共,兲其
,=−ᐉ,. . .,ᐉgeneralize Eq.共18兲,S共i兲共,兲being
the parametrized Schwinger operator basis. Unfortunately, the implementation of such unitary operations in a realistic quantum-computer technology encounters an almost unsur-mountable obstacle: the degrading and ubiquitous decoher-ence due to the unavoidable coupling with the environment 关33兴. However, recent progress关34兴has developed the idea of protecting or even creating a decoherence-free subspace for processing quantum information.
B. Marginal distributions, Radon transforms, and discrete phase-space tomography
The marginal distributions associated with the generalized probability distribution function F共s兲共,兲 are obtained through the usual mathematical procedure
Q共s兲共兲 ⬅
冑
1N
兺
=−ᐉ ᐉF共s兲共,兲=
兺
=−ᐉᐉ
exp
冉
−2iN
冊
⌶共s兲共,0兲,
共20兲
R共s兲共兲 ⬅
冑
1N
兺
=−ᐉ ᐉF共s兲共,兲=
兺
=−ᐉ ᐉ
exp
冉
−2iN
冊
⌶共s兲共0,兲.
共21兲 Note that the second equality in both definitions has been attained with the help of Eq. 共14兲. Consequently, the mar-ginal distributions are obtained by means of discrete Fourier transforms of thes-ordered characteristic function calculated in specific slices of the dual plane 共,兲. Now, if one con-siders the hierarchical process established by Eqs. 共10兲and 共11兲, alternative expressions for the marginal distributions associated with the Wigner and Husimi functions can also be derived,
Q共0兲共兲=
兺
⬘=−ᐉ ᐉ
E共
⬘
−兲Q共1兲共⬘
兲,R共0兲共兲=
兺
⬘=−ᐉ ᐉ
E共
⬘
−兲R共1兲共⬘
兲,Q共−1兲共兲=
兺
⬘=−ᐉ ᐉ
E共
⬘
−兲Q共0兲共⬘
兲,R共−1兲共兲=
兺
⬘=−ᐉ ᐉ
E共
⬘
−兲R共0兲共⬘
兲,where the smoothing functionE共兲is given by
E共兲= 1
冑2
N3共0兩ia兲3共2a兩ia兲+4共0兩ia兲4共2a兩ia兲
3共0兩ia兲3共0兩4ia兲+4共0兩ia兲2共0兩4ia兲 .
Thus, a sequential smoothing process is immediately established among the discrete marginal distributions: Q共1兲共兲→ Q共0兲共兲→ Q共−1兲共兲 and R共1兲共兲→ R共0兲共兲 → R共−1兲共兲. The importance of the quantum-mechanical mar-ginal distributions fors= 0 in the context of quantum tomog-raphy in discrete phase space has been stressed by Leonhardt 关16兴, where measurements on subensembles of a given quan-tum state are necessary in the reconstruction process.
con-sist of Bogoliubov-type unitary transformations generated by
J共⍀1,⍀2,⍀3兲=M共⍀3兲N共⍀2兲C共⍀1兲, where
C共⍀1兲= 1
冑
N,兺
=−ᐉᐉ
exp
冋
−iN共1 +⍀1兲
册
S„,共1 −⍀1兲…,N共⍀2兲= 1
冑
N,兺
=−ᐉᐉ
exp
冋
iN共⍀2− 2兲
册
S共,0兲,M共⍀3兲= 1
冑
N,兺
=−ᐉᐉ
exp
冋
−iN共⍀3+ 2兲
册
S共0,兲are unitary operators written in terms of the symmetrized Schwinger basis S共,兲, with ⍀1=4共1 +23兲−1, ⍀2=24−1共1 +
23兲, and ⍀3=34共1 +23兲−1. Here, the dis-crete elements of the set兵i其i=1,. . .,4assume integer values in the closed interval 关−ᐉ,ᐉ兴 and satisfy the relation
14−23= 1 mod共N兲. It is worth mentioning that this con-straint implies the existence of the inverse elements since
1=4−1共1 +23兲. Now, let us initially apply the unitary trans-formationJ共⍀1,⍀2,⍀3兲 on the parametrized Schwinger ba-sisS共s兲共,兲. Thus, after some algebra we obtain
J共⍀1,⍀2,⍀3兲S共s兲共,兲J†共⍀1,⍀2,⍀3兲
=
冋
K共1+2,3+4兲 K共,兲册
s
S共s兲共1+2,3+4兲.
共22兲
Using this auxiliary result in the calculation of
J共⍀1,⍀2,⍀3兲T共s兲共,兲J†共⍀1,⍀2,⍀3兲, we promptly obtain the intermediate expression
1
冑
N,兺
=−ᐉᐉ
exp
冋
i⌽共,;N兲−2iN 共+兲
册
⫻
冋
K共1+2,3+4兲K共,兲
册
sS共s兲共1+2,3+4兲.
The next step consists in replacing the dummy discrete vari-ables and by 4
⬘
−2⬘
and 1⬘
−3⬘
in the double sum, respectively, with the aim of establishing the compact expressionT共s兲共
⬘
,⬘
兲=冑
1N⬘,
兺
⬘=−ᐉ ᐉexp
冋
i⌽共⬘
,⬘
;N兲−2iN 共
⬘
⬘
+
⬘
⬘
兲册
⫻冋
K共⬘
,⬘
兲K共4
⬘
−2⬘
,1⬘
−3⬘
兲册
s⫻S共s兲共
⬘
,⬘
兲, 共23兲
⬘
=4−3 and⬘
=1−2being the new discrete vari-ables written as a linear combination of the old ones. In particular, this transformed mod共N兲-invariant operator basis can be used to derive the marginal distributions through the standard mathematical procedureQ共s兲共;1,3兲= 1
冑
N兺
⬘,⬘=−ᐉ ᐉ
F共s兲共
⬘
,⬘
兲␦,1⬘+3⬘
关N兴
, 共24兲
R共s兲共;2,4兲=
冑
1N
兺
⬘,⬘=−ᐉ ᐉ
F共s兲共
⬘
,⬘
兲␦,2⬘+4⬘
关N兴
. 共25兲
These results characterize the Radon transform in the present context and say that the sum of the parametrized function
F共s兲共
⬘
,⬘
兲 on specific lines in the N2-dimensional phase space represented by the discrete variables ⬘
and ⬘
are equal to the marginal distributions for any value of the pa-rameter s 共when s= 0, the marginal distributions coincide with probabilities兲. In terms of the discretes-ordered charac-teristic function, Eqs.共24兲and共25兲can be written asQ共s兲共;1,3兲=
兺
=−ᐉ ᐉ
exp
冉
−2iN
冊
冋
K共1,3兲 K共,0兲
册
s
⫻⌶共s兲共1,3兲,
R共s兲共;2,4兲=
兺
=−ᐉ ᐉ
exp
冉
− 2iN
冊
冋
K共2,4兲 K共0,兲
册
s
⫻⌶共s兲共2,4兲, whose inverse expressions are given by
⌶共s兲共1,3兲= 1
N
冋
K共,0兲 K共1,3兲册
s
兺
=−ᐉ ᐉ
exp
冉
2iN
冊
⫻Q共s兲共;1,3兲, 共26兲
⌶共s兲共2,4兲= 1
N
冋
K共0,兲 K共2,4兲
册
s
兺
=−ᐉ ᐉ
exp
冉
2iN
冊
⫻R共s兲共;2,4兲. 共27兲 Note that Eqs. 共26兲 and共27兲 establish a link between mea-surable quantities 关right-hand side 共RHS兲兴 and discrete
s-ordered characteristic functions共LHS兲; moreover, they can be used to construct, for instance, the quasiprobability distri-bution functions in finite-dimensional spaces. In summary, we have established a set of important theoretical results which constitute a discrete version of that obtained by Vogel and Risken关36兴for the continuous case.
From the theoretical point of view, the ideas of quantum computation can nowadays be used for illuminating some fundamental processes in quantum mechanics 关37兴. In this sense, Paz and co-workers关17兴have shown that tomography and spectroscopy are dual forms of the same quantum com-putation 共represented by a “scattering” circuit兲, since the state of a quantum system can be modeled on a quantum computer. Furthermore, using different versions of program-mable gate arrays, the authors have been capable not only of evaluating the expectation value of any operator acting on an
a slightly modified version of the scattering circuit to mea-sure the discrete Wigner function W共,兲. Basically, we modify this circuit by inserting a controlled-Uoperation be-tween the Hadamard gates, with U=
冑
NS共,兲 acting on a quantum system described by some unknown density opera-tor, and also a controlled Fourier transform共FT兲after thesecond Hadamard gate. This is illustrated in Fig. 1, where a set of measurements on the polarizations along thez and y
axes of the ancillary qubit兩0典 yields the expectation values 具z典=
冑
NRe关W共,兲兴 and 具y典=冑
NIm关W共,兲兴,respec-tively. In the absence of the controlled-FT operation, these measurements lead us to obtain the characteristic function
⌶共s兲共,兲 for s= 0—namely, 具
z典=
冑
NRe关⌶共0兲共,兲兴 and 具y典=冑
NIm关⌶共0兲共,兲兴. However, to construct the discreteHusimi function, some modifications must be included in the primary circuit 共see Ref.关17兴 for more details兲 or the link established by Eq. 共11兲 between the Wigner and Husimi functions should be employed. Both situations deserve a de-tailed theoretical investigation since their operational costs can be prohibitive from the experimental point of view. Next, we will present a phase-space description of the process in-herent to quantum teleportation for a system with an
N-dimensional space of states.
C. Discrete phase-space representation of quantum teleportation
In the last years, great advance has been reached in the quantum teleportation arena. In particular, we observe that共i兲 different theoretical schemes for teleportation of quantum states involving continuous and discrete variables have been proposed and investigated in the literature 关18–21,38兴 and 共ii兲 its experimental feasibility has been demonstrated in simple systems through pairs of entangled photons produced by the process of parametric down-conversion关39兴. More-over, the essential resource in both theoretical and experi-mental approaches is directly associated with the concept of entanglement, which naturally appears in quantum mechan-ics when the superposition principle is applied to composite systems. An immediate consequence of this important effect
has its origin in the theory of quantum measurement 关40兴, since the entangled state of the multipartite system can reveal information about its constituent parts.
Recently, the quasiprobability distribution functions have represented important tools in the phase-space description of the quantum teleportation process for a system with an
N-dimensional space of states. For instance, Koniorczyk et al. 关19兴 have presented a unified approach to quantum tele-portation in arbitrary dimensions based on the Wigner-function formalism, where the finite- and infinite-dimensional cases can be treated in a conceptually uniform way. Paz关20兴has extended the results obtained by Koniorc-zyket al.to the case where the space of states has arbitrary dimensionality. To this end, the author has used a different definition for the discrete Wigner function which permits us to analyze situations where entanglement among subsystems of arbitrary dimensionality is an important issue. Here, we use the mod共N兲-invariant operator basisT共s兲共,兲in order to obtain a discrete phase-space representation of quantum tele-portation which permits us to extend the results reached by Paz in the discrete Wigner-function context for any discrete quasiprobability distribution functions.
1. Generalized Bell states
The generalized Bell states were first introduced by Ben-nett et al. 关18兴 in the study of quantum teleportation for systems withN⬎2 orthogonal states. Basically, these states can be defined as兩⌿
1,2典=V1
1丢U
2 −2兩⌿
0,0典, where
兩⌿0,0典= 1
冑
N⑀兺
=−ᐉᐉ
兩v⑀典1丢兩v⑀典2
represents the pure state maximally entangled for a bipartite system关in this case, the reduced density matrix of each con-stituent part is equal to 共1 /N兲1i for i= 1 , 2兴, 兵兩v␣典i其␣=−ᐉ,. . .,ᐉ and 兵兩u典j其=−ᐉ,. . .,ᐉ being the eigenstates of the Schwinger
unitary operators Vi and Uj, respectively. Furthermore, the generalized Bell states satisfy the following properties:
共i兲 具⌿
1,2兩⌿1⬘,2⬘典=␦1⬘,1
关N兴
␦
2
⬘,2
关N兴
共orthogonality relation兲,
共ii兲
兺
1,2=−ᐉ
ᐉ
兩⌿1,2典具⌿1,2兩=11丢12 共identity relation兲, 共iii兲U+兩⌿1,2典=U1丢U2兩⌿1,2典
= exp关−共2i/N兲1兴兩⌿
1,2典,
V−兩⌿
1,2典=V1丢V2
−1兩⌿
1,2典= exp关共2i/N兲2兴兩⌿1,2典.
Note that U+ displaces both systems in coordinates by the same amount1, whileV−displaces them in momentum by the quantity 2 in the opposite direction. In addition, as 兵兩⌿1,2典其1,2=−ᐉ,. . .,ᐉare common eigenstates ofU+andV−, such states can be interpreted as corresponding to the eigen-states of the total momentum and relative coordinate opera-tors 关19,20兴; indeed, these states are the discrete version of FIG. 1. Slightly modified version of the “scattering circuit” used
to evaluate the real and imaginary parts of the expectation value Tr共U兲for a unitary operatorU, where兩0典represents the ancillary qubit state which acts as a probe particle in a scattering experiment and H denotes a Hadamard transform. In particular, for a controlled-Uoperation given by U=
冑
NS共,兲, the measurements of the ancillary qubit polarizations along thezandyaxes allow us to construct the discrete Wigner functionW共,兲= Tr关T共0兲共,兲兴(discrete characteristic function ⌶共0兲共,兲= Tr关S共0兲共,兲兴) in the
the continuous ones used by Einstein, Podolsky, and Rosen 关41兴. Thus, the generalized Bell measurements will be char-acterized in our context by the set of diagonal projection operators兵兩⌿1,2典具⌿1,2兩其1,2=−ᐉ,. . .,ᐉ.
Now, let us establish some further results related to the generalized Bell states and their discrete phase-space representation. The first one corresponds to the mapping of 兩⌿
1,2典具⌿1⬘,2⬘兩 in terms of the basis
兵Ti共si兲共
i,i兲其i,i=−ᐉ,. . .,ᐉfor each subsystem—i.e.,
兩⌿1,2典具⌿1⬘,2⬘兩=
1
N2
兺
1,1,2,2=−ᐉᐉ
T1共s1兲共
1,1兲
丢T2共s2兲共
2,2兲⌼共−s1,−s2兲
⫻共1,2,1
⬘
,2⬘
兩1,1,2,2兲 共28兲 with the coefficients of the expansion given by⌼共−s1,−s2兲共
1,2,1
⬘
,2⬘
兩1,1,2,2兲 = Tr关T1共−s1兲共1,1兲丢T2 共−s2兲共
2,2兲兩⌿1,2典具⌿1⬘,2⬘兩兴.
Consequently, the second one refers to the inverse mapping of Eq.共28兲, which can be directly reached with the help of property共ii兲as follows:
T1共s1兲共
1,1兲丢T2 共s2兲共
2,2兲
=
兺
1,2,1⬘,2⬘=−ᐉ
ᐉ
⌰共s1,s2兲共
1,1,2,2兩1,2,1
⬘
,2⬘
兲⫻兩⌿
1,2典具⌿1⬘,2⬘兩, 共29兲
being
⌰共s1,s2兲共
1,1,2,2兩1,2,1
⬘
,2⬘
兲 = Tr关T1共s1兲共1,1兲丢T2 共s2兲共
2,2兲兩⌿1⬘,2⬘典具⌿1,2兩兴.
It is worth mentioning that a general connection between the coefficients of both expansions 共28兲 and 共29兲 can also be promptly established for any values of兵s1,s2其苸R,
⌼共−s1,−s2兲共
1,2,1
⬘
,2⬘
兩1,1,2,2兲 =⌰共−s1,−s2兲共1,1,2,2兩1
⬘
,2⬘
,1,2兲 =关⌰共−s1,−s2兲共1,1,2,2兩1,2,1
⬘
,2⬘
兲兴*. The analytical expression of these coefficients will be omit-ted here due to its apparent irrelevance in the phase-space description of the quantum teleportation process.However, some useful results derived from these coeffi-cients deserve to be mentioned and discussed in detail. For instance, Eq. 共29兲 allows us to calculate the parametrized function
F 1,2
共s1,s2兲共
1,1,2,2兲= Tr关T1 共s1兲共
1,1兲丢T2 共s2兲共
2,2兲 ⫻兩⌿1,2典具⌿1,2兩兴, 共30兲
which coincides with⌰for particular values ofiandi
⬘
. In this situation, the analytical expressionF 1,2
共s1,s2兲共
1,1,2,2兲= 1
N2,
兺
=−ᐉ ᐉexp
再
2iN 关共1+2+1兲
+共1−2−2兲兴
冎
关K共,兲兴−共s1+s2兲 can be reduced to the following discrete quasiprobability dis-tribution functions:共i兲s1=s2= 0共Wigner function兲
W
1,2共1,1,2,2兲=␦1,−共1+2兲
关N兴 ␦
2,1−2
关N兴 ,
共ii兲s1=s2= − 1共Husimi function兲
H
1,2共1,1,2,2兲=N
−1兩K共
1+2+1,1−2−2兲兩2. To measure the discrete Wigner function associated with the generalized Bell states, some minor modifications should be implemented in the scattering circuit 共see Fig. 1兲: the first one concerns to the controlled-U operation between the Hadamard gates, since it must be replaced by
U=关
冑
NS1共1,1兲兴丢关冑
NS2共2,2兲兴 in order to process operations for bipartite systems, while the second one consists in preparing the input density operator in the generalized Bell states—namely, =兩⌿1,2典具⌿1,2兩. This
procedure leads us to obtain the expectation value 具z典 =NW
1,2共1,1,2,2兲 through a set of measurements on
the polarization along thez axis of the ancillary qubit. Fur-thermore, these minor modifications on the scattering circuit can also be used to measure any discrete Wigner function associated with a general bipartite system.
2. Quantum teleportation
Basically, the quantum teleportation process consists in a sequence of events that allows us to transfer the quantum state of a particle onto another particle through an essential feature of quantum mechanics: entanglement关18,39兴. In this sense, let us introduce a tripartite system described by
=1丢共兩⌿0,0典具⌿0,0兩兲23, where subsystems 2 and 3 were ini-tially prepared in one of the Bell states. The plan is to tele-port the initial state of subsystem 1 through the protocol established in关20兴.
共i兲We initiate the protocol considering the density opera-tor associated with the tripartite system written in terms of the new basis 兵Ti共si兲共
i,i兲其i,i=−ᐉ,. . .,ᐉ for each subsystem
i= 1 , 2 , 3 as follows:
=
1
N3
兺
1,1,2,2,3,3=−ᐉ
ᐉ
F1共−s1兲共
1,1兲F23 共−s2,−s3兲
⫻共2,2,3,3兲T1 共s1兲
共1,1兲丢T2 共s2兲
共2,2兲丢T3 共s3兲
⫻共3,3兲, where
F1共−s1兲共
1,1兲= Tr1关T1 共−s1兲共
F23共−s2,−s3兲共
2,2,3,3兲= Tr23关T2 共−s2兲共
2,2兲丢T3 共−s3兲共
3,3兲 ⫻共兩⌿0,0典具⌿0,0兩兲23兴.
Next, we perform a measurement on subsystems 1 and 2 that projects them into the Bell states共this procedure corresponds to a collective measurement which determines the total mo-mentum and relative coordinate for composite subsystem 1-2兲. For convenience, before the generalized Bell measure-ment, let us express the phase-space operators T1共s1兲共
1,1兲
丢T
2 共s2兲共
2,2兲according to Eq.共29兲,
=
1
N3
兺
1,1,2,2,3,3=−ᐉ
ᐉ
F1共−s1兲共
1,1兲F23共− s2,−s3兲共
2,2,3,3兲
⫻
兺
1,2,1⬘,2⬘=−ᐉ
ᐉ
⌰共s1,s2兲共
1,1,2,2兩1,2,1
⬘
,2⬘
兲⫻共兩⌿
1,2典具⌿1⬘,2⬘兩兲12
丢T
3 共s3兲共
3,3兲.
Thus, after the measurement on the first two subsystems, only the terms with1=1
⬘
=␣and2=2⬘
=survive. Con-sequently, a reduced density operator for the third subsystem can be promptly obtained,3R= 1
N
兺
3,3=−ᐉᐉ
⌳␣共−,s1,−s3兲共
3,3兲T3 共s3兲共
3,3兲, 共31兲
which does not depend on the complex parameters2. Here, the coefficients are given by
⌳␣共−,s1,−s3兲共
3,3兲=
兺
1,1=−ᐉ
ᐉ
R␣共s,3−s1兲共
1,1,3,3兲F1 共−s1兲共
1,1兲,
with
R␣共s,3−s1兲共
1,1,3,3兲= 1
N2,
兺
=−ᐉ ᐉexp
再
2iN 关共1−3+␣兲
−共1−3−兲兴
冎
关K共,兲兴s3−s1. Note that⌳␣共−,s1,−s3兲共3,3兲simply tells us how to construct, independently of parameterss1ands3, the final parametrized function for the third subsystem from the initial parametrized function of the first one.
共ii兲 Now, let us analyze the particular case s1=s3=s. In this situation, Eq.共31兲assumes the simplified form
3R= 1
N
兺
3,3=−ᐉᐉ
F1共−s兲共3−␣,3+兲T3共s兲共3,3兲, 共32兲
where the coefficientsF1共−s兲共3−␣,3+兲play a central role in the phase-space description of the quantum teleportation process. In fact, they allow us to conclude that, after the generalized Bell measurement, the third subsystem has a pa-rametrized function which is displaced in phase space by an amount 共−␣,兲 with respect to the initial state of the first subsystem—namely, F3共−Rs兲共3,3兲=F1
共−s兲共
3−␣,3+兲. Therefore, the recovery operation basically depends on the
calibration process of the generalized Bell measurements performed on the first two subsystems: for instance, when
␣== 0, we reach a complete recovery operation.
In short, we have presented a quantum teleportation pro-tocol that leads us to obtain a phase-space description of this process for any discrete quasiprobability distribution func-tions associated with physical systems described by the
N-dimensional space of states.
IV. CONCLUSIONS
In this paper we have employed the mod共N兲-invariant op-erator basis 兵T共s兲共,兲其,=−ᐉ,. . .,ᐉ recently proposed in 关26兴
with the aim of obtaining s-parametrized phase-space func-tions which are responsible for the mapping of bounded op-erators acting on a finite-dimensional Hilbert space on their discrete representatives in anN2-dimensional phase space. In fact, we have established a set of important formal results that allows us to reach a discrete analog of the continuous one developed by Cahill and Glauber关1兴. As a consequence, the discrete Glauber-Sudarshan 共s= 1兲, Wigner 共s= 0兲, and Husimi共s= −1兲functions emerge from this formalism as spe-cific cases ofs-parametrized phase-space functions describ-ing density operators associated with physical systems whose space of states has a finite dimension. In addition, we have also established a hierarchical order among them that con-sists of a well-defined smoothing process where, in particu-lar, the kernelK共,兲performs a central role. Next, we have applied our formalism to the context of quantum information processing, quantum tomography, and quantum teleportation in order to obtain a phase-space description of some topics related to unitary depolarizers, discrete Radon transforms, and generalized Bell states. Indeed such descriptions have allowed us to attain important results, within which some deserve to be mentioned:共i兲we have shown that the symme-trized version of the Schwinger operator basis 兵S共,兲其,=−ᐉ,. . .,ᐉ can be considered a unitary depolarizer;
共ii兲 we have also established a link between measurable quantities and discretes-ordered characteristic functions with the help of Radon transforms, which can be used to construct any quasiprobability distribution functions in finite-dimensional spaces; and finally, 共iii兲 we have presented a quantum teleportation protocol that leads us to obtain a gen-eralized phase-space description of this important process in physics. It is worth mentioning that the mathematical formal-ism developed here opens possibilities of future investiga-tions in similar physical systems关42兴or in the study of dis-sipative systems, where the decoherence effect has a central role in the quantum information processing 共e.g., see Ref. 关23兴兲. These considerations are under current research and will be published elsewhere.
ACKNOWLEDGMENTS
关1兴K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857共1969兲; 177, 1882共1969兲.
关2兴A new calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics have been discussed by G. S. Agarwal and E. Wolf, Phys. Rev. D2, 2161 共1970兲; 2, 2187共1970兲; 2, 2206共1970兲. For practical applica-tions of quasiprobability distribution funcapplica-tions in continuous phase space see, for instance, M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121共1984兲; N. L. Balazs and B. K. Jennings,ibid. 104, 347 共1984兲; and also H.-W. Lee,ibid. 259, 147共1995兲.
关3兴For a detailed discussion on the atomic coherent-state repre-sentation and its practical applications in multitime-correlation functions and generalized phase-space distributions involving quantum statistical systems characterized by dynamical vari-ables of the angular momentum type, see, for example, F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A6, 2211共1972兲; L. M. Narducci, C. M. Bowden, V. Bluemel, G. P. Garrazana, and R. A. Tuft,ibid. 11, 973共1975兲; G. S. Agarwal, D. H. Feng, L. M. Narducci, R. Gilmore, and R. A. Tuft, ibid. 20, 2040 共1979兲; G. S. Agarwal, ibid. 24, 2889 共1981兲; and also J. P. Dowling, G. S. Agarwal, and W. P. Schleich,ibid. 49, 4101共1994兲.
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