Phase-Space Formulation of Quantum Mechanics and Quantum State

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C. Brif and A. Mann

Department of Physics, Technion { Israel Institute of Technology, Haifa 32000, Israel

We present a detailed discussion of a general theory of phase-space distributions, intro duced

recently by the authors [J. Phys. A 31, L9 1998]. This theory provides a uni ed phase-space for-

mulation of quantum mechanics for physical systems p ossessing Lie-group symmetries. The concept

of generalized coherent states and the metho d of harmonic analysis are used to construct explicitly

a family of phase-space functions which are p ostulated to satisfy the Stratonovich-Weyl corresp on-

dence with a generalized traciality condition. The symb ol calculus for the phase-space functions

is given by means of the generalized twisted pro duct. The phase-space formalism is used to study

the problem of the reconstruction of quantum states. In particular, we consider the reconstruction

metho d based on measurements of displaced pro jectors, which comprises a numb er of recently pro-

p osed quantum-optical schemes and is also related to the standard metho ds of signal pro cessing.

A general group-theoretic description of this metho d is develop ed using the technique of harmonic

expansions on the phase space.

03.65.Bz, 03.65.Fd

I. INTRODUCTION P function [?,?] is asso ciated with the normal order-

ing and the Husimi Q function [?] with the antinormal

ordering of a and a . Moreover, a whole family of s-

The phase-space formulation of quantum mechanics

parameterized functions can be de ned on the complex

has a long history. In 1932 Wigner [?] intro duced his

plane which is equivalent to the q -p at phase space. The

famous function which has found numerous applications

index s is related to the corresp onding ordering pro cedure

in many areas of physics and electronics. In 1949 Moyal

of a and a ; the values +1, 0, and 1of s corresp ond to

[?] discovered that the Weyl corresp ondence rule [?] can

the P , W , and Q functions, resp ectively. These phase-

be inverted by the Wigner transform from an op erator

space functions are referred to as quasiprobability distri-

on the Hilb ert space to a function on the phase space.

butions QPDs, as they play in the Moyal formulation

As a result, the quantum exp ectation value of an op-

of quantum mechanics a role similar to that of genuine

erator can b e represented by the statistical-likeaverage

probability distributions in classical statistical mechan-

of the corresp onding phase-space function with the sta-

ics. Various QPDs has b een extensively used in many

tistical density given by the Wigner function asso ciated

quantum-optical applications [?,?]. Most recently, there

with the density matrix of the quantum state. In this

is great interest in the s-parameterized distributions b e-

way quantum mechanics can b e formally represented as

cause of their role in mo dern schemes for measuring the

a statistical theory on classical phase space. It should

quantum state of the radiation eld [?].

b e emphasized that this phase-space formalism do es not

The mathematical framework and the conceptual back-

replace quantum mechanics by a classical or semiclassical

ground of the Moyal quantization have b een essentially

theory. In fact, the phase-space formulation of quantum

enlarged and generalized in two imp ortant pap ers by

mechanics also known as the Moyal quantization is in

Bayen et al. [?]. Sp eci cally,itwas shown that noncom-

principle equivalenttoconventional formulations due to

mutative deformations of the algebra of classical phase-

Heisenb erg, Schrodinger, and Feynman. However, the

space functions de ned by the ordinary multiplication

formal resemblance of quantum mechanics in the Moyal

give rise to op erator algebras of quantum mechanics.

formulation to classical statistical mechanics can yield

This fact means that intro ducing noncommutative sym-

deep er understanding of di erences between the quan-

b ol calculus based on the so-called twisted pro duct also

tum and classical theories. Extensive lists of the litera-

known as the star or Moyal pro duct, one obtains a com-

ture on this sub ject can b e found in reviews and b o oks

pletely autonomous reformulation of quantum mechanics

[?,?,?,?,?,?].

in terms of phase-space functions instead of Hilb ert-space

states and op erators. This program of \quantization by

The ideas of Moyal were further develop ed in the late

deformation" has b een develop ed in a numberofworks

sixties in the works of Cahill and Glaub er [?] and Agar-

[?,?,?,?,?].

wal and Wolf [?]. As mentioned, the Wigner function

is related to the Weyl symmetric ordering of the p osi- For a long time applications of the Moyal formulation

tion and momentum op erators q and p or, equivalently, were restricted to description of systems like a spinless

of the b osonic annihilation and creation op erators a and non-relativistic quantum particle or a mo de of the quan-

a . However, there exist other p ossibilities of ordering. In tized radiation eld mo deled by a quantum harmonic

particular, it was shown [?] that the Glaub er-Sudarshan oscillator, i.e., to the case of the at phase space. There- 1

fore, an imp ortant problem is the generalization of the This description can b e useful not only for measurements

standard Moyal quantization for quantum systems p os- of quantum states but also in the eld of signal pro cess-

sessing an intrinsic group of symmetries, with the phase ing.

space b eing a homogeneous manifold on which the group

of transformations acts transitively [?,?]. It has b een

recently understo o d that this problem can b e solved us-

I I. BASICS OF MOYAL QUANTIZATION

ing the Stratonovich-Weyl SW corresp ondence. The

idea of the SW corresp ondence is that the linear bijec-

A. Generalized coherent states and the de nition of

tive mapping b etween op erators on the Hilb ert space and

quantum phase space

functions on the phase space can be implemented by a

kernel which satis es a number of physically sensible p os-

Given a sp eci c physical system, the rst thing one

tulates, with covariance and traciality b eing the two most

needs to do for the Moyal quantization i.e., for con-

imp ortant ones. This idea rst app eared in a pap er by

structing phase-space functions is to determine what is

Stratonovich[?] in 1956, but it was almost forgotten for

the related phase space. This can often b e done by anal-

decades. The SW corresp ondence, that has b een restated

ogy with the corresp onding classical problem, thereby

some years ago by Gracia-Bond a and V arilly [?,?], has

providing a direct route for the quantum-classical corre-

given a new impulse to the phase-space formulation of

sp ondence. From the technical p oint of view, the phase

quantum theory. The SW metho d of the Moyal quan-

space can b e conveniently determined using the concept

tization has b een applied to a numb er of imp ortant sit-

of coherent states [?]. The coherent-state approach is not

uations: a nonrelativistic free particle with spin, using

just a convenient mathematical to ol, but it also helps to

the extended Galilei group [?]; a relativistic free particle

understand howphysical prop erties of the system are re-

with spin, using the Poincar e group [?]; the spin, using

ected by the geometrical structure of the related phase

the SU2 group [?]; compact semisimple Lie groups [?];

space. It is p ossible to say that the concept of coher-

one- and two-dimensional kinematical groups [?,?,?,?];

ent states constitutes a bridge b etween the Moyal phase-

the two-dimensional Euclidean group [?,?]; systems of

space quantization and the Berezin geometric quantiza-

identical quantum particles [?]. For a review of basic

tion [?].

results see Ref. [?].

Let G b e a Lie group connected and simply connected,

Notwithstanding the success of the SW metho d in the

with nite dimension n, which is the dynamical symme-

Moyal quantization of many imp ortantphysical systems,

try group of a given quantum system. Let T be a uni-

the theory su ered from a serious problem. Sp eci cally,

tary irreducible representation of G acting on the Hilb ert

it was the absence of a simple and e ective metho d for

space H . By cho osing a xed normalized reference state

the construction of the SW kernel which should imple-

j i 2 H , one can de ne the system of coherent states

ment the mapping b etween Hilb ert-space op erators and

fj ig:

phase-space functions. The construction pro cedures for

the SW kernels, considered during the last decade see,

j i = T g j i; g 2 G: 2.1

g 0

e.g., Ref. [?], did not guarantee that the kernel will sat-

isfy all the SW p ostulates. Only very recently a general

The isotropy subgroup H G consists of all the group

algorithm for constructing the SW kernel for quantum

elements h that leave the reference state invariantupto

systems p ossessing Lie-group symmetries was prop osed

a phase factor,

[?]. It has b een shown that the constructed kernel ex-

plicitly satis es all the desired prop erties the SW p ostu-

ih ih

T hj i = e j i; je j =1; h 2 H: 2.2

0 0

lates and that in the particular cases of the Heisenb erg-

Weyl group and SU2 our general expression reduces to

For every element g 2 G, there is a decomp osition of g

the known results.

into a pro duct of two group elements, one in H and the

In the present pap er we essentially extend the results

other in the coset space X = G=H ,

of Ref. [?] and present a self-consistent theory of the

SW metho d for the phase-space formulation of quan-

g = h; g 2 G; h 2 H; 2 X: 2.3

tum mechanics. This theory makes use of the concept

of generalized coherent states and of some basic ideas

It is clear that group elements g and g with di erent

of harmonic analysis. Like the Cahill-Glaub er formal-

h and h but with the same pro duce coherent states

ism for the Heisenb erg-Weyl group, we construct the s-

which di er only by a phase factor: j i = e j i,

g g

parameterized family of functions on the phase space of

where = h h . Therefore a coherent state

a quantum system whose dynamical symmetry group is

j i j i is determined by a p oint = g in the

an arbitrary nite-dimensional Lie group. Accordingly,

coset space X . Avery imp ortant prop erty is the identity

weintro duce s-generalized versions of the traciality con-

resolution in terms of the coherent states:

dition and the twisted pro duct. The develop ed phase-

space formulation is used for a general group-theoretic

d j ih j = I; 2.4

description of the quantum state reconstruction metho d.

X 2

iii Covariance: where d is the invariantintegration measure on X ,

the integration is over the whole manifold X , and I is

F ; s=F g ; s; 2.5c

the identity op erator on H . The natural action of G on

X will b e denoted by g .

where Ag T g AT g .

An imp ortant class of coherent-state systems corre-

sp onds to the coset spaces X = G=H which are homo-

iv Traciality:

geneous Kahlerian manifolds. Then X can be consid-

ered as the phase space of a classical dynamical system,

d F ; sF ; s=TrAB : 2.5d

A B

and the mapping ! j ih j is the geometric quanti-

zation for this system [?]. The standard or maximum-

symmetry systems of the coherent states corresp ond to

If the function F ; s satis es the SW corresp ondence,

the cases when an `extreme' state of the representation

it is called the SW symb ol of the op erator A.

Hilb ert space e.g., the vacuum state of an oscillator or

The ab ove conditions have a clear physical meaning.

the lowest/highest spin state is chosen as the reference

The linearity and the traciality conditions are related to

state. This choice of the reference state leads to systems

the statistical interpretation of the theory. If B is the

consisting of states with prop erties \closest to those of

density matrix the state op erator of a system, then the

classical states" [?,?]. In what follows we will consider

traciality condition ?? assures that the statistical aver-

the coherent states of maximal symmetry and assume

age of the phase-space distribution F coincides with the

that the phase space of the quantum system is a homoge-

quantum exp ectation value of the op erator A. O'Connell

neous Kahlerian manifold X = G=H , each p oint of which

and Wigner [?] have shown that the traciality condi-

corresp onds to a coherent state j i. In particular, the

tion for density matrices of a spinless quantum parti-

Glaub er coherent states of the Heisenb erg-Weyl group

cle there it app ears as an overlap relation is necessary

H are de ned on the complex plane C =H =U1, and

3 3

for the uniqueness of the de nition of the Wigner func-

the spin coherent states are de ned on the unit sphere

tion. It has b een also shown [?] that the traciality con-

S = SU 2=U1. In the more rigorous mathematical

dition is necessary for the uniqueness of the de nition of

language of Kirillov's theory [?], the phase space X is

the symb ol calculus twisted or \star" pro ducts of the

de ned as the coadjoint orbit asso ciated with the uni-

phase-space functions and for the validity of the related

tary irreducible representation T of the group G on the

non-commutativeFourier analysis. Equation ?? is ac-

Hilb ert space H .

tually a generalization of the usual traciality condition

[?,?,?], as it holds for any s and not only for the Wigner

case s = 0. The reality condition ?? means that if

B. The Stratonovich-Weyl corresp ondence

A is self-adjoint, then F ; s is real. The condition

?? is a natural normalization, which means that the

image of the identity op erator I is the constant function Once the phase space of a quantum system is deter-

1. The covariance condition ?? means that the phase-

mined, the Moyal quantization pro ceeds in the following

space formulation must explicitly express the symmetry way. Let A b e an op erator on H . Then A can b e mapp ed

of the system. by a family of functions F ; s onto the phase space

The linearity is taken into accountifwe implement the

X the index s lab els functions in the family. If A is

map A ! F ; sby the generalized Weyl rule the density matrix of a quantum system, the corre-

sp onding phase-space functions F ; s P ; s are

F ; s=Tr[A ; s]; 2.6

called QPDs. Of course, the phase-space formulation

of the quantum theory for a given physical system can

where f ; sg is a family lab eled by s of op erator-

b e successful only if the functions F ; s p ossess some

valued functions on the phase space X . These op erators

physically motivated prop erties. These prop erties were

are referred to as the SW kernels. The generalized tra-

formulated by Stratonovich[?] and are referred to as the

ciality condition ?? is taken into accountif we de ne

SW corresp ondence:

the inverse of the generalized Weyl rule ??as

0 Linearity: A ! F ; s is one-to-one linear map.

d F ; s ; s: 2.7 A =

i Reality:

Now, the conditions ?? -?? of the SW corresp ondence

F ; s=[F ; s] : 2.5a

for F ; s can b e translated into the following condi-

tions on the SW kernel ; s:

ii Standardization:

i ; s = [ ; s] 8 2 X: 2.8a

ii d ; s=I: 2.8b

d F ; s=Tr A: 2.5b

iii g ; s=T g ; sT g : 2.8c 3

A. Necessary instruments: harmonic functions,

invariant co ecients, and tensor op erators

Substituting the inverted maps ?? for A and B into

the generalized traciality condition ??, we obtain the

relation between functions with di erent values of the

index s:

Our problem is to nd the explicit form of the SW

kernel ; s that satis es the conditions ??-?? and

0 0 0 0

d K F ; s= ; F ; s ; 2.9

??. In order to accomplish this task, we need three

s;s A A

basic ingredients: harmonic functions, invariant co e-

0 0 0

K ; Tr [ ; s ; s ]: 2.10

s;s

cients, and tensor op erators. The coherent states serve

here as the glue that binds them together.

If we take s = s in Eq. ?? and takeinto account the

We start by considering the Hilb ert space L X; of

arbitrariness of A,we obtain the following relation

square-integrable functions u on X with the invariant

measure d. The representation T of the Lie group G on

0 0 0

; s= d K ; ; s; 2.11

L X; is de ned as

where the function

T g u = ug : 3.3

0 0

K ; =Tr [ ; s ; s] 2.12

The eigenfunctions Y of the Laplace-Beltrami op er-

b ehaves like the delta function on the manifold X .

ator [?] form a complete orthonormal basis in L X; :

0 0

Y Y = ; 3.4a

I I I. CONSTRUCTION OF THE

STRATONOVICH-WEYL KERNEL

0 0

d Y Y = : 3.4b

It is clear that the Moyal quantization for aphysical

system is accomplished by constructing the SW kernel

The functions Y are called the harmonic functions,

; s that satis es the SW p ostulates. Although the

form of the SW kernel has b een known for many systems, and is the delta function in X with resp ect to

the measure d. Note that the index is multiple; it

a general construction metho d was not known. A pro-

has one discrete part, while the other part is discrete for

cedure that was applied in many works [?,?,?,?] is as

compact manifolds and continuous for noncompact man-

follows. An arbitrary p oint 2 X is xed and then an

ifolds. In the latter case the summation over includes

Ansatz is made for a self-adjoint op erator usu-

an integration with the Plancherel measure d and the

ally only the case s =0was considered that satis es the

symbol includes some Dirac delta functions for more

standardization condition ?? and the following prop-

details see Ref. [?]. For conciseness, we omit these de-

erty:

tails in our formulas. The harmonic functions Y are

=T T ; 8 2 H ; 3.1

0 0

0 linear combinations of matrix elements T g . There-

fore, the transformation rule for the harmonic functions

where H = f 2 G j = g is the isotropy sub-

0 0

is [?]

group for . For any 2 X there exists g 2 G such

that g = , and then the SW kernel is de ned by

0 0

: 3.5 T g Y T g Y = Y g =

= g =T g T g : 3.2

0 0

This kernel automatically satis es the covariance condi-

It should b e understo o d that the summation in Eq. ??

tion ??, but the problem is that the traciality is not

is only on the part of that lab els functions within an

guaranteed. Of course, in the describ ed pro cedure the

irreducible subspace.

form of the kernel dep ends on the Ansatz and often no

Next, we once again use the coherent states, in order

kernel satisfying the traciality condition is found.

to intro duce the concept of invariant co ecients. The

We prop ose here a simple and general algorithm for

0 2

p ositive-valued function jh j ij is symmetric in and

constructing the SW kernels the whole s-parameterized

. Therefore, its expansion in the orthonormal basis

family which explicitly satisfy all the SW p ostulates,

must b e of the form

including b oth the covariance and the traciality. Our

metho d makes use of Perelomov's concept of coherent

0 0 2

Y jh j ij = Y

states and of only some basic ideas from harmonic anal-

ysis. Hop efully, the simplicity and generality of our

metho d will draw more attention to the ideas of the

Y Y ; 3.6 =

phase-space quantization.

where are real p ositive co ecients. Using the invari- B. Explicit form of the kernel

0 0

ance h j i = hg jg i and the unitarity of the rep-

resentation T ,we obtain

Using the ab ove preliminary results, we are able to nd

the SW kernel ; s with all the desired prop erties.

0 2 0

Sp eci cally, let us de ne

jh j ij = Y g Y g

X X

; s f s; Y D : 3.14

Y T g Y g : 3.7 =

We will show that the construction of the generalized

kernel ?? satis es the SW corresp ondence. In equation

In order to satisfy this equality, the co ecients must

?? f s; is a function of and of the index s. We

be invariant under the index transformation of equation

assume that f p ossesses the invariance prop erties of .

??: = . This means that do not dep end on

Using the invariance of under the index transforma-

the part of which lab els functions within an irreducible

tion of Eq. ??, we see that the reality condition ??is

subspace. Since the Laplace-Beltrami op erator is self-

satis ed if f s; is a real-valued function. Therefore,

adjoint, one nds that

we can consider only real values of the index s.

Next we consider the standardization condition ?? .

Y = e Y ; 3.8

Using the de nition ?? , we obtain

Z Z

where Y is another harmonic function, with the same

0 2

d ; s= f s; D d Y ;

eigenvalue as Y . Since jh j ij is real, the co e-

X X

cients must b e invariant under the index transforma-

tion of equation ?? : = .

3.15

Next we use the coherent states, harmonic functions,

while Eq. ?? can b e used to write

and invariant co ecients for de ning the set of op erators

fD g on H :

X X

y 1=2

I = TrD D = D d Y :

1=2

D d Y j ih j: 3.9

3.16

The standardization condition is satis ed if the expres-

Using the expression ?? and the orthonormality rela-

sions ?? and ?? are equal. Using the identity resolu-

tion ?? for the harmonic functions, we obtain the or-

tion ?? and Eq. ?? , we can write

thonormality condition for the op erators D :

0 0 2

1=h j i = d jh j ij

TrD D = : 3.10

0 0

d Y : 3.17 = Y

1=2

Note that the factor in front of the integral in

Eq. ?? serves just for the prop er normalization. Us-

Multiplying the left and right sides of this equation by

ing ?? , we also obtain the relation

and integrating over d , we obtain Y

1=2

Z Z

h jD j i = Y : 3.11

d Y = d Y : 3.18

X X

The invariance of the co ecients implies that D are

Since is not identically 1, this relation can b e satis ed

the tensor op erators whose transformation rule is the

only if there exists some such that = 1 and

same as for the harmonic functions Y :

: 3.19 d Y /

0 0

T g D T g = T g D : 3.12 0

As was already mentioned, for noncompact manifolds

A useful prop erty of the tensor op erators is that any op- the symbol actually includes some Dirac delta func-

erator A on H can b e expanded in the orthonormal basis tions. It can be easily seen from Eqs. ?? , ?? , and

fD g: ?? that the standardization condition is satis ed if

1=2

= , i.e., f s;

A = TrAD D : 3.13

f s;1=1; 8s: 3.20

The covariance condition ?? can b e rewritten as 5

IV. PHASE-SPACE FUNCTIONS AND THE

f s; D Y g

SYMBOL CALCULUS

= f s; T g D T g Y : 3.21

A. Typ es of phase-space function

Using the transformation rules ?? and ??, Eq. ??

can b e transformed into

X X

As the explicit form of the SW kernels is known, we

f s; D T g Y

can write the SW symb ols on the phase space as

X X

0 0

s=2

= f s; T g D Y : 3.22

F ; s= A Y

s=2

Changing the summation indexes $ on either side

= A Y ; 4.1

of Eq. ?? , we immediately see that the covariance con-

dition is satis ed by the virtue of the invariance of

under the index transformation of Eqs. ?? and ?? .

where wehave de ned

In order to satisfy the relation ?? , the function

K ; of equation ??must b e the delta function in

A TrAD ; A TrAD : 4.2

X with resp ect to the measure d,

0 0 0

K ; = Y Y = : 3.23

For a self-adjoint op erator A, we get A = A . It can

b e easily veri ed that substituting expressions ?? and

?? into the inverse Weyl rule ??, one indeed ob-

This result is valid if

tains A = A D . We also note that the function

f s; f s; =1: 3.24

K ; of equation ?? is given by

s;s

This prop ertyis satis ed only by the exp onential func-

0 ss =2 0

K ; = Y Y ; 4.3

tion of s, i.e.,

s;s

f s; =[f ] : 3.25

and it clearly satis es Eq. ?? which connects the func-

Note that the standardization condition ?? then reads

tions with di erentvalues of the index s. In general, let

f 1 = 1.

F and H b e two phase-space functions such that

The exact form of the function f can b e determined

if we de ne [?] for s = 1

F = F Y ; 4.4

; 1 j ih j: 3.26

Then we obtain

H = H Y : 4.5

j ih j = [f ] Y D : 3.27

Then they are related through the transformation

Multiplying b oth sides of this equation by Y and

1=2

integrating over d , we nd f = , i.e.,

0 0 0

F = d K ; H ; 4.6

s=2

f s; = : 3.28

0 0

Y Y : 4.7 K ; =

Obviously, the standardization condition f 1 = 1 is sat-

is ed. Finally,we obtain

s=2

; s= Y D

Let fj ig be a complete orthonormal basis in the

Hilb ert space H . Using the generalized Weyl rule ??

y s=2

for the op erator A = j ih j,we obtain

n m

: 3.29 Y D =

F ; s=h j ; sj i ; s: 4.8

A m n mn

It is evident that this kernel is completely determined by

the harmonic functions on the corresp onding manifold

Using Eq. ?? , we nd

and by the coherent states which form this manifold. We

will see that the SW kernel ?? is a generalization of the

s=2 y

Cahill-Glaub er kernel for a harmonic oscillator [?,?] and

h jD j iY : 4.9 ; s=

m n mn

of the Agarwal kernel for spin [?].

The standardization and traciality conditions ?? and Let us rst consider the case of the Wigner function

?? can b e used to show that s = 0. The twisted pro duct of two functions is denoted

by W W and is determined by the condition

A B

d ; s= ; 4.10

mn mn

W W = W 4.18

A B AB

d ; s ; s= : 4.11

for anytwo op erators A and B . Note that the condition

mn kl ml nk

?? assures the asso ciativity of the twisted pro duct. On

the other hand, this pro duct is, in general, noncommu-

The functions ; s form a useful orthonormal basis

tative. In this way the algebra of op erators is mapp ed

in L X; .

onto the algebra of phase-space functions. If one starts

The SW symb ols obtained for some sp ecial values of

from a classical phase-space description, the intro duction

s are frequently used in numerous applications. In par-

of the twisted pro duct can b e viewed as the quantization

ticular, for s = 1, we obtain the Q function Berezin's

realized by a deformation of the algebra of functions [?].

covariant symbol [?]:

Using the Weyl rule ?? and its inverse ??, we ob-

tain

Q F ; 1 = h jAj i: 4.12

A A

W = Tr[ AB ]

AB W

Equation ?? can be easily obtained by recalling [see

Eqs. ?? and ??] that

0 0 0

=Tr d W

W W

1=2

; 1 = j ih j = Y D : 4.13

00 0 00

d W : 4.19

For s =1,we obtain the P function Berezin's contravari-

Intro ducing the function trikernel

ant symbol [?]:

0 00 0 00

L ; ; =Tr[ ]; 4.20

1=2

W W W

P F ; 1 = A Y ; 4.14

A A

we obtain the following de nition of the twisted pro duct:

Z Z

whose de ning prop ertyis

0 00 0 00

W W d d L ; ;

A B

X X

A = d P j ih j: 4.15

0 00

W W : 4.21

A B

The functions P and Q are counterparts in the traciality

The so-called Moyal bracket is de ned as

condition ??. Perhaps the most imp ortant SW sym-

[W ;W ] = iW W W W : 4.22

b ol corresp onds to s = 0, b ecause this function is \self-

A B M A B B A

conjugate" in the sense that it is the counterpart of itself

The twisted pro duct can b e easily generalized for ar-

in the traciality condition ?? . It is natural to call the

bitrary values of s. The s-parameterized twisted pro d-

function with s = 0 the generalized Wigner function:

uct F F ; s of any two functions F ; s and

A B A

F ; s is once again determined by the condition

W F ; 0 = A Y : 4.16

A A

0 00

F ; s F ; s =F ; s: 4.23

A B AB

The corresp onding SW kernel is

Analogously to the Wigner function case, this leads to

the de nition

D : 4.17 ; 0 = Y

Z Z

0 00 0 00

0 00

F F ; s ; ; d d L

A B s;s ;s

X X

0 0 00 00

F ; s F ; s ; 4.24

A B

B. The generalized twisted pro duct

where the generalized trikernel is given by

The phase-space formulation of quantum mechanics

0 00 0 0 00 00

0 00

L ; ; =Tr [ ; s ; s ; s ]

s;s ;s

can be made completely autonomous if one intro duces

0 0 00 00

a symb ol calculus for the functions on the phase space,

; s ; s ; s : 4.25 =

mn nk km

which replaces the usual manipulations with op erators

m;n;k

on the Hilb ert space. This symb ol calculus is based on

the so-called twisted pro duct or Moyal pro duct which Using the standardization condition ?? and the def-

corresp onds to the usual pro duct of op erators [?,?,?]. inition ??, we obtain 7

0 00 0 0 00 00

eld mo deled by a quantum harmonic oscillator. The

0 00

d L ; ; =Tr [ ; s ; s ]

s;s ;s

Wigner function [?] and the Moyal quantization [?] were

0 00

originally intro duced for such systems. The kernel im-

0 00

= K ; : 4.26

s ;s

plementing the mapping between Hilb ert-space op era-

This result together with the relation ?? can b e used

tors and s-parameterized families of phase-space func-

to obtain the so-called tracial identity for the generalized

tions the SW kernel in our notation for H was intro-

twisted pro duct,

duced by Cahill and Glaub er [?]. The generalization of

the formalism to the many-dimensional case is straight-

forward see, e.g., Ref. [?].

d F F ; s

A B

The nilp otent Lie algebra of H is spanned by the basis

y y

fa; a ;Ig, where a and a are the b oson annihilation and

0 0

= d F ; s F ; s ; 4.27

A B

creation op erators, satisfying the canonical commutation

relation, [a; a ]=I . Group elements can b e parameter-

which holds for any s and s . Equation ?? is the phase-

ized in the following way:

space version of the tracial identity for the op erators,

a a i'I

g = g ; '; T g = e e ; 5.1

TrAB = A B : 4.28

where 2 C and ' 2 R.

The phase space is the complex plane C = H =U1,

Using the covariance condition ?? and the de nition

and the Glaub er coherent states are

??, we nd the invariance prop erty of the trikernel

j i j i = D j0i; 2 C ; 5.2

0 00 0 00

L g ;g ;g =L ; ; : 4.29

s;s ;s s;s ;s

where

This prop erty implies the equivariance of the twisted

pro duct:

D = exp a a 5.3

g g

g 0 00

F F ; s=F ; s F ; s ; 4.30

A B

is the displacement op erator. The invariant measure is

where

1 2

d d ; 5.4

; s F g ; s: 4.31 F

and the corresp onding delta function is

0 2 0

: 5.5

V. EXAMPLES

The harmonic functions on C are the exp onentials:

The general formalism presented ab ove can b e under-

Y Y Y ; = exp : 5.6

stood much b etter by illustrating it with a number of

simple examples. We will consider two simple physi-

Here 2 C with the Plancherel measure given by

cal systems: a nonrelativistic spinless quantum parti-

1 2 2 0

d d and with . Note that

;

cle and spin, whose dynamical symmetry groups are the

for the Heisenb erg-Weyl group b oth the phase-space co-

Heisenb erg-Weyl group H and SU2, resp ectively. It

ordinate and the index are complex num-

should b e emphasized that the SW kernels for these ba-

b ers, and the Plancherel measure is similar to the invari-

sic systems have b een known for a long time [?], so the

ant measure on C .

novelty here is not the result itself but the metho d of

The invariant co ecients can be found in

derivation. Our aim is to demonstrate how the general

the following way. In the present context Eq. ?? takes

algorithm works by applying it to a numb er of relatively

the form

simple and well-known problems. We will show that by

identifying harmonic functions, invariant co ecients, and

0 0 0 2

0 2 j j

e : jh j ij = e =

tensor op erators for a given system, one can readily de-

rive the explicit form of the SW kernel.

Taking into account that the Fourier transform of a Gaus-

sian function is once again a Gaussian, it is not dicult

A. The Heisenb erg-Weyl group

to obtain

First, we consider the Heisenb erg-Weyl group H which

= expj j : 5.7

is the dynamical symmetry group for a spinless quan-

tum particle and for a mo de of the quantized radiation Then we deduce that the tensor op erator 8

[J ;J ]=i J : 5.11

j j =2

p r pr t t

D D =e e j ih j 5.8

The unitary irreducible representations are lab eled by

a a

the index j j = 0; 1=2; 1;:::, and the Hilb ert space

is just the displacement op erator D =e . The

H is spanned by the orthonormal basis jj; i =

natural orthonormal basis in the Hilb ert space is the Fock

j; j 1;:::;j . Group elements can b e parameterized

basis fjnig, a ajni = njni n =0; 1; 2;:::. The matrix

using the Euler angles ; ; :

elements of the tensor op erator are given by[?]

i J i J i J

z y z

g = g ; ; =e e e : 5.12

hmjD jni

j j =2 mn mn 2

n!=m! e L j j ; m n

The phase space is the unit sphere S = SU2=U1,

j j =2 nm nm 2

and each coherent state is characterized by the unit vec-

m!=n! e L j j ; m n;

tor

where L x are the asso ciated Laguerre p olynomials.

n = sin cos ; sin sin ; cos : 5.13

Using the parameterization ?? of group elements, one

can easily nd the transformation rule

Sp eci cally, the coherent states j i jj ; ni are given by

the action of the group element

T g D T g =D D D

= exp D : 5.9

iJ i J

z y

g = g ; =e e 5.14

Therefore, the index do es not change under the group

on the highest-weight state jj; j i:

transformation, as D and Y ; are just multiplied

by a phase factor. Corresp ondingly, there is no index

jj ; ni = jj ; ; i = g ; jj; j i

transformation, induced by the action of group elements,

1=2

to which should be invariant. On the other hand,

j + j

= cos =2 sin =2

Y ; =Y ; , and isobviously invariant un-

j +

der the index transformation $ .

e jj; i: 5.15

Finally, the harmonic functions Y ; , the invariant

co ecients , and the tensor op erators D can be

The invariant measure is

substituted into the general formula ??. Then one ob-

tains the SW kernel for the Heisenb erg-Weyl group:

2j +1 2j +1

dn = sin d d; 5.16 d

4 4

2 y

sj j =2 a a

e e e ; 5.10 ; s=

and the corresp onding delta function is

which is exactly the kernel intro duced by Cahill and

0 0

n n

Glaub er [?].

2j +1

0 0

= cos cos : 5.17

2j +1

B. The SU2 group

The harmonic functions on S are the familiar spherical

As another example, we consider SU2 which is the

harmonics:

dynamical symmetry group for the angular momentum

or spin and for many other systems, for example, a

Y ; : 5.18 Y

collection of two-level atoms, the Stokes op erators de-

2j +1

scribing the p olarization of the quantized light eld,

In this context is the double discrete index fl; mg with

two light mo des with a xed total photon numb er, etc.

l =0; 1; 2;::: and m = l; l 1;:::;l . The transforma-

A number of authors have used di erent approaches

tion rule for the spherical harmonics reads

to the construction of the Wigner function for spin

[?,?,?,?,?,?,?,?,?,?,?,?]. The explicit expressions for the

Q, W , and P functions for arbitrary spin were rst ob-

g ; ; Y ; = D ; ; Y ; ;

lm lm

m m

tained by Agarwal [?], who used the spin coherent-state

m =l

representation [?,?,?] and the Fano multip ole op erators

[?]. V arilly and Gracia-Bond a [?] have shown that the

5.19

spin coherent-state approach is equivalent to the formal-

ism based on the SW corresp ondence.

where

The simple Lie algebra of SU2 is spanned by the basis

D jg ; ; jl; mi 5.20 ; ; =hl; m fJ ;J ;J g,

x y z

m m 9

VI. RECONSTRUCTION OF QUANTUM STATES is the matrix representation of SU2 elements and

g ; ; isgiven by Eq. ?? . Another prop erty of the

spherical harmonics is

A. Basic systems and metho ds

Y ; =1 Y ; 5.21

l;m

A great amountofwork has b een devoted in the last

The invariant co ecients can be found using the fol-

few years to the problem of determining the quantum

lowing expansion [?]:

state from information obtained by a set of measure-

ments p erformed on an ensemble of identically prepared

1+n n

0 2

systems. The task is to reconstruct the density matrix

jhj; njj; n ij =

which, according to the principles of quantum physics,

contains all available information ab out the state of a

2l +1

2 0

system. Of course, the question arises which set of mea-

= hj; j ; l; 0jj; j i P n n ; 5.22

2j +1

surements provides information sucient for the state re-

l=0

construction. This question rst app eared in early works

where P x are the Legendre p olynomials and

byFano [?] and Pauli [?] and was discussed in a number

of pap ers [?,?,?,?,?,?].

j j j

1 2

hj ;m ; j ;m jj; mi C 5.23

1 1 2 2

m m m

1 2

Recently, signi cant theoretical and exp erimental

progress has b een achieved in the reconstruction of quan-

are the Clebsch-Gordan co ecients. Using the addition

tum states of the light eld see, e.g., a recent book [?].

formula for the spherical harmonics,

One of the most successful reconstruction metho ds in this

context is the optical homo dyne tomography. A tomo-

2l +1

0 0

P n n = Y nY n ; 5.24

graphic approachto the Wigner function was discussed

l lm

m=l

by Bertrand and Bertrand [?] and a quantum-optical

scheme was prop osed byVogel and Risken [?]. The recon-

Eq. ?? can b e rewritten as

struction of quantum states of the light eld by means of

homo dyne tomographywas realized in a series of intrigu-

X X

ing exp eriments [?,?]. Various metho ds for data analysis

0 2 2

jhj; njj; n ij = hj; j ; l; 0jj; j i

in optical homo dyne tomography measurements were re-

2j +1

l=0 m=l

cently discussed [?,?,?,?,?]. The tomographic schemes

Y nY n : 5.25

were also generalized for the reconstruction of the joint

density matrix for two- and multi-mo de optical elds

Comparing this result with the general formula ??, we

[?,?,?,?,?]. Among other approaches to the reconstruc-

readily nd that the invariant co ecients are given by

tion of quantum states of lightwewould like to mention

the symplectic tomography [?] and the photon count-

2j + 1[2j !]

: 5.26 = hj; j ; l; 0jj; j i =

ing metho ds [?,?,?] also known as the photon number

2j + l + 1!2j l !

tomography[?].

In the case of a single-mo de microwave eld inside a

Note that = 0 for l > 2j . The invariance of is

l l

high-Q cavity, a direct measurement on the system it-

ensured by the fact that they are indep endentofm.

self is imp ossible. Instead, one can prob e the state of the

The tensor op erators for spin are the well-known Fano

intra-cavity eld via the detection of atoms after their in-

multip ole op erators [?], which can b e written in the form

teraction with the eld mo de [?,?,?]. Similar ideas were

also applied to the reconstruction of the quantum mo-

2l +1

hj; k ; l; mjj; q ijj; q ihj; k j: 5.27 D =

tional state of a laser-co oled ion trapp ed in a harmonic

2j +1

k;q =j

p otential [?,?,?,?,?,?], including a b eautiful exp erimental

realization [?].

Substituting expressions ??, ?? , and ?? into the

State reconstruction pro cedures were prop osed for var-

general formula ??, we nd that the SW kernel for spin

ious quantum systems, for example, one-dimensional

is given by

wave packets [?,?], harmonic and anharmonic molecular

vibrations [?,?], motional states of atom b eams [?], Bose-

Einstein condensates [?], cyclotron states of a trapp ed

; ; s= hj; j ; l; 0jj; j i

2j +1

electron [?], atomic Rydb erg wave functions [?], etc.

l=0

State reconstruction metho ds for systems with a nite-

dimensional state space e.g., for spin were also dis-

; ; 5.28 D Y

cussed [?,?,?,?,?,?]. Exp erimental reconstructions were

m=l

also rep orted for electronic angular-momentum states of

which coincides for s =0; 1 with the results by Agarwal hydrogen [?] and for vibrational quantum states of a

[?] and byV arilly and Gracia-Bond a [?]. diatomic molecule [?]. 10

B. Displaced pro jectors =D jnihnjD 6.6

is obtained for jui = jni b eing the Fock state. In par-

It turns out that the ma jority of schemes used for

ticular, measuring the probability to nd the displaced

the reconstruction of quantum states are related to the

oscillator in the ground state j0i, one obtains the Husimi

phase-space formalism. Frequently, the Q function, the

function Q = h jj i. On the other hand, if one

Wigner function, or other phase-space QPDs represent-

knows the functions p for all values of n, then the

ing the density matrix of the system can be either

Wigner function can b e built as [?]

measured directly or deduced in some way from mea-

sured data. In particular, in many prop osed and realized

schemes the measured quantity is the exp ectation value

W =2 1 p : 6.7

n=0

p =h i =Tr[ ] 6.1

u u u

This formula can be generalized for QPDs with other

of a self-adjoint op erator

values of s [?]:

=U juihujU ; 6.2

s +1 2

p : 6.8 F ; s P ; s=

1 s s 1

which is a transformed pro jector on a quantum state jui.

n=0

The unitary op erator U represents the corresp onding

These metho ds for determining the Husimi function

transformation, and the measurements are made for a

and the Wigner function and thus reconstructing the

range of values of the transformation parameter .

quantum state of the system were rst discussed by

We will distinguish here between two p ossibilities. If

Royer [?] in 1985. Recently, such a scheme for mea-

U = T is the phase-space displacement op erator

suring the Q function was prop osed in the context of

which represents an elementofX = G=H , with G b eing

trapp ed ions [?]. Another metho d for the reconstruction

the dynamical symmetry group of given quantum system,

of the motional state of a trapp ed ion, prop osed and ex-

we will call the observable = the prop erly

u u

p erimentally realized by the NIST group [?], employs the

transformed pro jector or the displaced pro jector. Oth-

interaction b etween the vibrational mo de of the ion and

erwise will be called the improp erly transformed

its internal electronic levels. The initial motional state

pro jector.

is displaced in the phase space, as in Eq. ??, and then

In order to illustrate these de nitions, let us consider

the interaction with the two-level internal subsystem is

a quantum harmonic oscillator which is the mo del sys-

induced for a time t. The p opulation P t; of the lower

tem for a single mo de of the quantized radiation eld, a

internal state ji is measured for di erentvalues of dis-

laser-co oled ion moving in a harmonic trap, or a har-

placement amplitude and time t this measurement can

monic vibrational mo de of a diatomic molecule. The

b e made with great accuracy by monitoring the uores-

corresp onding symmetry group is the Heisenb erg-Weyl

cence pro duced in driving a resonant dip ole transition.

group H , and the phase space is the complex plane C =

If ji is the internal state at t = 0, the signal averaged

H =U1 see section ??. In this context U =D is

over many measurements is

the Glaub er displacement op erator, and the exp ectation

value of the displaced pro jector,

1+ p cos2 te ; 6.9 P t; =

n n;n+1

p =Tr[ ] = Tr[D D ]; 6.3

u u u

n=0

is called the op erational phase-space probability distribu-

where are the Rabi frequencies and are the

n;n+1 n

tion [?,?,?]. Here, is the density matrix of the quantum

exp erimentally determined decay constants. This rela-

state of the system and is the density matrix given

tion allows to determine the p opulations p of the

by the pro jector juihuj for a pure state of the so-called

displaced motional eigenstates. As one can see from

\quantum ruler" state whichcharacterizes the measure-

Eq. ??, the functions p in their turn can b e used to

ment device. For example, displacing the state of the

calculate the QPDs P ; s e.g., the Wigner function.

oscillator,

Alternatively, the density matrix in the Fock representa-

tion can b e deduced directly from p .

! =D D ; 2 C ; 6.4

In the optical domain, the function p can b e de-

termined in principle as the probability of recording n

and measuring the probability of nding it in the Fock

counts with an ideal photo detector exp osed to the dis-

state jni, one obtains the op erational phase-space prob-

placed light eld. In practice, one could use the unbal-

ability distribution,

anced homo dyning detection [?,?,?,?], in which the signal

eld is mixed in a b eam splitter with the lo cal oscillator

p =hnj jni =Tr[ ]: 6.5

n n

of coherent amplitude and the photon statistics of the

sup erimp osed eld is then counted by a photo detector The displaced pro jector 11

of nding the displaced system of quantum eciency . The resulting counting statis-

tics is denoted by p ; , where = R =T is the

1 2

n=g ng n; n 2 S 6.16

e ective displacement amplitude T and R are the trans-

mission and re ection co ecients of the b eam splitter

in the state jj; i. These ideas for spin are conceptu-

and = jT j is the overall quantum eciency. In this

ally very similar to the prop osals in the context of op-

realistic situation formula ?? should b e replaced by the

tical elds or trapp ed ions. Therefore, it seems natural

following result [?]:

to apply the phase-space formalism develop ed ab oveto

the general group-theoretic description of the state recon-

struction metho d based on the measurement of displaced

2+ s 1 2

p ; : 6.10 P ; s=

pro jectors.

1 s s 1

n=0

This metho d of state reconstruction is sometimes called

C. General reconstruction formalism

the photon numb er tomography.

As an example of measurements with improp erly trans-

From the practical p oint of view, the reconstruction

formed pro jectors, we mention the optical homo dyne to-

pro cedure consists of two steps. First, the system de-

mography [?,?] in which one measures the probability

scrib ed by the density matrix is displaced in the phase

distribution P x; for the rotated eld quadrature

space:

x = x cos + p sin = U xU : 6.11

! = T T ; 2 X: 6.17

The second step is the measurement of the probabilityto

The eld quadratures x and p can b e viewed as the scaled

nd the displaced system in a quantum state jui,

p osition and momentum op erators of the harmonic oscil-

1=2 y

lator, with a = 2 x + ip, and U = expi a a

p = huj jui: 6.18

is the rotation op erator known in optics as the phase

Rep eating this pro cedure for a large number of phase-

shifter on the phase plane. U represents an element

space p oints , one can in principle determine the func-

of the SO2U1 subgroup of the oscillator group H

y y

tion p .

whose algebra is spanned by fI ; a; a ;a ag. The improp-

erly transformed pro jector is given by

1. Moreabout displacedprojectors

=U jxihxjU ; 6.12

where jxi are the p osition eigenstates. The measured dis-

The information contained in the function p is

tribution P x; can b e used for determining the Wigner

enough for the reconstruction of the density matrix. It is

function via the inverse Radon transform [?,?,?]. Alter-

convenient to analyze this problem with the help of the

natively, the density matrix in some basis e.g., in the

displaced pro jector,

Fock basis can b e deduced directly from P x; byaver-

= T juihujT ; 6.19

aging a set of pattern functions [?,?,?,?]. Another exam-

ple of measurements with improp erly transformed pro-

whose exp ectation value gives the measured function

jectors is the symplectic tomography [?], in which the

p , as in Eq. ?? . The displaced pro jector satis es a

phase-space rotation is accompanied by the squeezing

numb er of useful prop erties:

transformation.

i It is a self-adjoint op erator,

In the case of measurements with improp erly trans-

formed pro jectors, a general group-theoretic approachis

= 8 2 X: 6.20

problematic, b ecause the numb er of p ossible transforma-

Since p is not only real but also non-negative

tions is very large and one should consider each situa-

this is a probability, is also a non-

tion separately. On the other hand, the metho d of prop-

negatively de ned op erator.

erly transformed pro jectors works uniformly for physical

systems with di erent symmetry groups. For example,

ii Provided that the state jui is normalized, is

in the case of the SU2 symmetry e.g., spin, two-level

a trace-class op erator of trace one, and the follow-

atoms, etc., prop osals app eared [?,?] for measuring the

ing standardization condition holds,

Q function,

d = I: 6.21

Qn=hj; njjj; ni =Trjj; nihj; nj; 6.13

This implies the normalization of p ,

or, more generally, for measuring the probability

d p =1: 6.22

p n=Tr[ n]; 6.14

n=g njj; ihj; jg n 6.15

iii The displaced pro jector is manifestly covariant, 2. Entropy

T g T g = g : 6.23

u u

A useful quantity for analyzing statistical prop erties

Consequently, if p corresp onds to the initial

of the system in particular, the amount of noise is the

density matrix , the function p g will cor-

entropy. A phase-space version of the entropy can be

resp ond to the transformed density matrix g =

intro duced in the following way,

T g T g .

Denoting the density matrix of the quantum ruler state

S = d p ln p : 6.30

u u u

by whichisjuihuj for a pure state, the op erational

phase-space probability distribution reads

For jui = j i, Eq. ?? gives

p = Tr[T T ]: 6.24

u u

Using the inverse Weyl rule ?? for the density matrix

S = d Q ln Q ; 6.31

,we obtain

0 0 1 0

whichis a generalization of the Wehrl entropy[?] that

p = d P ; sTr[T ; sT ];

u u

was de ned originally on the at phase space of the Weyl-

Heisenb erg group. The entropy?? can b e useful in the

where P ; s F ; s is the SW symbol of . Now,

reconstruction pro cedure, as it is a sensitive measure of

the covariance prop erty?? can b e used to obtain the

the noise added to the system during the displacement

following expression:

and detection pro cesses. A similar situation exists also

0 0 0

in the eld of signal pro cessing [?]. There jui represents

p = d P ; sP ; s; 6.25

u u

the test signal and p is a distribution on the time-

frequency space. One can pro duce various test signals jui

where P ; s is the SW symbol of . Therefore, the

u u

and compute the corresp onding entropies S . Cho osing

op erational phase-space probability distribution p is

jui that minimizes the entropy, one obtains the optimal

given by a convolution of the two QPDs representing the

form of pattern analysis in particular, this metho d al-

quantum state of the system and the quantum ruler state

lows to achieve data compression.

of the measurement apparatus. In the particular case of

the Heisenb erg-Weyl group and for s = 0, the general

expression ?? reduces to the known result [?]

3. Harmonic expansions

2 0

0 0

W + W : 6.26 p =

u u

A useful expression for p can be derived in the

If the quantum ruler state jui = j i is the reference

following way. Using the expansion

state of the coherent-state basis, then

= j ih j = ; 1 6.27

; 6.32 = R D ; R TrD

is the SW kernel with s = 1, and

we obtain

p = h jj i = Q 6.28

is the Q function. However, except for this coincidence,

p = R hujT D T jui

the displaced pro jectors are not the SW kernels, as they

X X

do not satisfy the traciality condition. On the other hand,

jui: 6.33 = R T hujD

the functions p di er from the ma jority of QPDs, as

they are p ositive on the whole phase space which re-

ects the fact that they are measurable probabilities.

Expanding p in the basis of harmonic functions,

Usually the state jui is chosen to b elong to some com-

plete orthonormal basis fj ig which consists of energy

R Y ; 6.34 p =

eigenstates of a natural Hamiltonian of the physical sys-

tem e.g., the Fock basis for a harmonic oscillator or J

eigenstates for spin. Then there exists the relation

we identify the co ecients by means of the following

p = 1; 6.29

relation

1 u

which follows from the completeness of the basis. D T jui: 6.35 Y = hujT

0 1 0 2

Tr[ ] = jhujT T jv ij Formally,we can write

u v

u v 0

Z Z

= Y Y : 6.44

0 u 1=2

d d Y =

X X

0 0 2

Y jhuj ij ; 6.36

Certainly, the most convenientway for deducing the den-

sity matrix from the measured functions p is by cal-

u but actually Eq. ?? is more convenient for calculating

culating the co ecients R via Eq. ??.

. the co ecients

The measured functions p can b e used also for the

Equation ?? for the functions p corresp onds to

reconstruction of various QPDs which represent the den-

the expansion

sity matrix in the phase-space formulation. According

X X

u u y

to the general expression ??, the QPDs for the density

= Y D = Y D 6.37

matrix are given by the harmonic expansion

for the displaced pro jectors. It follows from the prop er-

s=2

F ; s P ; s= R Y : 6.45

ties of that the co ecients are p ositive and

p ossess the same invariance prop erties as . Using the

general result ??, we obtain the relation between the

functions p and p , corresp onding to di erent

u v

Therefore, one can just use the co ecients R calculated

quantum ruler states jui and jv i,

via Eq. ??. On the other hand, Eq. ?? can b e used

to obtain the relation b etween the QPDs P ; s and the

0 0 0

measured functions p :

p = d K ; p ; 6.38

u uv v

0 0

0 0 0

Y Y : 6.39 K ; =

P ; s= d K ; p ; 6.46 uv

0 + 0 0

p = d K ; P ; s; 6.47

4. Deducing the density matrix and quasiprobabilities

where the transformation kernels are

Knowledge of the phase-space function p allows to

i h

reconstruct the density matrix in a simple way:

0 s=2 0 u

: 6.48 Y Y ; = K

h i

: 6.40 R = d p Y

+ 0 1 0

It can b e easily shown that K ; =P ; s

Formally, we can also represent the density matrix by

where P ; s is the SW symbol of juihuj, so Eq. ?? is

means of an integral transform of the displaced pro jec-

consistent with the relation ??.

tor:

As was already mentioned, if the state jui is the refer-

ence state j i of the coherent-state basis, then =

= d r : 6.41

u u

j ih j and the measured function p coincides with

the function Q = P ; 1. Comparing the har-

This relation gives the density matrix in terms of a phase-

monic expansions ?? and ?? for the case jui = j i,

space function r , and in this sense it is the inverse of

we nd the following relation:

Eq. ??. The function r is de ned by its harmonic

expansion,

1=2

= : 6.49

u 1

r = [ ] R Y : 6.42

In this case we also obtain that the function r of

Eq. ?? is just the P function,

We also obtain the following relation b etween the func-

tions r and p ,

u u

r = P = P ; 1: 6.50

0 0 0

d r Tr[ ]; 6.43 p =

u u u u

Note that in the case of the Heisenb erg-Weyl group one

where can also calculate the QPDs using the formula ?? . 14

we nally obtain that the co ecients are indep endent 5. Examples

of the index m:

We see that the mathematical pro cedure of the recon-

= hj; ; l; 0jj; i: 6.58

struction of the density matrix and its QPDs P ; s

from the measured probability p actually consists of

j 1=2

the simple transformation ??. The mathematical to ols

For = j , one nds = . Indeed, according to

l l

one needs for this pro cedure are the harmonic functions

the de nition ?? of the SU2 coherent states, the func-

Y and the invariant co ecients and . In what

tion Q ; =hj; njjj; ni coincides with the probability

p ; to nd the displaced system in the highest spin

follows we compute the explicit form of for simple

state jj; j i. It is not dicult to see that the probability

but instructive examples of the Heisenb erg-Weyl group

p ; to nd the displaced system in the lowest spin

with jui b eing a Fock state and the SU2 group with

state jj; j i is equal to Q + ; . As an application of

jui b eing a J eigenstate.

the relation ?? , we also obtain the following expression

In the case of the Heisenb erg-Weyl group, we consider

for the SU2 Wigner function in terms of the measured

the probability p to nd the displaced initial state

probability p n,

in the Fock state jni n =0; 1; 2;:::. Then Eq. ?? can

b e rewritten in the form

4 2l +1

0 0 0

dn P n n p n ; W n=

Y ; =hnjD D D jni: 6.51

hj; ; l; 0jj; i

l=0

Using Eq. ?? , we obtain

6.59

D D D =Y ; D : 6.52

where P x are the Legendre p olynomials.

Therefore, the co ecients are given by

2 2 n

j j L j j : 6.53 =hnjD jni = exp

D. Informational completeness and unsharp

measurements

0 1=2

Of course, for n = 0 one gets =[ ] .

In the case of the SU2 group, we consider the prob-

When the question of the state reconstruction arises,

ability p ; to nd the displaced initial state in the

it is understo o d that the set of measurements one makes

J eigenstate jj; i = j; j 1;:::;j . Then Eq. ??

on an ensemble of identically prepared systems should

takes the following form

give complete information ab out the quantum state. In

particular, if one measures exp ectation values of some ob-

2j +1

Y ; = hj; jg ; D g ; jj; i:

lm lm

servables, it is natural to ask how many such observables

are needed to characterize completely the state of the

6.54

system. In this sense a set of observables, whose exp ec-

tation values are sucient to reconstruct the quantum

Using the parameterization ?? for g ; and the trans-

state or, equivalently, to distinguish between di erent

formation rule ??, we can write

states, can be considered as informationally complete.

A formal de nition is as follows [?]: A set of b ounded

op erators A = fAg on H is said to be informationally

g ; D g ; = D 0; ;D : 6.55

lm lm

m m

complete if for density matrices ; the equality of ex-

m =l

p ectation values,

Since the matrix element of the tensor op erator,

TrA=Tr A 8A 2 A; 6.60

2l +1

hj; jD jj; i = hj; ; l; m jj; i; 6.56

implies = .

2j +1

The informational completeness of p ositive op erator-

valued measures covariant with resp ect to Heisenb erg-

vanishes unless m = 0, Eq. ?? reads

Weyl, ane, and Galilei groups was recently discussed in

Ref. [?]. This sub ject was shown [?] to b e of imp ortance

2l +1

Y ; = D 0; ;hj; ; l; 0jj; i:

not only in quantum mechanics but also in signal pro- lm

cessing where a problem exists of extracting information

from nonstationary signals and images. Another inter-

Taking into account the fact that

esting feature is that b oth in quantum mechanics and in

signal pro cessing the phase-space formulation is of great

Y ; ; 6.57 D ; ; =

imp ortance for approaching this kind of problems.

2l +1 15

It would be interesting to analyze the results of the fZ B j B 2 Borel sets of X g of lo calization op erators.

present pap er from the p oint of view of informational Therefore, in the case of realistic unsharp measurements,

completeness. First, it is evident from the expansion the lo calization op erators maybe conveniently used for

analysis and reconstruction of quantum states or elec-

= TrD D 6.61

tronic signals and images.

that the orthonormal set fD g of the tensor op erators

is informationally complete. Corresp ondingly, the set

VI I. CONCLUSIONS

f ; s j 2 X g of the SW kernels for each s is also

informationally complete. This fact is re ected by the

In the present pap er we prop ose a simple algorithm for

inverse Weyl rule written as

constructing the SW kernels which implement the lin-

ear bijective mapping between Hilb ert-space op erators

= d Tr[ ; s] ; s: 6.62

and phase-space functions for physical systems p ossess-

ing Lie-group symmetries. The constructed kernels are

In other words, the density matrix can be uniquely re-

manifestly covariant under the action of the corresp ond-

constructed from its s-parameterized QPD P ; s =

ing dynamical symmetry group and satisfy the traciality

Tr[ ; s]. From the practical p oint of view, the im-

condition which ensures that quantum exp ectations can

p ortant thing is the informational completeness of the

b e represented by statistical-likeaverages over the phase

set f j 2 X g of the displaced pro jectors for

space. Adding the noncommutativetwisted pro duct that

any jui 2 H . This fact was formally proved in Ref.

equips phase-space functions with the algebraic structure

[?]. Here, we presented a simple algorithm based on

of quantum op erators, an autonomous phase-space for-

the metho d of harmonic expansion for the reconstruc-

mulation of quantum mechanics is develop ed.

tion of the density matrix from the measurable probabil-

It turns out that the concept of phase space naturally

ities p = Tr[ ]. This reconstruction pro cedure

u u

emerges in the ma jorityofschemes prop osed for the re-

clearly implies the informational completeness of the set

construction of quantum states as well as in the standard

f j 2 X g.

metho ds of signal analysis. In particular, we fo cus on the

One of the useful features of the metho d of displaced

metho d based on measurements of displaced pro jectors

pro jectors is the ability to take into account in a sim-

and develop its general group-theoretic description. We

ple way the unsharpness of a realistic measurement. Of

do so by applying the same technique of harmonic ex-

course, it is imp ossible in practice to make a completely

pansions on the phase space that was used for the con-

accurate displacement to a sp eci ed p oint on the phase

struction of the SW kernels. The problem of the state

space. For example, one should take into account the

reconstruction is also approached using the concept of in-

phase and intensity uctuations of a classical microwave

formational completeness, and the role of lo calization op-

source that displaces the quantum state of the radiation

erators in describing realistic measurements is discussed.

eld in a cavity, or instabilities of a classical driving eld

that displaces the motional state of a trapp ed ion. Simi-

larly, the so-called coarse graining problem arises in radar

ACKNOWLEDGMENTS

analysis due to frequency instabilities of the test signal

or uncertainties in timing of signal initiation. As a re-

sult, the probabilities p should be integrated over

This work was supp orted by the Fund for Promotion of

the variation range. This yields the exp ectation value of

Research at the Technion, by the Technion VPR Fund|

the lo calization op erator de ned by[?]

Promotion of Sp onsored Research, and by the Technion

VPR Fund|The Harry Werksman Fund.

Z f = d f ; 6.63

u u

where f is a lo calization function. Z f has a purely

discrete sp ectrum, is b ounded when f 2 L X; , k 1,

and is self-adjoint when f is real. In particular, let B be

[email protected]

a region more sp eci cally, a Borel set in X , with the

[email protected]

characteristic function that equals 1 for 2 B

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