PHYSICAL REVIEW A
VOLUME 57, NUMBER 2
FEBRUARY 1998
Generalized coherent states are unique Bell states of quantum systems with Lie-group symmetries C. Brif,* A. Mann,† and M. Revzen‡ Department of Physics, Technion—Israel Institute of Technology, Haifa 32000, Israel ~Received 9 July 1997! We consider quantum systems, whose dynamical symmetry groups are semisimple Lie groups, which can be split or decay into two subsystems of the same symmetry. We prove that the only states of such a system that factorize upon splitting are the generalized coherent states. Since Bell’s inequality is never violated by the direct product state, when the system prepared in the generalized coherent state is split, no quantum correlations are created. Therefore the generalized coherent states are the unique Bell states, i.e., the pure quantum states preserving the fundamental classical property of satisfying Bell’s inequality upon splitting. @S1050-2947~98!07001-2# PACS number~s!: 03.65.Bz, 03.65.Fd
Entanglement is the main quantum mechanical feature that distinguishes between quantum and classical physics. A state of a quantum system, consisting of two subsystems, is called entangled if it cannot be represented as a direct product of states of the subsystems. The nonclassicality of such a situation is expressed mathematically by the violation of Bell’s inequality @1,2#. More specifically, let us consider a quantum system A ~e.g., a mode of the quantized radiation field or a spin-j particle!, which is split or decays into two subsystems B and C @3#. A pure quantum state that upon splitting will not violate Bell’s inequality ~i.e., a quantum state that possesses the basic attribute of classical physics! has been called the Bell state @4#. It has been proven @4–6# that a state that factorizes upon splitting into a direct product cannot violate Bell’s inequality, while a state that gives rise to an entangled ~i.e., nonproduct! state can always, by a suitable choice of local apparatus, be made to violate the inequality. Therefore the Bell state of the system A must factorize upon splitting into the direct product of states of the subsystems. In 1966 Aharonov et al. @7# showed that the only states of the single-mode quantized radiation field that factorize upon splitting are the Glauber coherent states ~CS! @8#. Therefore the Glauber CS are the unique Bell states of the quantized radiation field @4#. The Glauber CS are known to have a number of classical characteristics. Actually, the CS were discovered in 1926 by Schro¨dinger @9#, who looked for a wave packet with minimum possible dispersions that will preserve its form while moving along a classical trajectory. The Glauber CS are such wave packets for a quantum harmonic oscillator ~which is the mathematical model of the single-mode quantized radiation field!. One of the criteria used by Glauber @8# was to choose the quantum state of the radiation field produced by a classically prescribed current. Also, the Glauber CS have the property of maximal coherence ~in accordance to their name!, i.e., their normal-ordered correlation functions of all orders factorize @10#. *Electronic address:
[email protected] † ‡
Electronic address:
[email protected] Electronic address:
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1050-2947/98/57~2!/742~4!/$15.00
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Formally, the Glauber CS ua& ( a PC) can be defined in two equivalent ways: ~i! as the eigenstates of the boson annihilation operator a, aua&5aua&
~1!
~this property actually implies another possible definition of ua& as minimum-uncertainty states!, and ~ii! as the states produced by the action of the displacement operator D( a )5exp(aa†2a*a) on the vacuum, u a & 5D ~ a ! u 0 & .
~2!
Various generalizations of the Glauber states to quantum systems other than the harmonic oscillator have been proposed. The most useful of them is the group-theoretic generalization of CS developed by Perelomov @11# and Gilmore and co-workers @12,13#. The generalized CS for a system, possessing a dynamical symmetry Lie group, are produced by the action of group elements on a reference state in the Hilbert space. For the Weyl-Heisenberg group W, which is the dynamical symmetry group of a quantum harmonic oscillator, group elements are represented by the displacement operator D( a ), the reference state is the vacuum, and the general definition of the CS reduces to Eq. ~2!. Under the action of group elements, the generalized CS transform among themselves. For the evolution operator which is an element of the dynamical symmetry group, the generalized CS will evolve along a classical trajectory in the phase space of the group parameters. While the generalized CS possess this classical feature, not all the properties of the Glauber states are preserved under the Gilmore-Perelomov generalization. For example, not every generalized coherent state is a minimum-uncertainty state. However, we will show in the present paper that the main classical attribute of the Glauber CS is preserved under the group-theoretic generalization. Specifically, we will consider quantum systems, whose dynamical symmetry groups are semisimple Lie groups, and prove that the only states of such a system that factorize upon splitting are the generalized CS. Therefore the generalized CS are the unique Bell states of quantum systems with semisimple Lie-group symmetries. In order to il742
© 1998 The American Physical Society
57
GENERALIZED COHERENT STATES ARE UNIQUE BELL . . .
lustrate this general result, we will consider the decay of a spin-j particle, with SU~2! as the dynamical symmetry group. The theory of the generalized CS is well developed and elaborated ~e.g., see Refs. @11–13#!. Here we recapitulate some basic results which are essential for our discussion. Let G be an arbitrary Lie group and G L its unitary irreducible representation acting on the Hilbert space HL . By choosing a fixed normalized reference state u C 0 & PHL , one can define the system of states $ u C g & % , u C g & 5G L ~ g ! u C 0 & ,
gPG
~3!
which is called the coherent-state system. The isotropy ~or maximum-stability! subgroup H,G consists of all the group elements h that leave the reference state invariant up to a phase factor, G L ~ h ! u C 0 & 5e i f ~ h ! u C 0 & ,
hPH.
~4!
For every element gPG, there is a unique decomposition of g into a product of two group elements, one in H and the other in the coset space G/H, gPG, hPH, VPG/H.
g5Vh,
~5!
It is clear that group elements g and g 8 with different h and h 8 but with the same V produce CS which differ only by a phase factor: u C g & 5e i d u C g 8 & , where d 5 f (h)2 f (h 8 ). Therefore a coherent state u V & [ u C V & is determined by a point V5V(g) in the coset space G/H. This coset space is actually the phase space of the system. One can see that the choice of the reference state u C 0 & determines the structure of the coherent-state system and of the phase space. In order to make our discussion more concrete, we will consider the case when G is a semisimple Lie group. ~Continuous representations of noncompact semisimple groups could be problematic from the point of view of mathematical rigorousness of our results. Therefore we restrict our discussion to compact groups and to discrete representations of noncompact groups.! The corresponding Lie algebra G has a Cartan subalgebra H, and the set of non-zero roots is denoted by D @14#. We use the standard Cartan-Weyl basis $ H i ,E a % ; for H i PH and a , b PD, the commutation relations are @14#
@ E a ,E b # 5
@ H i ,H j # 50,
~6a!
@ H i ,E a # 5 a ~ H i ! E a ,
~6b!
H
0 Ha
if a 1 b 50
N a,bE a1b
the dynamical symmetry group of a spin-j particle. The su~2! Lie algebra is spanned by the three operators $ J 0 ,J 1 ,J 2 % , @ J 0 ,J 6 # 56J 6 ,
~6c!
if a 1 b PD.
The operators H i , which constitute the Cartan subalgebra H, may be taken as diagonal in any unitary irreducible representation G L , while E a are the ‘‘shift operators.’’ One can always choose the representation G L such that H †i 5H i and E a† 5E 2 a . Every group element gPG can be written as the exponential of an anti-Hermitian complex linear combination of H i and E a . As an illustrative example, we consider the most elementary compact non-Abelian simple Lie group SU~2!, which is
@ J 1 ,J 2 # 52J 0 .
~7!
The Casimir operator J 2 5J 20 1(J 1 J 2 1J 2 J 1 )/2 for any unitary irreducible representation is the identity operator times a number: J 2 5 j( j11)I. Thus an irreducible representation G j of SU~2! is determined by a single number j that can be a non-negative integer or half integer: j50, 21 ,1, 32 ,2,... . The representation Hilbert space H j is u j,m & spanned by the orthonormal basis (m5 j, j21, . . . ,2 j). The Hermitian operator J 0 is diagonal, J 0 u j,m & 5m u j,m & , while J 1 and J 2 are the raising and lowering operators, respectively, J 6 u j,m & 5 @ ( j7m)( j 6m11) # 1/2u j,m61 & and J †1 5J 2 . It is physically sensible to choose the reference state u C 0 & to be the ground state of the system, which is mathematically an ‘‘extremal state’’ in the Hilbert space HL . We will see that this choice of the reference state determines the classical properties of the CS. This is related to the fact that the coset space G/H in this case is a symmetric space that can be considered as the phase space of a classical dynamical system. This ‘‘extremal state’’ is the lowest-weight state u L,2L & @15#, which is annihilated by all the lowering operators E a with a ,0, E a u L,2L & 50,
a ,0
~8!
and mapped into itself by all the diagonal operators H i , H i u L,2L & 5L i u L,2L & ,
H i PH.
~9!
The last equation means that the Lie algebra of the maximum-stability subgroup H is just the Cartan subalgebra H. Therefore elements V of the coset space G/H are the generalized displacement operators of the form
F(
V5exp
a .0
G
~ h a E a 2 h a* E 2 a ! .
~10!
Here the parameters h a are complex numbers and the summation ( a .0 is restricted to those raising operators E a ( a .0) which do not annihilate the reference state, E a u L,2L & Þ0. We also use the Baker-Campbell-Hausdorff formula @16#,
F(
exp
if a 1 b Þ0 and a 1 b ¹D
743
a .0
~ h a E a 2 h a* E 2 a !
G
F ( G F( G F
5exp
a .0
t a E a exp
i
G
g i H i exp 2 ( t a* E 2 a . a .0
The relation between t a , g i , and h a can be derived from the matrix representation of G; for details see Ref. @13#. The generalized CS uL,V& are given by
F( G
u L,V & 5V u L,2L & 5N exp
where
a .0
t a E a u L,2L & , ~11!
744
C. BRIF, A. MANN, AND M. REVZEN
F( G
N5exp
i
g iL i
F(
~12!
is the normalization factor. In the case of the SU~2! group, the lowest-weight state is u j,2 j & , which is annihilated by J 2 . The maximum-stability subgroup H5U(1) consists of all group elements h of the form h5exp(idJ3). The coset space is SU~2!/U~1! ~the sphere!, and the coherent state is specified by a unit vector n5 ~ sin u cos w ,sin u sin w ,cos u ! .
~13!
Then an element V of the coset space can be written as V5exp~ j J 1 2 j * J 2 ! ,
~14!
where j 52( u /2)e 2i w . The SU~2! CS are given by u j, z & 5V u j,2 j & 5 ~ 11 u z u 2 ! 2 j exp~ z J 1 ! u j,2 j & , ~15!
where z 5( j / u j u )tanuju52tan(u/2)e 2i w . Now we consider a quantum system A, whose dynamical symmetry group is G, that is split or decays into two subsystems B and C of the same symmetry. The requirement that B and C must have the same symmetry as A is of crucial importance. For example, if one considers a boson realization of a Lie group G, then boson splitting a †A 5 m a †B 1 n a †C ( u m u 2 1 u n u 2 51) cannot describe a situation when the system A, with G as the symmetry group, is split into two subsystems B and C, which also possess the symmetry of G. The commutation relations ~6! are satisfied by operators describing each of the three systems, if and only if E A a 5E B a ^ I C 1I B ^ E C a ,
~16a!
H Ai 5H Bi ^ I C 1I B ^ H Ci ,
~16b!
where a PD and H Xi PH, X5A,B,C. In the case of the SU~2! group, conditions ~16! are equivalent to the rule for the addition of angular momenta, JA 5JB ^ I C 1I B ^ JC .
~17!
First, it is necessary that the lowest-weight state ~which usually is the ground state of the quantum system! factorize: u L A ,2L A & A 5 u L B ,2L B & B ^ u L C ,2L C & C .
^ exp
~19!
~20!
If the representations of the subsystems B and C satisfy the condition ~19!, then any generalized coherent state factorizes upon splitting. Indeed, using Eqs. ~16a! and ~18!, we obtain
F( F(
u L A ,V ~ h ! & A 5exp
5exp
a .0
a .0
G G
~ h a E A a 2 h a* E A† a ! u L A ,2L A & A ~ h a E B a 2 h a* E B† a ! u L B ,2L B & B
G
~ h a E C a 2 h a* E C† a ! u L C ,2L C & C ,
where we also use the basic property of the exponential function: exp(x1y)5exp(x)exp(y), if @ x,y # 50 ~of course, operators describing different systems commute!. Then we get u L A ,V ~ h ! & A 5 u L B ,V ~ h ! & B ^ u L C ,V ~ h ! & C .
~21!
We see that the coherent amplitudes h a are the same for the system A and for the subsystems B and C. In particular, we have for the spin u j A , z & A5u j B , z & B ^ u j C , z & C ,
~22!
i.e., z A 5 z B 5 z C . Note that the situation is rather different for the Glauber coherent state u a & A of the radiation field, split by a half-silvered mirror. Then one obtains @7# u a & A 5 u ma & B ^ u n a & C ,
u m u 2 1 u n u 2 51.
~23!
The reason for this distinction lies in the differing structures of the nilpotent Weyl-Heisenberg group W and a semisimple group G. The CS cannot be defined and the condition ~20! cannot be satisfied for a spin-zero particle, so its decay cannot be described in the framework of our formalism. The case of a spin-half particle, decaying into spin-half and spin-zero particles, is trivial. For j>1, the lowest and highest spin states u j,2 j & and u j, j & , which factorize upon splitting, provided the condition ~20! is satisfied, are actually particular cases of the SU~2! CS. In order to illustrate the general result, we consider the most elementary nontrivial example of a spinone particle ~the system A! decaying into two spin-half particles ~the subsystems B and C!. In this elementary case u 1,1& A 5 u 21 , 12 & B ^ u 21 , 12 & C ,
~24a!
u 1,0& A 5 ~ u 21 , 12 & B ^ u 21 ,2 21 & C 1 u 21 ,2 12 & B ^ u 21 , 12 & C )/&, ~24b! u 1,21 & A 5 u 21 ,2 12 & B ^ u 21 ,2 21 & C .
~24c!
The SU~2! CS can be written as u 21 , z & 5
u 21 ,2 12 & 1 z u 21 , 12 &
A11 u z u 2
~25!
for j51/2 and u 1,z & 5
In particular, for the SU~2! group this condition reads j A5 j B1 j C .
a .0
~18!
Acting on both sides of this equation with H Ai PH and using Eq. ~16b!, we obtain the necessary condition for the factorization: L Ai 5L Bi 1L Ci .
57
u 1,21 & 1& z u 1,0& 1 z 2 u 1,1& 11 u z u 2
~26!
for j51. Let us assume that the spin-one particle A was prepared before the decay in the coherent state ~26!. Then, substituting Eqs. ~24! into Eq. ~26!, we obtain u 1,z & A 5
u 21 ,2 12 & B 1 z u 21 , 12 & B
A11 u z u 2
5 u 21 , z & B ^ u 21 , z & C .
^
u 21 ,2 21 & C 1 z u 21 , 12 & C
A11 u z u 2 ~27!
The direct products ~24a! and ~24c! are obtained as particular cases of Eq. ~27! for z →` and z 50, respectively.
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GENERALIZED COHERENT STATES ARE UNIQUE BELL . . .
We can also prove that the generalized CS are the only states which factorize upon splitting. We first note that the states of the orthonormal basis are obtained by applying the raising operators to the lowest-weight state one or more times: ~ E a ! p u L,2L & 5 u L,2L1p a & 3factor,
~28!
where a .0 and pPN. Therefore any state uF& in the Hilbert space can be obtained by applying a function of the raising operators to the lowest-weight state: u F & 5 f ~ $ E a % ! u L,2L & ,
a .0.
~29!
For example, for the SU~2! group one has
S D
2j u j,m & 5 j1m
1/2
~ J 1 ! j1m u j,2 j & , ~ j1m ! !
~30!
and u F & 5 f (J 1 ) u j,2 j & . If the state u F A & A factorizes upon splitting, u F A& A5 u F B& B ^ u F C& C ,
~31!
the following functional equation must be satisfied: f A~ $ E Aa% ! 5 f B~ $ E Ba% ! f C~ $ E Ca% !
~32!
~here and in what follows we consider only the raising operators, i.e., a .0!. Using Eq. ~16a! ~for the sake of simplicity, we omit the identity operators!, we obtain f A ~ $ E B a 1E C a % ! 5 f B ~ $ E B a % ! f C ~ $ E C a % ! .
~33!
Using the same method as in Ref. @7#, we can easily prove that Eq. ~33! has the unique solution—the three functions f X (X5A,B,C) must be exponentials:
F(
f X ~ $ E X a % ! 5 f X ~ 0 ! exp
a .0
G
t aE Xa ,
~34!
@1# J. S. Bell, Physics ~Long Island City, NY! 1, 195 ~1964!. @2# There exist many variants of Bell’s inequality; the one used in Refs. @4–6# is due to J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 ~1969!. @3# The radiation field can be split by a half-silvered mirror. Such a lossless beam splitter actually has two input and two output ports. Therefore, when speaking about the radiation field splitting, we assume that the vacuum enters the second input port. @4# A. Mann, M. Revzen, and W. Schleich, Phys. Rev. A 46, 5363 ~1992!. @5# A. Peres, Am. J. Phys. 46, 745 ~1978!. @6# N. Gisin, Phys. Lett. A 154, 201 ~1991!. @7# Y. Aharonov, D. Falkoff, E. Lerner, and H. Pendleton, Ann. Phys. ~N.Y.! 39, 498 ~1966!. @8# R. J. Glauber, Phys. Rev. 131, 2766 ~1963!. @9# E. Schro¨dinger, Naturwissenschaften 14, 664 ~1926!.
745
where f X (0) is a normalization factor. Here t a are complex parameters which are the same for the three systems A, B, and C. For example, in the case of the SU~2! group, we find f X ~ J X1 ! 5 ~ 11 u z u 2 ! 2 j X exp~ z J X1 ! ,
~35!
where X5A,B,C and z PC. @For compact groups, e.g., for SU~2!, the representation spaces HL are finite dimensional and t a can acquire any complex value. For noncompact groups the representation spaces HL are infinite dimensional and admissible values of the parameters t a are determined by the normalization condition; e.g., for SU~1,1! the coherent-state amplitude z is restricted to the unit disk, u z u ,1.# Recalling Eq. ~11! @or Eq. ~15! for the SU~2! case#, we see that the operator-valued function f X ( $ E X a % ) for each of the three systems ~A, B, and C! is precisely the function that produces the generalized CS. @The constant factor f X (0) is recognized as the normalization factor N of Eq. ~12!#. This completes the proof of uniqueness. In conclusion, we have proven that, for quantum systems with semisimple Lie-group symmetries, the generalized CS are the only pure states which factorize upon splitting. Therefore the generalized CS are the unique Bell states for these systems, i.e., the unique quantum states which, when split, do not give rise to any quantum correlations, and so Bell’s inequality is never violated. Our result shows that the group-theoretic generalization of the Glauber CS preserves their main classical attribute—fulfillment of Bell’s inequality upon splitting. C.B. gratefully acknowledges the financial support from the Technion and the Gutwirth family. A.M. and M.R. were supported by the Fund for Promotion of Research at the Technion, by the Technion VPR Fund, R. and M. Rochlin Research Fund, and by GIF ~German-Israeli Foundation! for Research and Development.
@10# However, the Glauber states are not the only maximally coherent states of the radiation field. @11# A. M. Perelomov, Commun. Math. Phys. 26, 222 ~1972!; Usp. Fiz. Nauk 123, 23 ~1977! @Sov. Phys. Usp. 20, 703 ~1977!#; Generalized Coherent States and Their Applications ~Springer, Berlin, 1986!. @12# R. Gilmore, Ann. Phys. ~N.Y.! 74, 391 ~1972!; Rev. Mex. Fis. 23, 142 ~1974!; J. Math. Phys. 15, 2090 ~1974!. @13# W.-M. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod. Phys. 62, 867 ~1990!. @14# A. O. Barut and R. Raczka, Theory of Group Representations and Applications, 2nd ed. ~World Scientific, Singapore, 1986!, Chap. 1. @15# For compact groups one could just as well choose the highestweight state uL,L&. @16# S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces ~Academic Press, New York, 1978!.
