- Email: [email protected]
Entanglement, information theory and Bell's inequality
2 February
1998
PHYSICS
ELSEWIER
LETTERS
A
Physics Letters A 238 ( 1998) 85-89
Entanglement, information theory and Bell’s inequality A. Mann a*1,M. Revzen a,b*2,E. Santos b,3 a Depariment of Physics, Technion, Haifa 32000, Israel h Departamento de Fisica Moderna. Universidad de Cantabria. Santander; Spain Received 2 December
1996; revised manuscript received 12 November 1997; accepted Communicated by P.R. Holland
for publication
17 November
1997
Abstract Information theory is used to quantify entanglement of two spin-l /2 particles or two photons. Noise contaminating a pure entangled (singlet) state is parametrized to allow convenient comparison with experimental studies of possible violation of Bell’s inequality for such states. @ 1998 Elsevier Science B.V. PACS: 03.65.-W:
03.65.8~
1. Introduction Information theory [ 11 quantifies the intuitive notion of information gained upon measuring. Thus when the result of a measurement can be predicted with certainty, the information gain is nil. If we consider two entries (e.g. two particles) A and B whose properties can be simultaneously measured yielding, say, possible discretized values a and b, respectively, with joint probability (for the state considered) p (a, b), the “self-information” gained upon obtaining aforAandbforBis[l-31 f(a,b)
=
-lnp(u,b).
(1)
The central quantity in information theory is the entropy, S, which is the mean value of I [ l-31. Within classical physics, where the joint probability can always be defined, we have the “obvious” result that a ’ E-mail: [email protected] * E-mail: [email protected] 1 E-mail: [email protected]. 03759601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00898-O
complete knowledge of the composite system, A and B, i.e. S( A, B) = 0, implies a complete knowledge of its constituents: S(A) = S(B) = 0. A strange peculiarity of the extension of the above to quantum physics via von Neumann’s density matrix [ 3,4] is that in quantum mechanics “complete knowledge” of the composite system (i.e. the system is in a pure state) does not imply complete knowledge of its constituents. Pure states with this property are called entangled states [ 3,5]. Fully entangled states contain no knowledge whatever about their constituents. Entangled pure states have the remarkable property of allowing violation of Bell’s inequality [ 3,6,7] It is, thus, of interest to extend this to non-idealized, i.e. non-pure (“mixed”) quanta1 states. In the next section we present our version of using information theory to extend the definition of entanglement to “contaminated states”. Since we deal primarily with photonic states, we simulate the contamination of the pure fully entangled state by random sources of one and two photons, viz., we consider the so-called Werner states [ 81 for the two photons’ contamination which, in addition,
86
A. Mann et al. /Physics
is admixed with single photon noise. In Section 3, we consider in detail such a “contaminated density matrix” where we parametrize the contaminating noise in a particuiar manner which allows, in this parameter space, to define a region where the state is entangled, the noise notwithstanding, thereby rendering such a state a candidate for observing Bell’s inequality violation. The boundary in the contaminating parameters’ space determined in this way, based as it is on information theory, turns out to be almost identical with a similarly aimed parametrization used earlier by one of us [9]. In Ref. [9] direct violation of Bell’s inequality was contemplated whereas in this Letter we initiate such a study on the basis of information theoretic approach. Section 4 contains the experimental meaning of our parameters with a brief summary and comparison with the results of Ref. [ 91. The last section contains some concluding remarks.
2. Correlations
in quantum
information
theory
= - xp(a,b) (u.h)
lnp(u.6).
(21
Here a (b) designates a possible value for the A (B) entry (e.g. the particles’ position in a discretized phase space). p( a, b) is the joint probability for a and b. We also have the entropy of A (and, analogously, of B) S(A)
= - xp(a) ‘,
hip(a).
(3)
with p(n)
= c
p(a.
6).
p(4b) = pfa>
S(A(B)
with
= S(AB)
- S(B)
3 0,
(7)
with a corresponding expression for S( B/A). Eq. (6) is recognized as Bayes’s theorem in probability theory with p( al b) the conditional probability of a, given 0. The conditional entropy (Eqs. (5) and (7) ) is obviously non-negative, because p( 6) 3 p(u, b) (cf. Eq. (4) ) , i.e. p( al 6) 6 1. Thus we have a classically valid inequality, 2S(AB)
3 S(A)
+ S(B).
(8)
The corresponding, or better perhaps, quanta1 formulae are formulated in terms mann’s [4] density matrix. Thus with density matrix of the composite system, (quantum) is given by
S(A)
= -Trp(AB) = -Trp(A)
lnp(AB), Inp(A),
the related, of von Neup( AB), the the entropy
(9) (10)
with (e.g., Wehrl in Ref. [ I ] ) p(A)
= Trsp(AB).
(11)
When considered in the representation is diagonal - recall it is a non-negative erator [ 3,4] - we have S(AB)
= -Cp(ub) ah
lnp(ab),
where p( AB) hermitian op-
(12)
where now the sum is over all the states of the composite system, designated ab, and the eigenvalues of the density operator, pfub), give the probability that the composite system is in the state labeled by (ab). In the trivia1 case, where A and B are uncorrelated, then, classically ~(a.
b) = p(a)p(b),
while, quantum p(AB)
= - c p(u, b) lnp(ulb) (1I.h)
(6)
(4)
h
p(a) is the probability for A yielding a regardless of the B reading. A conditional entropy is defined by SCAIB)
b)lp(b).
Hence,
S(AB)
Consider our previously defined composite system consisting of two entries: A and B. Classically some basic information theoretic relations hold. Thus the basic quantity in information theory is the entropy of the composite system. It is given by [ I] S(A,B)
Letters A 238 (I 998) 85-89
(5)
(13)
mechanically,
= p(A)p(B).
(14)
In this factorized case all classical considerations hold for the quantum case - inclusive of the existence of joint probabilities and conditional entropy.
A. Mann
For both the classical and the quantum entropies we have [ 1,3,101 S(AB)
< S(A)
+ S(B).
et al. /Physics
mechanical
(15)
When A and B are uncorrelated the maximal value is attained both in the classical and the quanta1 cases [ lo]. In the correlated case, although classically S( Al B) is still well defined via Eq. (5)) the quanta1 expression for S( AIB) cannot be, generally, simply related to S(A) or S(B). A relation such as Eq. (5) has no simple quanta1 transcription (such as Eq. (9) ) when p( AB) is not diagonal in a product basis [ 111. In this case quantum mechanics harbours subtle correlations that have no clear classical counterparts. Thus, in quantum mechanics, generally, S(AIB) is a complicated entity. Indeed such is the case whenever we consider quantities for which classical information analysis requires the use of Bayes’s theorem (Eq. (6) ) . In the case under study here, correspondence to classical theory requires a “conditional” density matrix, p( AIB), which, by formal analogy with Eq. (5), should allow an expression such as S(AIB)
= -Trp(AB)
lnp(A(Z3).
I$(A,B))
= lrcI(A,B))(rll(A,B)j, = (I TA~B) - I Ms))/~~
(17) (18)
in obvious notation. In this case S(AB) = 0, it is the entropy of a pure state. But S(A) = ln2, as p(A) is completely random; hence the inequality (7) is violated. Classical physics implies this inequality (Eq. (7)), hence a sufficient condition for a state (i.e. a density matrix) to have distinctly quantum correlations (i.e. exhibiting entanglement which has no classical analogue) is 2S(AB)
< S(A)
+ S(B),
87
i.e. to have a negative S( Al B). This may thus be used as a measure for entanglement. Hence our criteria for entanglement with S(A) = S(B) = S(I), S( AB) = S(Zf) is S(I)
> S(II).
(20)
Now the violation of Bell’s inequality is possible only for states with distinctly quantum correlations [ 6,7]. In fact the violation may be taken as the fingerprints of quantum correlations. We are thus motivated to study states where “pure” quantum entanglement is admixed with some noise but still allowing a violation of Bell's inequality. In, perhaps, its simplest form this leads to states involving two parameters designated CYand L’ allowing admixtures of singles ((Y) and pairs (u) of photons. We now use our criterion, Eq. (20), to determine the boundary in the a,~: plane within which pure quantum correlations are assured. This boundary, based on information theory analysis, gives (as is shown in the next section) results essentially similar to the boundary obtained previously [ 91 by a different reasoning.
(16)
Such a quantity was indeed defined, e.g. in the Oviedo workshop [ lo], but via a rather unwieldy formula. Perhaps a more physical manifestation of this difficulty is seen by noting that S(AIB) as defined by Eq. (7) can, in quantum mechanics, be negative [ lo], thereby precluding a “density matrix” interpretation to p( A(B), since the eigenvalues of a density matrix must be non-negative and bounded by unity, implying (Eq. ( 16)) S(A(B) 3 0. An example for this is the case of two spin-half entries p(AB)
Letters A 238 (I 998) 85-89
(19)
3. Parametrized
source state
To allow a convenient comparison with experimental setups [ 121 and previous work [ 91 we consider a source state from which photons are propagating to a left detector, A, and a right detector, B. The idealized source is a maximally entangled state given by (cf. Eq. (18)) Ps =
by
(21)
IS)N~
1s)= (IRA&l)
-
ILA’h))/fi>
(22)
where R (L) designates right (left) handed polarization and the subscripts A and B indicate which way the photon is headed. These subscripts will be omitted henceforth when no confusion is likely to arise, always keeping the first symbol for the A photon and the second for the B. The contaminated source is given by the state p(AB) +
= alups (1
-~>[(pApVS
+
(1
-
u)pmI
+pVAPB)/21.
Here the random source of two photons is
(23)
A. Mann et al. /Physics
88
Letters A 238 (1998) 85-89 1
+ ]LR)(RL] + ILL)(LLl)/4.
v
(24)
0.8.
The random sources for single photons are PA
=
(IR)(RI
+
and the vacuum (no photons) PVA
term is
0.4.
(35)
IOA)(OAI.
=
The last two terms have, of course, corresponding terms indexed B. When (Y= cy*, u = u*, and CX,u < 1. cy=u= 1 is the “pure” entangled state while LY= 0 gives the one photon random state, CY= 1, u = 0 give a random two photons state. Thus our Hilbert space is nine-dimensional, e.g. IRR), (LL), IRL), )LR), IRO), ILO), IOR), IOL), 100). The last state, IOO), is not included in the density matrix as it gives rise to no readings in either detector. Direct evaluation of the matrix elements of p (Eq. (23)) and then diagonalizing it gives for the entropy of the composite system -S(AB) +
=
3cu(l -u) 4
lna(l
-u) 4
(u( 1 + 3u) ,n cy( 1 + 3u) 4 4
+(1
-CU)lnF,
while evaluation yields -S(A)=2
.71
0.6.
(251
If3(‘51)/2,
(1 -al
(27)
I
0’.
0.2
0.4
0.6
0.8
a
1
Fig. 1. Parameter space of a spin singlet contaminated by random two particle and single particle states. The region to the right of the continuous (dashed) curve represents source states violating the classical entropy inequality (the Bell inequality [ 91 ). The limiting values are indicated.
right in another direction, ~$2. We assume ideal polarizers and detectors, because we wish to focus on noise contaminating the source (i.e. in the preparation of the correlated two particle system) - not on the noise that enters in the measurement. The calculations of the probabilities are straightforward [ 91: We evaluate the trace of the product of the corresponding projection operator with the density matrix of the source state, Eq. (23). We get for the coincidence detection probability
of the entropy of the A subsystem
l+alnl+Cr
fAB(h~2b’y[~
-
4
+
I-ff 2
= -S(B). The boundary, in the parameter glement is obtained by equating is depicted in Fig. 1: the region curve is the quanta1 region, i.e. analogue exists.
1 --cy In 2
+~cos(2h
-2’$2)1/4,
(2%
and for the single probabilities (28)
space, for the entanS( AB) = S(A). This above the continuous to which no classical
PA = PB = ( 1 + C-k,)/4 (independent
of 41 and 42 ) .
If these results are put into Bell’s inequality PA
+&l
>
+PAB(~IY(P;)
pAB(4I.42)
+ -
~A&#+&).
(30) [ 131
pAB(d’:.4’2)
(31)
4. Bell’s inequality In order to relate to experiments we should calculate the results of measuring the polarization of the photon in a given direction, say at an angle 41 with the vertical, and the polarization of the photon on the
We get that the inequality may be violated for an optimum choice of the angles provided the parameters fulfill [9] ffu > I/&-.
(32)
A. Mann
et al. /Physics
The region of violation is shown in Fig. 1. We see that this region almost coincides with the region where the classical entropy inequality, Eq. (8)) is violated.
Lefters A 238 (1998)
III
C.E.
89
85-89
Shannon, W. Weaver,
Communication
The Mathematical
Theory
(Univ. of Illinois Press, Urbana,
1949)
of
;
A. Wehrl. Rev. Mod. Phys. 50 ( 1978) 221; W.T. Grandy Jr.: Resource Lett. ITP-I.
5. Conclusions
(1997) SM.
The concept of entanglement for pure quantum states was introduced already in the early days of quantum mechanics, but the use of the concept for mixed states is relatively recent. Several definitions have been put forward to generalize entanglement to mixed states (some of them may be found in Refs. [ 5,141). Most obvious are the violation of the entropy inequality (advocated most recently in Ref. [ IO] ) and the violation of Bell’s inequality. Here we showed that these two definitions yield similar results, at least for a system of two photons (or spin l/2 particles) correlated via their polarizations (or spins). We have considered a density matrix for a particle pair, which represents realistically what is produced in actual experiments.
121S.L.
Bamett, S.J.D. Phoenix, Phys. Rev. A 40 Braunstein, CL.
662; Ann. Phys. (N.Y)
202 (1990)
2404.
22.
141 J. von Neumann. The Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, 1955). ISI
A. Shimony, in: The Dilemma
of Einstein.
Podolsky and
Rosen - 60 Years Later, ed. A. Mann, M. Revzen (Institute of Phys. Publ., Bristol, 1996). 195. 161 J.S. Bell, Physics I (1964) 171 V. Capasso, D. Fortunato, E Selleri, Int. J. Theor. Phys. 7 (1973)
319;
N. Gisin, Phys. Lett. A 154 (1991)
201:
A. Mann, M. Revzen, W. Schleich. Phys. Rev. A 46 ( 1992) S363. ISI
R.F. Werner, Phys. Rev. A 40 ( 1989) 4277.
191
E. Santos, Phys. Rev. A 46 ( 1992) 3646; Phys. Lett. A 2 12 (1996)
IO.
N.J. Cerf, C. Adami,
in: 2nd Int. Symp. on Fundamental
Problems in Quantum Physics. Oviedo, A. van der Metwe; Phys. Rev. A 55 A. Mann. K. Nakamura.
We acknowledge financial support from the Spanish Ministry of Education, through the programme of Sabbatical in Spain, which allowed a four months stay of one of us (M.R.) in the University of Cantabria. A.M and M.R. were supported by the Fund for Promotion of Research at the Technion, by the VPR Fund - R.&M. Rochlin Research Fund and by GIF - German-Israeli Foundation for Research and Development. M.R. acknowledges with thanks the help of Dr. A. Shiekh.
( 1989)
Caves, Phys. Rev. Lett. 61 ( 1988)
131 A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993).
1101
Acknowledgement
Am. J. Phys. 65
466;
1996, ed. M. Ferrero,
( 1997)
337 I,
M. Revzen. J. Phys. A 25 ( 1992)
L8Sl. A. Aspect, I? Grangier. G. Roger. Phys. Rev. Lett. 47 ( I98 I ) 460. J.F. Clauser, M.A.
Home, Phys. Rev. D IO (1974)
S26.
J. Schlienz, G. Mahler, Phys. Rev. A S2 ( 1995) 4396; S. Popescu, Phys. Rev. Lett. 74 (1995) N. Gisin. Phys. Lett. A 210 (1996) A. Peres. Phys. Rev. Lett. 77 ( 1996) M. Horodecki, P. Horodecki.
2619;
1.51; 1413;
R. Horodecki,
78 ( 1997) 574, and references therein.
Phys. Rev. Len.
1998
PHYSICS
ELSEWIER
LETTERS
A
Physics Letters A 238 ( 1998) 85-89
Entanglement, information theory and Bell’s inequality A. Mann a*1,M. Revzen a,b*2,E. Santos b,3 a Depariment of Physics, Technion, Haifa 32000, Israel h Departamento de Fisica Moderna. Universidad de Cantabria. Santander; Spain Received 2 December
1996; revised manuscript received 12 November 1997; accepted Communicated by P.R. Holland
for publication
17 November
1997
Abstract Information theory is used to quantify entanglement of two spin-l /2 particles or two photons. Noise contaminating a pure entangled (singlet) state is parametrized to allow convenient comparison with experimental studies of possible violation of Bell’s inequality for such states. @ 1998 Elsevier Science B.V. PACS: 03.65.-W:
03.65.8~
1. Introduction Information theory [ 11 quantifies the intuitive notion of information gained upon measuring. Thus when the result of a measurement can be predicted with certainty, the information gain is nil. If we consider two entries (e.g. two particles) A and B whose properties can be simultaneously measured yielding, say, possible discretized values a and b, respectively, with joint probability (for the state considered) p (a, b), the “self-information” gained upon obtaining aforAandbforBis[l-31 f(a,b)
=
-lnp(u,b).
(1)
The central quantity in information theory is the entropy, S, which is the mean value of I [ l-31. Within classical physics, where the joint probability can always be defined, we have the “obvious” result that a ’ E-mail: [email protected] * E-mail: [email protected] 1 E-mail: [email protected]. 03759601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00898-O
complete knowledge of the composite system, A and B, i.e. S( A, B) = 0, implies a complete knowledge of its constituents: S(A) = S(B) = 0. A strange peculiarity of the extension of the above to quantum physics via von Neumann’s density matrix [ 3,4] is that in quantum mechanics “complete knowledge” of the composite system (i.e. the system is in a pure state) does not imply complete knowledge of its constituents. Pure states with this property are called entangled states [ 3,5]. Fully entangled states contain no knowledge whatever about their constituents. Entangled pure states have the remarkable property of allowing violation of Bell’s inequality [ 3,6,7] It is, thus, of interest to extend this to non-idealized, i.e. non-pure (“mixed”) quanta1 states. In the next section we present our version of using information theory to extend the definition of entanglement to “contaminated states”. Since we deal primarily with photonic states, we simulate the contamination of the pure fully entangled state by random sources of one and two photons, viz., we consider the so-called Werner states [ 81 for the two photons’ contamination which, in addition,
86
A. Mann et al. /Physics
is admixed with single photon noise. In Section 3, we consider in detail such a “contaminated density matrix” where we parametrize the contaminating noise in a particuiar manner which allows, in this parameter space, to define a region where the state is entangled, the noise notwithstanding, thereby rendering such a state a candidate for observing Bell’s inequality violation. The boundary in the contaminating parameters’ space determined in this way, based as it is on information theory, turns out to be almost identical with a similarly aimed parametrization used earlier by one of us [9]. In Ref. [9] direct violation of Bell’s inequality was contemplated whereas in this Letter we initiate such a study on the basis of information theoretic approach. Section 4 contains the experimental meaning of our parameters with a brief summary and comparison with the results of Ref. [ 91. The last section contains some concluding remarks.
2. Correlations
in quantum
information
theory
= - xp(a,b) (u.h)
lnp(u.6).
(21
Here a (b) designates a possible value for the A (B) entry (e.g. the particles’ position in a discretized phase space). p( a, b) is the joint probability for a and b. We also have the entropy of A (and, analogously, of B) S(A)
= - xp(a) ‘,
hip(a).
(3)
with p(n)
= c
p(a.
6).
p(4b) = pfa>
S(A(B)
with
= S(AB)
- S(B)
3 0,
(7)
with a corresponding expression for S( B/A). Eq. (6) is recognized as Bayes’s theorem in probability theory with p( al b) the conditional probability of a, given 0. The conditional entropy (Eqs. (5) and (7) ) is obviously non-negative, because p( 6) 3 p(u, b) (cf. Eq. (4) ) , i.e. p( al 6) 6 1. Thus we have a classically valid inequality, 2S(AB)
3 S(A)
+ S(B).
(8)
The corresponding, or better perhaps, quanta1 formulae are formulated in terms mann’s [4] density matrix. Thus with density matrix of the composite system, (quantum) is given by
S(A)
= -Trp(AB) = -Trp(A)
lnp(AB), Inp(A),
the related, of von Neup( AB), the the entropy
(9) (10)
with (e.g., Wehrl in Ref. [ I ] ) p(A)
= Trsp(AB).
(11)
When considered in the representation is diagonal - recall it is a non-negative erator [ 3,4] - we have S(AB)
= -Cp(ub) ah
lnp(ab),
where p( AB) hermitian op-
(12)
where now the sum is over all the states of the composite system, designated ab, and the eigenvalues of the density operator, pfub), give the probability that the composite system is in the state labeled by (ab). In the trivia1 case, where A and B are uncorrelated, then, classically ~(a.
b) = p(a)p(b),
while, quantum p(AB)
= - c p(u, b) lnp(ulb) (1I.h)
(6)
(4)
h
p(a) is the probability for A yielding a regardless of the B reading. A conditional entropy is defined by SCAIB)
b)lp(b).
Hence,
S(AB)
Consider our previously defined composite system consisting of two entries: A and B. Classically some basic information theoretic relations hold. Thus the basic quantity in information theory is the entropy of the composite system. It is given by [ I] S(A,B)
Letters A 238 (I 998) 85-89
(5)
(13)
mechanically,
= p(A)p(B).
(14)
In this factorized case all classical considerations hold for the quantum case - inclusive of the existence of joint probabilities and conditional entropy.
A. Mann
For both the classical and the quantum entropies we have [ 1,3,101 S(AB)
< S(A)
+ S(B).
et al. /Physics
mechanical
(15)
When A and B are uncorrelated the maximal value is attained both in the classical and the quanta1 cases [ lo]. In the correlated case, although classically S( Al B) is still well defined via Eq. (5)) the quanta1 expression for S( AIB) cannot be, generally, simply related to S(A) or S(B). A relation such as Eq. (5) has no simple quanta1 transcription (such as Eq. (9) ) when p( AB) is not diagonal in a product basis [ 111. In this case quantum mechanics harbours subtle correlations that have no clear classical counterparts. Thus, in quantum mechanics, generally, S(AIB) is a complicated entity. Indeed such is the case whenever we consider quantities for which classical information analysis requires the use of Bayes’s theorem (Eq. (6) ) . In the case under study here, correspondence to classical theory requires a “conditional” density matrix, p( AIB), which, by formal analogy with Eq. (5), should allow an expression such as S(AIB)
= -Trp(AB)
lnp(A(Z3).
I$(A,B))
= lrcI(A,B))(rll(A,B)j, = (I TA~B) - I Ms))/~~
(17) (18)
in obvious notation. In this case S(AB) = 0, it is the entropy of a pure state. But S(A) = ln2, as p(A) is completely random; hence the inequality (7) is violated. Classical physics implies this inequality (Eq. (7)), hence a sufficient condition for a state (i.e. a density matrix) to have distinctly quantum correlations (i.e. exhibiting entanglement which has no classical analogue) is 2S(AB)
< S(A)
+ S(B),
87
i.e. to have a negative S( Al B). This may thus be used as a measure for entanglement. Hence our criteria for entanglement with S(A) = S(B) = S(I), S( AB) = S(Zf) is S(I)
> S(II).
(20)
Now the violation of Bell’s inequality is possible only for states with distinctly quantum correlations [ 6,7]. In fact the violation may be taken as the fingerprints of quantum correlations. We are thus motivated to study states where “pure” quantum entanglement is admixed with some noise but still allowing a violation of Bell's inequality. In, perhaps, its simplest form this leads to states involving two parameters designated CYand L’ allowing admixtures of singles ((Y) and pairs (u) of photons. We now use our criterion, Eq. (20), to determine the boundary in the a,~: plane within which pure quantum correlations are assured. This boundary, based on information theory analysis, gives (as is shown in the next section) results essentially similar to the boundary obtained previously [ 91 by a different reasoning.
(16)
Such a quantity was indeed defined, e.g. in the Oviedo workshop [ lo], but via a rather unwieldy formula. Perhaps a more physical manifestation of this difficulty is seen by noting that S(AIB) as defined by Eq. (7) can, in quantum mechanics, be negative [ lo], thereby precluding a “density matrix” interpretation to p( A(B), since the eigenvalues of a density matrix must be non-negative and bounded by unity, implying (Eq. ( 16)) S(A(B) 3 0. An example for this is the case of two spin-half entries p(AB)
Letters A 238 (I 998) 85-89
(19)
3. Parametrized
source state
To allow a convenient comparison with experimental setups [ 121 and previous work [ 91 we consider a source state from which photons are propagating to a left detector, A, and a right detector, B. The idealized source is a maximally entangled state given by (cf. Eq. (18)) Ps =
by
(21)
IS)N~
1s)= (IRA&l)
-
ILA’h))/fi>
(22)
where R (L) designates right (left) handed polarization and the subscripts A and B indicate which way the photon is headed. These subscripts will be omitted henceforth when no confusion is likely to arise, always keeping the first symbol for the A photon and the second for the B. The contaminated source is given by the state p(AB) +
= alups (1
-~>[(pApVS
+
(1
-
u)pmI
+pVAPB)/21.
Here the random source of two photons is
(23)
A. Mann et al. /Physics
88
Letters A 238 (1998) 85-89 1
+ ]LR)(RL] + ILL)(LLl)/4.
v
(24)
0.8.
The random sources for single photons are PA
=
(IR)(RI
+
and the vacuum (no photons) PVA
term is
0.4.
(35)
IOA)(OAI.
=
The last two terms have, of course, corresponding terms indexed B. When (Y= cy*, u = u*, and CX,u < 1. cy=u= 1 is the “pure” entangled state while LY= 0 gives the one photon random state, CY= 1, u = 0 give a random two photons state. Thus our Hilbert space is nine-dimensional, e.g. IRR), (LL), IRL), )LR), IRO), ILO), IOR), IOL), 100). The last state, IOO), is not included in the density matrix as it gives rise to no readings in either detector. Direct evaluation of the matrix elements of p (Eq. (23)) and then diagonalizing it gives for the entropy of the composite system -S(AB) +
=
3cu(l -u) 4
lna(l
-u) 4
(u( 1 + 3u) ,n cy( 1 + 3u) 4 4
+(1
-CU)lnF,
while evaluation yields -S(A)=2
.71
0.6.
(251
If3(‘51)/2,
(1 -al
(27)
I
0’.
0.2
0.4
0.6
0.8
a
1
Fig. 1. Parameter space of a spin singlet contaminated by random two particle and single particle states. The region to the right of the continuous (dashed) curve represents source states violating the classical entropy inequality (the Bell inequality [ 91 ). The limiting values are indicated.
right in another direction, ~$2. We assume ideal polarizers and detectors, because we wish to focus on noise contaminating the source (i.e. in the preparation of the correlated two particle system) - not on the noise that enters in the measurement. The calculations of the probabilities are straightforward [ 91: We evaluate the trace of the product of the corresponding projection operator with the density matrix of the source state, Eq. (23). We get for the coincidence detection probability
of the entropy of the A subsystem
l+alnl+Cr
fAB(h~2b’y[~
-
4
+
I-ff 2
= -S(B). The boundary, in the parameter glement is obtained by equating is depicted in Fig. 1: the region curve is the quanta1 region, i.e. analogue exists.
1 --cy In 2
+~cos(2h
-2’$2)1/4,
(2%
and for the single probabilities (28)
space, for the entanS( AB) = S(A). This above the continuous to which no classical
PA = PB = ( 1 + C-k,)/4 (independent
of 41 and 42 ) .
If these results are put into Bell’s inequality PA
+&l
>
+PAB(~IY(P;)
pAB(4I.42)
+ -
~A&#+&).
(30) [ 131
pAB(d’:.4’2)
(31)
4. Bell’s inequality In order to relate to experiments we should calculate the results of measuring the polarization of the photon in a given direction, say at an angle 41 with the vertical, and the polarization of the photon on the
We get that the inequality may be violated for an optimum choice of the angles provided the parameters fulfill [9] ffu > I/&-.
(32)
A. Mann
et al. /Physics
The region of violation is shown in Fig. 1. We see that this region almost coincides with the region where the classical entropy inequality, Eq. (8)) is violated.
Lefters A 238 (1998)
III
C.E.
89
85-89
Shannon, W. Weaver,
Communication
The Mathematical
Theory
(Univ. of Illinois Press, Urbana,
1949)
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;
A. Wehrl. Rev. Mod. Phys. 50 ( 1978) 221; W.T. Grandy Jr.: Resource Lett. ITP-I.
5. Conclusions
(1997) SM.
The concept of entanglement for pure quantum states was introduced already in the early days of quantum mechanics, but the use of the concept for mixed states is relatively recent. Several definitions have been put forward to generalize entanglement to mixed states (some of them may be found in Refs. [ 5,141). Most obvious are the violation of the entropy inequality (advocated most recently in Ref. [ IO] ) and the violation of Bell’s inequality. Here we showed that these two definitions yield similar results, at least for a system of two photons (or spin l/2 particles) correlated via their polarizations (or spins). We have considered a density matrix for a particle pair, which represents realistically what is produced in actual experiments.
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We acknowledge financial support from the Spanish Ministry of Education, through the programme of Sabbatical in Spain, which allowed a four months stay of one of us (M.R.) in the University of Cantabria. A.M and M.R. were supported by the Fund for Promotion of Research at the Technion, by the VPR Fund - R.&M. Rochlin Research Fund and by GIF - German-Israeli Foundation for Research and Development. M.R. acknowledges with thanks the help of Dr. A. Shiekh.
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