- Email: [email protected]
Entanglement and unambiguous discrimination between non-orthogonal states
8 December 1997
PHYSICS
ELSEWIER
LETTERS
A
Physics Letters A 236 (1997) 177-179
Entanglement and unambiguous discrimination between non-orthogonal states Anthony Chefles ‘, Stephen M. Barnett Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 ONG, Scotland, UK
Received 29 September 1997; accepted for publication 13 October 1997 Communicated by P.R. Holland
Abstract We study the effect of unambiguous state-discrimination measurements on entangled quantum systems. These are shown to he equivalent to certain methods of entanglement concentration. We also examine the consequences of performing such
measurements within the context of the Hardy-Goldstein demonstration of nonlocality without inequalities. The correlations induced by such measurements are, unlike those associated with von Neumann measurements, compatible with local-realism. @ 1997 Elsevier Science B.V. PACS: 03.65.82
In classical mechanics, there is no fundamental limit to the accuracy with which we can determine the state of a physical system of which we have no prior knowledge. This contrasts with the situation in quantum mechanics. To be certain of identifying the state of a quantum system, we must already know an orthogonal basis to which its state vector belongs. No measurement scheme can therefore discriminate between non-orthogonal states with complete success. The optimal measurement strategy for discriminating between a pair of states was devised by Helstrom [ 1] and generalised to any number of linearly independent states by Holevo [ 21. By optimal, we mean that the Helstrom measurement results in the lowest average probability of error. There do, however, exist state-discrimination measurements for which the error probability is zero. One strategy of this kind was ’ E-mail: [email protected] ‘E-mail: [email protected]
developed by Ivanovic [3] and subsequently refined by Peres [ 41. The Ivanovic-Peres (IP) measurement can discriminate between any pair of quantum states without errors, although with a non-zero probability of yielding an inconclusive result. In this Letter, we examine the effect of performing IP measurements on part of a larger system in an entangled state. We are concerned, in particular, with imperfectly entangled pure states of a pair of spin- l/2 particles. Such states can always be written so that orthogonal states of one system are correlated with non-orthogonal states of the other. We examine the nature of the correlations induced by IP measurements on the latter. In particular, we show that if some of the standard measurements used in the demonstration of nonlocality without inequalities due to Hardy [ 51 and Goldstein [6] are replaced by IP measurements, the resulting correlations are compatible with localrealism.
0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00802-5
178
A. Chejles. SM. BarnettIPhysics
We denote the pair of single-particle states by ]I,&). There exists an orthonormal basis I*) for the space spanned by ]I,&) such that we can write ]I,&) = cos f3 I+) f sin 8 I-), for some angle 0. The nonuniqueness of this basis implies that we may, without any loss of generality, choose cosd > sin0 > 0. The IP measurement is a generalised measurement and may be represented by a positive operator-valued measure (POVM) operation [ 71, here being described by the Hermitian operators Ao=$[tan8+l+(tan8-l)b,], Ar=
;JiQ&c1
(1) +LF-,),
(2)
where ii + ,@ = i and cz has the states 15) as its eigenstates with eigenvalues f 1. Applying these operators to the states I$*), we find &I+*>
=sinB((+)
[email protected]*)
= &zQ+).
f I-)>,
(3) (4)
As a consequence of the orthogonality of the states on the right-hand side of (3)) one can discriminate between II,&) with certainty when the outcome operator is &. The discrimination failure operator & maps both states I+*) onto I+), erasing all information about the initial state, giving an inconclusive result. If the particle is in either of the states ]I,&) with equal a priori probabilities of l/2, then the probability P of unambiguously determining the state of the system is 1 - cos 28. Let us now consider performing a state-discrimination measurement on part of a correlated system. Consider a general state of the form
IW = c+b+)[email protected]+) + c-b->I+-)9
(5)
where the pairs of states 1~) and II,&) are normalised but not necessarily orthogonal. We wish to examine how a measurement which discriminates between the states in each pair affects the correlations between the particles. If a conclusive result is obtained for this state-discrimination measurement then the states I&) are mapped onto ( I+) f I-) ) / &. If we then measure the observable i @ &,, the results fl imply that the first particle has been projected onto the states ]a*), respectively. This argument is symmetrical, in the sense that successful discrimination of the states Iu*) of the first
Letters A 236 (1997) 177-l 79
particle will project the second onto I$*), unambiguously, irrespective of the possible non-orthogonality of the latter pair. Although, in the form (5)) the state Ily) is not expressed in its Schmidt basis, the symmetry of the correlations enforced by successful IP measurements is reminiscent of that revealed upon making von Neumann measurements of related Schmidt observables [ 81. It is interesting to note that any state of the form (5) may be written so that the coefficients cf are equal to l/a and that the states of one subsystem, say the la*), are orthogonal. Then a successful IP measurement which discriminates between the states II+&) will convert IY) into a maximally entangled state. When the discrimination attempt fails, we obtain a product state. Thus, given a number of imperfectly entangled states, we can use the IP measurement on one of the particles to produce a smaller number of maximally entangled pairs. This method of entanglement concentration is equivalent to the “Procrustean” technique proposed by Bennett et al. [ 91. There are situations in which the inability of standard von Neumann measurements to distinguish between non-orthogonal states has interesting consequences. This is the case in the two-particle demonstration of nonlocality without inequalities due to Hardy [S] and subsequently simplified by Goldstein [ 61, which relies upon the lack of perfect correlations in states which are not maximally entangled to show a non-statistical violation of local-realism. In Goldstein’s version of the nonlocality proof, we have the non-maximally entangled state
IV) = QI~I)I~Z)+ ~14)1~2)+ 4~I)l~2>?
(6)
with ubc f 0 and (u,~u~) = 0, where the index r = 1,2 distinguishes the particles. We also have the operators 0, = ]u~)(z+l and tir = ]wr)(w,], where the Iw,) are definedby Iwt) = (l~(~+~6~~)-‘/~(ulu~)+~~~~)) and (~2) = (]u(~+ ]c]~)-‘/*(u(u~) +clu$). It follows that the state Ip) may also be expressed as
2 l/2 IW = (Ial2 + I4 1 IWl)I~2) + 4Q)I~Z) 2 1/2 = (Ial + ICI ) IUIJIW2)+ 4w)lu2).
(7)
Standard von Neumann measurements reveal that one cannot regard the non-commuting operator observables 0, and I!‘,. as corresponding to local “elements of reality” in the sense of Einstein, Podolsky and Rosen
A. Chejes. S.M. Barneft/Physics Letters A 236 (1997) 177-l 79
(EPR) [ lo]. This follows from the following observations. We first see that 01 &IT) = 0. If 01 and 02 reveal the preexisting values of elements of reality al and ~42,then either or both ui and 4 must be zero. Furthermore, if we measure the 0 operator for one particle and obtain the value 0, the state of the other particle is projected onto its (w) state. Thus, assuming that the @, operators similarly correspond to elements of reality wr, at least one of these w, must have the value 1. The incompatibility with local-realism comes from the fact that, with non-zero probability, we can obtain zero results for measurements of both mr. This incompatibility follows from the fact that the 1w) and 1u) states defined for each particle are not orthogonal; a von Neumann measurement of [email protected] with outcome 1 does not mean that we were not in state 1~). If, however, we use an IP measurement to discriminate between 1w) and 1u), then finding we are in state 1w) does mean we were not in state 1u) . Successful IP measurements discriminate between non-orthogonal states, so when one of the particles is found to be in its Iw) (IL])) state the other will be projected onto its Iu) ( Iu) ) state. If we then measure the I!? operator for the latter particle we are sure to find the result 0( 1) . Such perfect correlations (one state found in state 10) implying that the other is in Iw)) are not in conflict with local-realism. It is illuminating to recall the symmetry of the correlations induced by successful IP measurements. Here, this means that when such a measurement is used to discriminate between the Iw) and 10) states for one particle, it will project the other onto its ]u) and Iu) states. and vice versa. The consistency of these
179
correlations with local-realism is closely related to the fact that a nonlocality proof of the Hardy-Goldstein type is unobtainable for maximally entangled states. For such states, every single-particle observable is perfectly, symmetrically correlated with some property of the other particle. It is the symmetry of the correlations observed for both von Neumann measurements on maximally entangled states and appropriate IP measurements on imperfectly entangled pure states which is responsible for their compatibility with localrealism. We thank Dr. John Jeffers for valuable discussions, and gratefully acknowledge financial support by the University of Strathclyde Research and Development Fund and the UK Engineering and Physical Sciences Research Council (EPSRC) .
References [ I ] C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). [2] AS. Holevo, J. Multivar. Anal. 3 (1973) 337. [3] I.D. Ivanovic, Phys. Lett. A 123 ( 1987) 257. [4] A. Peres, Phys. Lett. A 128 (1987) 19. [5] L. Hardy, Phys. Rev. Lett. 71 (1993) 1665. [6] S. Goldstein, Phys. Rev. Lett. 72 ( 1994) 1951. [ 71 K. Kraus, States, Effects and Operations, No. 190, in: Lecture Notes in Physics (Berlin, 1983). [Sj A. Eke& l?L. Knight, Am. J. Phys. 63 (1995) 415. [9] C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53 (1996) 2046. [ 101 A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 ( 1935) 777.
PHYSICS
ELSEWIER
LETTERS
A
Physics Letters A 236 (1997) 177-179
Entanglement and unambiguous discrimination between non-orthogonal states Anthony Chefles ‘, Stephen M. Barnett Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 ONG, Scotland, UK
Received 29 September 1997; accepted for publication 13 October 1997 Communicated by P.R. Holland
Abstract We study the effect of unambiguous state-discrimination measurements on entangled quantum systems. These are shown to he equivalent to certain methods of entanglement concentration. We also examine the consequences of performing such
measurements within the context of the Hardy-Goldstein demonstration of nonlocality without inequalities. The correlations induced by such measurements are, unlike those associated with von Neumann measurements, compatible with local-realism. @ 1997 Elsevier Science B.V. PACS: 03.65.82
In classical mechanics, there is no fundamental limit to the accuracy with which we can determine the state of a physical system of which we have no prior knowledge. This contrasts with the situation in quantum mechanics. To be certain of identifying the state of a quantum system, we must already know an orthogonal basis to which its state vector belongs. No measurement scheme can therefore discriminate between non-orthogonal states with complete success. The optimal measurement strategy for discriminating between a pair of states was devised by Helstrom [ 1] and generalised to any number of linearly independent states by Holevo [ 21. By optimal, we mean that the Helstrom measurement results in the lowest average probability of error. There do, however, exist state-discrimination measurements for which the error probability is zero. One strategy of this kind was ’ E-mail: [email protected] ‘E-mail: [email protected]
developed by Ivanovic [3] and subsequently refined by Peres [ 41. The Ivanovic-Peres (IP) measurement can discriminate between any pair of quantum states without errors, although with a non-zero probability of yielding an inconclusive result. In this Letter, we examine the effect of performing IP measurements on part of a larger system in an entangled state. We are concerned, in particular, with imperfectly entangled pure states of a pair of spin- l/2 particles. Such states can always be written so that orthogonal states of one system are correlated with non-orthogonal states of the other. We examine the nature of the correlations induced by IP measurements on the latter. In particular, we show that if some of the standard measurements used in the demonstration of nonlocality without inequalities due to Hardy [ 51 and Goldstein [6] are replaced by IP measurements, the resulting correlations are compatible with localrealism.
0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00802-5
178
A. Chejles. SM. BarnettIPhysics
We denote the pair of single-particle states by ]I,&). There exists an orthonormal basis I*) for the space spanned by ]I,&) such that we can write ]I,&) = cos f3 I+) f sin 8 I-), for some angle 0. The nonuniqueness of this basis implies that we may, without any loss of generality, choose cosd > sin0 > 0. The IP measurement is a generalised measurement and may be represented by a positive operator-valued measure (POVM) operation [ 71, here being described by the Hermitian operators Ao=$[tan8+l+(tan8-l)b,], Ar=
;JiQ&c1
(1) +LF-,),
(2)
where ii + ,@ = i and cz has the states 15) as its eigenstates with eigenvalues f 1. Applying these operators to the states I$*), we find &I+*>
=sinB((+)
[email protected]*)
= &zQ+).
f I-)>,
(3) (4)
As a consequence of the orthogonality of the states on the right-hand side of (3)) one can discriminate between II,&) with certainty when the outcome operator is &. The discrimination failure operator & maps both states I+*) onto I+), erasing all information about the initial state, giving an inconclusive result. If the particle is in either of the states ]I,&) with equal a priori probabilities of l/2, then the probability P of unambiguously determining the state of the system is 1 - cos 28. Let us now consider performing a state-discrimination measurement on part of a correlated system. Consider a general state of the form
IW = c+b+)[email protected]+) + c-b->I+-)9
(5)
where the pairs of states 1~) and II,&) are normalised but not necessarily orthogonal. We wish to examine how a measurement which discriminates between the states in each pair affects the correlations between the particles. If a conclusive result is obtained for this state-discrimination measurement then the states I&) are mapped onto ( I+) f I-) ) / &. If we then measure the observable i @ &,, the results fl imply that the first particle has been projected onto the states ]a*), respectively. This argument is symmetrical, in the sense that successful discrimination of the states Iu*) of the first
Letters A 236 (1997) 177-l 79
particle will project the second onto I$*), unambiguously, irrespective of the possible non-orthogonality of the latter pair. Although, in the form (5)) the state Ily) is not expressed in its Schmidt basis, the symmetry of the correlations enforced by successful IP measurements is reminiscent of that revealed upon making von Neumann measurements of related Schmidt observables [ 81. It is interesting to note that any state of the form (5) may be written so that the coefficients cf are equal to l/a and that the states of one subsystem, say the la*), are orthogonal. Then a successful IP measurement which discriminates between the states II+&) will convert IY) into a maximally entangled state. When the discrimination attempt fails, we obtain a product state. Thus, given a number of imperfectly entangled states, we can use the IP measurement on one of the particles to produce a smaller number of maximally entangled pairs. This method of entanglement concentration is equivalent to the “Procrustean” technique proposed by Bennett et al. [ 91. There are situations in which the inability of standard von Neumann measurements to distinguish between non-orthogonal states has interesting consequences. This is the case in the two-particle demonstration of nonlocality without inequalities due to Hardy [S] and subsequently simplified by Goldstein [ 61, which relies upon the lack of perfect correlations in states which are not maximally entangled to show a non-statistical violation of local-realism. In Goldstein’s version of the nonlocality proof, we have the non-maximally entangled state
IV) = QI~I)I~Z)+ ~14)1~2)+ 4~I)l~2>?
(6)
with ubc f 0 and (u,~u~) = 0, where the index r = 1,2 distinguishes the particles. We also have the operators 0, = ]u~)(z+l and tir = ]wr)(w,], where the Iw,) are definedby Iwt) = (l~(~+~6~~)-‘/~(ulu~)+~~~~)) and (~2) = (]u(~+ ]c]~)-‘/*(u(u~) +clu$). It follows that the state Ip) may also be expressed as
2 l/2 IW = (Ial2 + I4 1 IWl)I~2) + 4Q)I~Z) 2 1/2 = (Ial + ICI ) IUIJIW2)+ 4w)lu2).
(7)
Standard von Neumann measurements reveal that one cannot regard the non-commuting operator observables 0, and I!‘,. as corresponding to local “elements of reality” in the sense of Einstein, Podolsky and Rosen
A. Chejes. S.M. Barneft/Physics Letters A 236 (1997) 177-l 79
(EPR) [ lo]. This follows from the following observations. We first see that 01 &IT) = 0. If 01 and 02 reveal the preexisting values of elements of reality al and ~42,then either or both ui and 4 must be zero. Furthermore, if we measure the 0 operator for one particle and obtain the value 0, the state of the other particle is projected onto its (w) state. Thus, assuming that the @, operators similarly correspond to elements of reality wr, at least one of these w, must have the value 1. The incompatibility with local-realism comes from the fact that, with non-zero probability, we can obtain zero results for measurements of both mr. This incompatibility follows from the fact that the 1w) and 1u) states defined for each particle are not orthogonal; a von Neumann measurement of [email protected] with outcome 1 does not mean that we were not in state 1~). If, however, we use an IP measurement to discriminate between 1w) and 1u), then finding we are in state 1w) does mean we were not in state 1u) . Successful IP measurements discriminate between non-orthogonal states, so when one of the particles is found to be in its Iw) (IL])) state the other will be projected onto its Iu) ( Iu) ) state. If we then measure the I!? operator for the latter particle we are sure to find the result 0( 1) . Such perfect correlations (one state found in state 10) implying that the other is in Iw)) are not in conflict with local-realism. It is illuminating to recall the symmetry of the correlations induced by successful IP measurements. Here, this means that when such a measurement is used to discriminate between the Iw) and 10) states for one particle, it will project the other onto its ]u) and Iu) states. and vice versa. The consistency of these
179
correlations with local-realism is closely related to the fact that a nonlocality proof of the Hardy-Goldstein type is unobtainable for maximally entangled states. For such states, every single-particle observable is perfectly, symmetrically correlated with some property of the other particle. It is the symmetry of the correlations observed for both von Neumann measurements on maximally entangled states and appropriate IP measurements on imperfectly entangled pure states which is responsible for their compatibility with localrealism. We thank Dr. John Jeffers for valuable discussions, and gratefully acknowledge financial support by the University of Strathclyde Research and Development Fund and the UK Engineering and Physical Sciences Research Council (EPSRC) .
References [ I ] C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). [2] AS. Holevo, J. Multivar. Anal. 3 (1973) 337. [3] I.D. Ivanovic, Phys. Lett. A 123 ( 1987) 257. [4] A. Peres, Phys. Lett. A 128 (1987) 19. [5] L. Hardy, Phys. Rev. Lett. 71 (1993) 1665. [6] S. Goldstein, Phys. Rev. Lett. 72 ( 1994) 1951. [ 71 K. Kraus, States, Effects and Operations, No. 190, in: Lecture Notes in Physics (Berlin, 1983). [Sj A. Eke& l?L. Knight, Am. J. Phys. 63 (1995) 415. [9] C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53 (1996) 2046. [ 101 A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 ( 1935) 777.