- Email: [email protected]
Nonlocality criteria for quantum teleportation
a _-
__ k!!!E!
ELSEVIER
8 January 1996
PHYSICS
Physics Letters A 210 (1996)
LETTERS
A
157-159
Nonlocality criteria for quantum teleportation N. Gisin ’ Group ofApplied Physics, University oj’Geneva, 121 I Geneva 4, Switzerland Received 20 September 1995; accepted for publication 17 October 1995 Communicated by P.R. Holland
Abstract An larger states. to be
upper bound for the fidelity of quantum teleportation explainable by local hidden variables is derived. This than the fidelity corresponding to product states, i.e. to local quantum states. This is relevant for the study In particular, the fidelity of Werner’s mixed state, known to be larger than the fidelity of product states, smaller than the fidelity explainable by local hidden variables. Hence the fidelity of Werner’s mixed state
bound is of mixed is found
does not
exhibit nonclassical aspects. PACT: 03.65.B~;
89.7O.+c;
42.50.Wm
Quantum teleportation is a recently discovered manifestation of quantum nonlocality [ 11. The aim of this Letter is to determine how good teleportation has to be in order to really exhibit nonlocality, i.e. how faithful teleportation must be in order to be incompatible with local hidden variables. In quantum teleportation, a pair of quantum systems in state D is distributed between two partners, Alice and Bob. We shall limit the discussion to spin i, hence D is a pure or mixed two-spin
ice’s side (pure state) and transferred to Bob’s side (possibly a mixed state), respectively. Clearly, @& may depend on #Alice, on D and on the strategy used by Alice and Bob. The fidelity _F( D) is the mean distance between #Alice and Pa& corresponding to the optimal strategy * , F( D> = M~strategies[M{ (~AliceIPBobIJIAlice)}l
(1)
i state. Alice has a test
spin i particle. The aim of the partners is to transfer the spin state of that test particle as faithfully as possible to Bob using only the common pair of particles in state D (quantum channel) and two classical bits (classical channel). No other channel is allowed, in particular Alice is not allowed to send the test particle to Bob. The faithfulness or fidelity of the transfer is measured as follows. Let $Aiice and Pa&, denote the (normalized) state of the test particle prepared on Al-
where the mean M is taken over all possible
[email protected].
0375-9601/96/$12.00
0
SSDI 0375-9601(95)00771-7
states
@Alice.
In their original article [ 1] Bennett and co-workers proved that if D is the singlet state, or any maximally entangled state, then the fidelity obtains the maximal possible value 1, i.e. Pa& = ($,&rice)($AiiceI for all @Alice
9
.F( singlet) *Note
’ E-mail:
7
= 1.
that with this definition, when ~a&
pendent of $Aticc the fidelity is not 0 but
1996 Elsevier Science B.V. All rights reserved
f
is completely inde-
158
N. Gisin/Physics
Leiters A 210 (1996) 157-159
On the other extreme, when D is a product state, Alice can only measure @Alicealong an arbitrary but given direction z, tell Bob the result, up or down, and Bob prepares Pa& in the up or down St&! [ 21. In this way, only one bit of classical information is transferred. The corresponding fidelity is .T( product state) = Frs = i, From this, Popescu [ 51 concluded that a state D with fidelity larger than 3 has “nonclassical aspects”. Assuming that “nonclassical aspects” means “not explainable by local hidden variables”, this raises the question what the fidelity for teleportation of an hypothetical theory with hidden local variables would be? The hidden varibale would correspond to all quantum states. In particular, the distributed state D would be local, hence useless for teleportation. Assumingly, within the local hidden variable paradigm, Alice could measure the State @Alice itI the classical sense of “measuring”: finding out what the state @Aliceis s . Certainly, Alice could do no better than know precisely the state of her test particle. She could use the two classical bits at her disposal to inform Bob in which one of the four quarters of the state space @Aliceis, but with Only two bits she could not send more information to her partner. Bob could then prepare Pa&, in the center of this quarter. Recalling that the (pure) state space of a spin i is isomorphic to the sphere, the optimal division into four parts is based upon the following four limiting vectors, ei = (O,O, I), e2 = (J-,0,77),
(2) (3)
e3=(J1_rl2cos(~~),~~sin(~~),r), (4) e4= (J~cos(~9r),--_~sin($r),r)),
(5) with r] = -f. The first vector points to the north pole, the three other vectors form a pyramid. The angle between any pair of vectors is the same: e;*ej = - 3 for 3Though the local variables are hidden at present, the most favorable for teleportation is that they would not remain hidden forever.
all i, j = 1,2,3,4. A not too complicated shows that in such a case T( local hidden variables) aictan Jz %-
computation
= Fijhv x 0.87.
(6)
Note that this protocol uses both of the allowed classical bits and that the corresponding fidelity is a larger value than F(product states). Accordingly, the fidelity of a state D reveals “nonclassical aspects” (i.e. is incompatible with local hidden variables) only if F(D) > 0.87. Let us now examine the implication of this bound .Tihv for some states. First, consider the following family of pure states,
qkl,p = aI + -) - PI -
+)t
(7)
where LY> /? > 0 are two real numbers satisfying cr2+ /3’ = 1. The corresponding fidelity is 4 3(Ga,p) = $(LY’ - /?“)/(LY - p). On the one hand, for (Yclose to l/d this fidelity can be larger than .&vr but this is no surprise, since it is well known that all nonproduct states are incompatible with local hidden variables [ 31. On the other hand, for (Yclose to 1, _F( t_ba,p)can be smaller than .Ti;hveven for nonproduct states, that is for states that do violate Bell’s inequalities. Next, following Werner [ 41 and Popescu [ 51, consider the following family of mixed two-spin i states, W ( A) =
APqinglet + ( 1 - A) 1.
The corresponding
fidelity
is F( W( A))
(8) = i( A +
1). For A < I/&’ such states do not violate any Bell-CHSH inequality, but their fidelity is larger than F( product state). This, however, does not imply that the Werner state W(A) contains some hidden nonlocality, since their fidelity is smaller than the upper bound & (though it neither does imply that W(A) does not contain hidden nonlocality). In conclusion, the main results of this Letter are: (1) If the teleportation fidelity satisfies 3(D) > fihv = k + fi
arctan &‘/T
M 0.87, then the state D is
nonlocal in thesense of incompatible with local hidden variables. Indeed, if D were local, then Alice could not 4 This result holds for the standard quantum teleportation protocol [ I 1. We assume that this is also the best strategy for &,p.
N. Gisin/Physics
Letters A 210 (1996) 157-159
make use of the quantum channel and with only two bits communicated over the classical channel, she can not transfer more information than the above bound, even if she knew the exact state of her test particles. (2) The teleportation fidelity compatible with local hidden variables differs from the teleportation tidelity compatible with local quantum states. Hence teleportation reveals two different aspects of nonlocality, one related to classical local variables, with upper bound 3ihv M 0.87, the other related to local quantum states, with upper bound 3ps = i. The difference is specially relevant for the characterization of nonlocal mixed quantum states.
159
It is a pleasure to acknowledge stimulating discussions with Sandu Popescu and the Horodeckis, who found a mistake in an earlier version of this Letter. Part of this work was supported by the Swiss National Science FNRS. References [ I] [2] [3] [4] [5]
C.H. Bennett et al., Phys. Rev. Len. 70 ( 1993) 1895. S. Popescu, Phys. Rev. Lett. 72 (I 994) 797. N. Gisin, Phys. Lett. A 154 (1991) 201. R.F. Werner, Phys. Rev. A 40 ( 1989) 4277. S. Popescu, Phys. Rev. Lett. 74 ( 1995) 2619.
__ k!!!E!
ELSEVIER
8 January 1996
PHYSICS
Physics Letters A 210 (1996)
LETTERS
A
157-159
Nonlocality criteria for quantum teleportation N. Gisin ’ Group ofApplied Physics, University oj’Geneva, 121 I Geneva 4, Switzerland Received 20 September 1995; accepted for publication 17 October 1995 Communicated by P.R. Holland
Abstract An larger states. to be
upper bound for the fidelity of quantum teleportation explainable by local hidden variables is derived. This than the fidelity corresponding to product states, i.e. to local quantum states. This is relevant for the study In particular, the fidelity of Werner’s mixed state, known to be larger than the fidelity of product states, smaller than the fidelity explainable by local hidden variables. Hence the fidelity of Werner’s mixed state
bound is of mixed is found
does not
exhibit nonclassical aspects. PACT: 03.65.B~;
89.7O.+c;
42.50.Wm
Quantum teleportation is a recently discovered manifestation of quantum nonlocality [ 11. The aim of this Letter is to determine how good teleportation has to be in order to really exhibit nonlocality, i.e. how faithful teleportation must be in order to be incompatible with local hidden variables. In quantum teleportation, a pair of quantum systems in state D is distributed between two partners, Alice and Bob. We shall limit the discussion to spin i, hence D is a pure or mixed two-spin
ice’s side (pure state) and transferred to Bob’s side (possibly a mixed state), respectively. Clearly, @& may depend on #Alice, on D and on the strategy used by Alice and Bob. The fidelity _F( D) is the mean distance between #Alice and Pa& corresponding to the optimal strategy * , F( D> = M~strategies[M{ (~AliceIPBobIJIAlice)}l
(1)
i state. Alice has a test
spin i particle. The aim of the partners is to transfer the spin state of that test particle as faithfully as possible to Bob using only the common pair of particles in state D (quantum channel) and two classical bits (classical channel). No other channel is allowed, in particular Alice is not allowed to send the test particle to Bob. The faithfulness or fidelity of the transfer is measured as follows. Let $Aiice and Pa&, denote the (normalized) state of the test particle prepared on Al-
where the mean M is taken over all possible
[email protected].
0375-9601/96/$12.00
0
SSDI 0375-9601(95)00771-7
states
@Alice.
In their original article [ 1] Bennett and co-workers proved that if D is the singlet state, or any maximally entangled state, then the fidelity obtains the maximal possible value 1, i.e. Pa& = ($,&rice)($AiiceI for all @Alice
9
.F( singlet) *Note
’ E-mail:
7
= 1.
that with this definition, when ~a&
pendent of $Aticc the fidelity is not 0 but
1996 Elsevier Science B.V. All rights reserved
f
is completely inde-
158
N. Gisin/Physics
Leiters A 210 (1996) 157-159
On the other extreme, when D is a product state, Alice can only measure @Alicealong an arbitrary but given direction z, tell Bob the result, up or down, and Bob prepares Pa& in the up or down St&! [ 21. In this way, only one bit of classical information is transferred. The corresponding fidelity is .T( product state) = Frs = i, From this, Popescu [ 51 concluded that a state D with fidelity larger than 3 has “nonclassical aspects”. Assuming that “nonclassical aspects” means “not explainable by local hidden variables”, this raises the question what the fidelity for teleportation of an hypothetical theory with hidden local variables would be? The hidden varibale would correspond to all quantum states. In particular, the distributed state D would be local, hence useless for teleportation. Assumingly, within the local hidden variable paradigm, Alice could measure the State @Alice itI the classical sense of “measuring”: finding out what the state @Aliceis s . Certainly, Alice could do no better than know precisely the state of her test particle. She could use the two classical bits at her disposal to inform Bob in which one of the four quarters of the state space @Aliceis, but with Only two bits she could not send more information to her partner. Bob could then prepare Pa&, in the center of this quarter. Recalling that the (pure) state space of a spin i is isomorphic to the sphere, the optimal division into four parts is based upon the following four limiting vectors, ei = (O,O, I), e2 = (J-,0,77),
(2) (3)
e3=(J1_rl2cos(~~),~~sin(~~),r), (4) e4= (J~cos(~9r),--_~sin($r),r)),
(5) with r] = -f. The first vector points to the north pole, the three other vectors form a pyramid. The angle between any pair of vectors is the same: e;*ej = - 3 for 3Though the local variables are hidden at present, the most favorable for teleportation is that they would not remain hidden forever.
all i, j = 1,2,3,4. A not too complicated shows that in such a case T( local hidden variables) aictan Jz %-
computation
= Fijhv x 0.87.
(6)
Note that this protocol uses both of the allowed classical bits and that the corresponding fidelity is a larger value than F(product states). Accordingly, the fidelity of a state D reveals “nonclassical aspects” (i.e. is incompatible with local hidden variables) only if F(D) > 0.87. Let us now examine the implication of this bound .Tihv for some states. First, consider the following family of pure states,
qkl,p = aI + -) - PI -
+)t
(7)
where LY> /? > 0 are two real numbers satisfying cr2+ /3’ = 1. The corresponding fidelity is 4 3(Ga,p) = $(LY’ - /?“)/(LY - p). On the one hand, for (Yclose to l/d this fidelity can be larger than .&vr but this is no surprise, since it is well known that all nonproduct states are incompatible with local hidden variables [ 31. On the other hand, for (Yclose to 1, _F( t_ba,p)can be smaller than .Ti;hveven for nonproduct states, that is for states that do violate Bell’s inequalities. Next, following Werner [ 41 and Popescu [ 51, consider the following family of mixed two-spin i states, W ( A) =
APqinglet + ( 1 - A) 1.
The corresponding
fidelity
is F( W( A))
(8) = i( A +
1). For A < I/&’ such states do not violate any Bell-CHSH inequality, but their fidelity is larger than F( product state). This, however, does not imply that the Werner state W(A) contains some hidden nonlocality, since their fidelity is smaller than the upper bound & (though it neither does imply that W(A) does not contain hidden nonlocality). In conclusion, the main results of this Letter are: (1) If the teleportation fidelity satisfies 3(D) > fihv = k + fi
arctan &‘/T
M 0.87, then the state D is
nonlocal in thesense of incompatible with local hidden variables. Indeed, if D were local, then Alice could not 4 This result holds for the standard quantum teleportation protocol [ I 1. We assume that this is also the best strategy for &,p.
N. Gisin/Physics
Letters A 210 (1996) 157-159
make use of the quantum channel and with only two bits communicated over the classical channel, she can not transfer more information than the above bound, even if she knew the exact state of her test particles. (2) The teleportation fidelity compatible with local hidden variables differs from the teleportation tidelity compatible with local quantum states. Hence teleportation reveals two different aspects of nonlocality, one related to classical local variables, with upper bound 3ihv M 0.87, the other related to local quantum states, with upper bound 3ps = i. The difference is specially relevant for the characterization of nonlocal mixed quantum states.
159
It is a pleasure to acknowledge stimulating discussions with Sandu Popescu and the Horodeckis, who found a mistake in an earlier version of this Letter. Part of this work was supported by the Swiss National Science FNRS. References [ I] [2] [3] [4] [5]
C.H. Bennett et al., Phys. Rev. Len. 70 ( 1993) 1895. S. Popescu, Phys. Rev. Lett. 72 (I 994) 797. N. Gisin, Phys. Lett. A 154 (1991) 201. R.F. Werner, Phys. Rev. A 40 ( 1989) 4277. S. Popescu, Phys. Rev. Lett. 74 ( 1995) 2619.