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Bell inequality, Bell states and maximally entangled states for n qubits
7 September 1998
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 246 ( 1998) l-6
Bell inequality, Bell states and maximally entangled states for n qubits N. Gisin, H. Bechmann-Pasquinucci Group of Applied Physics, University of Geneva I21 I Geneva 4, Swilzerland
Received 26 May 1998; accepted for publication 29 June 1998 Communicated
by P.R. Holland
Abstract First, we present a Bell-type inequality for n qubits, assuming that M out of the n qubifs are independent. Quantum mechanics violates this inequality by a ratio that increases exponentially with m. Hence an experiment on n qubits violating this inequality sets a lower bound on the number m of entangled qubits. Next, we propose a definition of maximally entangled states of n qubits. For this purpose we study five different criteria. Four of these criteria are found compatible. For any number n of qubits, they determine an orthogonal basis consisting of maximally entangled states generalizing the Bell states. @ 1998 Elsevier Science B.V.
1. Introduction We consider n qubits, all spatially separated from each other. The purpose of this paper is to find characterizations of maximally entangled states of the n qubits. We assume that the n qubits are equivalent (there is no privileged qubit(s)), hence we restrict ourself to symmetric states. We test the following five criteria that could, a priori, characterize such symmetric maximally entangled (SME) states: ( 1) maximally entangled states violate Bell inequality [ l] maximally; (2) maximally entangled states are maximally fragile; (3) whenever m < n qubits are measured (in some appropriate basis), the outcomes determine a maximally entangled state of the m - n remaining qubits; (4) the mutual information of measurement outcomes is maximal; (5) all partial traces of maximally entangled states
are maximally mixed. The Bell inequality we use is presented in the next section and the above criteria are made more precise and analysed in Section 3. The result of our analysis is that the last one of the above criteria is not a good characterization. However, we find that all the other four properties are compatible and define SME states. By local transformation, the SME states generate a basis of the entire Hilbert space.
2. Bell-Klyshko
inequality for n qubits
In this section we briefly present a generalization to n qubits of the Bell-CHSH [ 21 inequality well known in the two-qubit case. This generalization was first presented by Klyshko and Belinskii [ 31. Let us start with a brief review of the well-known two-qubit case. Let a = f 1 and a’ = &I denote two possible outcomes of two possible measurements on
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N. Gisin. H. Bechmann-Pasquinucci/Physics
2
the first qubit and similarly b = f 1 and b’ = fl for the second qubit. Let us consider the following linear combination of joint results,
ietlers A 246 (1998) 1-6
again the linearity of the mean, one obtains the BellKlyshko inequality for n qubits, Eindep.(F,(at,...,a,))
F2 = ab + ab’ + a’b - a’b’ =(a+a’)b+(a_a’)b’<2.
(1)
The above inequality holds because either a = a’ and the second term vanishes or a = -a’ and the first term vanishes. In the first case the inequality follows because la + a’1 6 2 and lb1 6 1, and similarly in the second case. Assuming “local realism”, the a’s and b’s are independent and the probabilities of joint resuits are simply means of products. Using the linearity of the mean operator M and denoting &n&r, (a, b) = M( a sb) the expectation value of the products of the results of the experiments a and b, one obtains the traditional Bell-CHSH inequality [ 21, &dep. -
(a,
6) + &dep.
Eindep.
(a,
b’)
-I Eindep. (a’,
(a’, b’) 6 2.
(2)
The quantum mechanical description associates to the first measurement the Pauli matrix au with normalized three-dimensional vectors a, and similarly for the other measurements. The mean correlation in state JI is given by the quantum expectations: EaM( a, b) = (au @Ibu)+. Hence, the Bell-CHSH inequality involves the Bell operator, t32 =aa~ba+au~b’o+a’a~ba-a’a@b’u. (3) A straightforward computation shows that B: < 8. Accordingly, the largest eigenvalue of & is 24 and the Bell-CHSH inequality can be violated by quantum state by a maximal factor of fi [4]. The above brief presentation of the standard BellCHSH inequality, valid for two qubits, motivates the following generalization for n qubits, which is defined recursively. Let a, = f 1 and a; = f 1 denote two possible outcomes of two possible measurements on the nth qubit and define F,, = ;(a,, + aL)F,,_l + $(a, - aa)FL_, < 2, where the aj cisely CHSH
Similarly to the two-qubit operator for n qubits,
(5) case, we define the Bell
a, = B,_~~~(a,u+a~u)+~~_,~~(a,u-a~u). (6) An upper bound on the eigenvalues the following lemma,
of t3, is given by
z?,” 6 2”f’.
(7)
Again, the proof follows closely the one for the twoqubit case, a,2 = a,‘_, @3;<1 +a,a;j + @_J2
6)
<2.
+
c9 ;(I
[B,-I,E$_,I
- a,a;>
8 ii0
< (1 + la, A a#2”
6 2”+l.
(8)
The maximally entangled states discussed in the next section saturate the above bound. Hence, the largest eigenvalue of f3, is 2(“+‘)j2 and the Bell-Klyshko inequality for n qubits can be violated by quantum states by a maximal factor of 2(n-1)/2. The maximal violation of inequality (5) is larger than the one presented by Mermin [ 51 for even numbers n of qubits and is larger than the inequality derived by Ardehali [ 61 for odd numbers of qubits. Inequality (5) is symmetric among the n qubits. For example, for three qubits it reads E( a, 6, c’) + E(a, b’, c) + E(a’, b, c) - E(a’, b’,c’) < 2. An interesting property of inequality (5) is that if one assumes that only m < n qubits have independent elements of reality, then, combining inequalities (4) and (7)) one obtains Em indep. qubits( Fn > < 2(n--n’f’)‘2. Indeed, applying recursively obtains (see Appendix B)
(9) the definition
(4)
one
(4)
F,!, denote the same expression F,, but with all and a$ exchanged. Inequality (4) holds for prethe same reason as the standard two-qubit Bellinequality, as presented below Eq. ( 1) . Using
Fn = $(F,-,
+ F;_,,,)F,,
+ ;(F,-,
- F;_,,)F;,. (10)
Hence, if the m last qubits are independent, then F,,, < 2 and Fh < 2, thus F,, f F,,_,. Finally, if
N. Gisin, H. Bechmann-Pasquinucci/Physics
the n - m first qubits are maximally entangled, then E nt tndep. qubits(Fn) = EQM(F,-,) < 2(n-n*+‘)‘2. Note that by symmetry the same result holds if any m qubits are independent. Accordingly, from an experimental measurement result of E( F,,) between 2(n-m+‘)/2 and 2(n-n’+2)/2 one can infer that at least m qubits are entangled [7,8]. As a first example, consider the following three-qubit mixed state with only twoqubit entanglement [9]: p = ~(Ps @ PT + Pt 8 Ps), where Ps denotes the singlet state of two qubits and PT = 1 T) (T 1the “up” state of a single qubit. Then the maximal violation of the generalized Bell-CHSH inequality (5) equals EQM( Fj) = ( 1 + fi). This value is reached for (Y= --LY’= y = -y’ = rrTr/4,,6 = n-, p’ = -7r/2, where (Yand cy’ are the angles defining the directions a and a’, respectively, in the xz plane (angle 0 corresponding to the z direction) and similarly for the four other angles. From this value one concludes rightly that the state p contains two-qubit entanglement. Finally, consider a second example of “twowise entanglement” [ 1o] : p = i (Ps @ Pt + PT 8 4 ), where PT denotes the triplet state 1 tl) + 1 Lt). The maximal violation of inequality (5) indicates again that p contains only two-qubit entanglement, though with the maximal value: EQM( F3) = 2& ( LY= 0, ayl = 7r/2, p = 3%-14, p’ = -3lr14, y = y’ = 0).
3. Maximally entangled states of n qubits In this section we consider several criteria to characterize symmetric maximally entangled (SME) states of n qubits. 3.1. Maximal
violation of Bell inequality
Maximally entangled states should maximally violate the Bell inequality. This criterion is not precise, because there are infinitely many versions of the Bell inequality. In this subsection we consider the Klyshko version presented in Section 2, because it is the natural generalization of the Bell-CHSH inequality which nicely characterizes the two-qubit Bell states. The generalized Bell-Klyshko inequality (5) for n qubits is maximally violated by the GHZ [ 111 state / 7 . . T) + 1 1 . _. 1) for aj regularly distributed in the xy plane (on the Poincare sphere) with angles ( j - I ) ( - 1 )nfl (7r/2n) with respect to the x-axis and
Letters A 246 (1998) 1-6
ai I a,j. More generally,
3
the state LYI1 . . 7) + /3 1
I . . 1) with real LYand p satisfying a2 + fi2 = 1 violate the ri-qubit inequality (5) by a factor ap2(“-‘)/‘. 3.2. Maximal entanglement
as maximal fragility
It seems natural to assume that entanglement is fragile. More specifically, if the qubits are subject to noise, acting independently on each of them, the more entangled states are more affected than the less entangled ones. Noise acting on a qubit can be modelled by a fluctuating Hamiltonian: H = /3u, where the three components of the vector /3 are independent Wiener processes (i.e. white noise). Accordingly, the density matrix of the n qubits follows the master equation n
Pt = C(u.jp,Ui
- 3p,),
(II)
./=I
where the Pauli matrices a, act on the jth qubit. Now, a state If, is fragile if under the above evolution it quickly drifts away, that is if 1(t,!,lp$ I$)1 = I Cy=, (Uj); - 3nl is large. Accordingly, a state $ is maximally fragile iff (Uj),$ = 0 for all j. This condition is equivalent to the one-qubit partial state being equal to the maximally mixed state, p1 - Trz..,(P$)
= l/2.
( 12)
Another type of fragility, natural for entanglement, is fragile under measurement interactions: if one or some qubits are measured (in the computational basis) on a maximally entangled state, then all the entanglement is destroyed, i.e. the measurement projects the state onto a product state. The GHZ states IO. . . 0) & 11 . . . 1) satisfy both of these two criteria of maximal fragility. Let us note that since maximally entangled states are maximally fragile, they are ideal candidates for ultrasensitive sensors: the drawback of fast decoherence can be turned into a potential application! 3.3. Distribution
of entangled stated
Another criterion for maximal entanglement could be the following. A state of n qubits is maximally entangled if any k < n holders of a qubit can distribute to the k-n other qubit holders a maximally entangled
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N. Gisin, H. Bechmann-Pasquinucci/Physics
state. Note that this criterion applies recursively to larger and larger number of qubits, starting from the well-known Bell states for two qubits. At first this criterion may seem in contradiction with the one of Section 3.2. However, this is not the case, provided the measurement is not done in the computational basis (the “z-basis”), but in the “x-basis”. Indeed, using the formula of the appendix the GHZ states can be rewritten as (0, n), + In, n), = (IO, k)z + Ik, k),) @ (I&n
- k)z + In - k,n - k),)
Letters A 246 (1998) 1-6
are equally probable, and H(a2,. . _, a,lai) is minimal if the outcome al fully determines all the results a2 to a,,. The GHZ states do clearly satisfy this criterion. Note that for n > 3 they are the only symmetric states satisfying this criterion (for n = 2, the third symmetric Bell state does also satisfy this criterion). Note also the similarity between this mathematical criterion and the more physical “maximally fragile” criterion of Section 3.2: maximizing H( al ) is equivalent to condition ( 12) which represents maximal fragility under noise and minimizing H(a2,. . . , an/al ) is equivalent to maximal fragility under one-qubit measurement in the computational basis.
+ (IO, k)z - IkYkj,) @ (10, n - k)z - In - k, n - k)z)
(13)
k/2 = c 12j, k)X @ (10, n - k)L + In - k, n - k)z) j=O (k+1)/2 +
12j+I,k).~(lO,n-k),-In-k,n-k),).
c
j=O (14) Hence, if k qubits are measured in the x-basis and an even number of 1 are obtained, then the remaining n-k qubits are in the GHZ state (0, n- k)z + In - k, n-k),. Else, if an odd number of 1 is obtained, the n-k qubits are in the other GHZ state, IO, n - k), - In - k, n - k),. 3.4. Mutual information of measurement outcomes on maximally entangled states Let aj denote the results of simultaneous measurements of all qubits in the computational basis. A natural criterion for maximal entanglement is that the mutual information of the n random variables aj is maximum. Recall that Z({aj}) E xi H( aj) - H(al , . _. a,), where H is the entropy function H(x) 3 -M(log(p(x))). Using the symmetry among the aj and the chain rule [ 121 H(al,..., a,) = H(al) + H(a2 ,..., a,lal), one obtains I({aj})
= (n-l>H(al)-H(a2,...,a,lal).
(15)
Accordingly, the mutual information I( {ai}) is maximal if H(al) is maximal and H(az, . . _ , a,lal) is minimal. Finally, H( ai ) is maximal if both outcomes
3.5. Partial states of maximally entangle states are maximally mixed? In Section 3.2 we found that the one-qubit partial state of a maximally fragile state is the maximally mixed state. A more general criterion for maximal entanglement could be that all partial states are maximally mixed. Since we assume all through this article symmetric states, maximally mixed states differ from states represented by a multiple of the identity matrix: maximally mixed states are homogenous mixtures of all the n + 1 symmetric states. For example, the maximally mixed state of two qubits is the mixture of the three symmetric Bell states, all three with the same weight. Note that the density matrices representing mqubit and (n - m)-qubit partial states of an n-qubit state have the same spectra. Hence, only the partial states for m = 1, . . . , [n/2] can be maximally mixed (where [n/2] is the largest integer smaller or equal to n/2). Consequently, the m-qubit partial states of a SME state should, according to this criterion, have exactly m + 1 identical eigenvalues different from zero, forallm= l,... , [n/2]. Note that the conditions for different m’s are not independent. Indeed, if the partial state for m = [n/2] is maximally mixed, then all partial states for smaller m’s are also necessarily maximally mixed. We found states of n qubits satisfying this criterion for n = 2,3,4 and 6. However, no such state exists for n = 5, nor for n 2 7! Hence, in general, no state satisfying this criterion exists. This is no surprise, since the number of constraints increases with n much faster than the number of parameters defining symmetric states. This criterion is thus not suitable to
N. Gisin, H. Bechmann-Pasquinucci/Physics
characterize maximal entanglement. Indeed, if entanglement can be measured, then maximally entangled states should exist for any number of qubits. For completeness we nevertheless list some examples of states satisfying this criterion, Jr.?,*1 = 1093) * 1333) , Ykt2
= 103
$4.11
=
3) k I1*3) - 1233) T 13,3) 3
-31(X4)
= IO.
‘b6.kl $6.2
$63
-3)(X6) = hjo,
f
1576)
(18) ( 19)
>
(20)
+ 1296) + 1496) - 3166) , 6) I!=
(17)
+ 124)
+ )4,4) ,
4) + $1274)
= 11,6) =
* &II,4)
- 3(4,4) )
f &13,4) $4.2
(16)
$13,6) + &\6,6)
,
(21) (22)
where Ij, n) denotes the sum of all product states with j 11) and n - j IO) (see Appendix A) and where the first index of the $‘s indicates the number of qubits and the second index labels the different states with maximally mixed partial states.
4. Conclusion The Bell-Klyshko inequality for n qubits has been presented. Maximally entangled quantum states violate this inequality by a factor that grows exponentially with II. For even and odd numbers of qubits, this maximal violation is larger than for the inequality devised by Mermin [ 51 and by Ardehali [ 61, respectively. If only n - m qubits are assumed independent, then the same inequality leads to a higher bound. Hence, from an experimental test of the Bell-Klyshko inequality for n qubits, one can infer a lower bound on the number of entangled qubits. Maximally entangled symmetric states of n qubits were analysed according to five different criteria. The criterion that the partial states should be maximally mixed is found to be of limited value, since no such states exist, neither for n = 5 nor for n 2 7. However, all four other criteria are shown to be compatible. The conclusion is that the two GHZ states [ 1 I] are the maximally entangled symmetric states: they violate Bell inequality maximally, they are maximally
Letters A 246 (I 998) 1-6
5
fragile and maximize the mutual information. Moreover, depending on the measurement bases, m holders of qubits can either distribute maximally entangled qubits to their II - m colleagues or leave them with product states. Clearly all states obtained by local transformation of a maximally entangled state are equally valid maximally states [ 131, though not necessarily symmetric states. Starting from the GHZ state one thus obtains 2” linearly independent maximally entangled states. Hence, the maximally entangled states form an orthogonal basis for any number of qubits, like the wellknown Bell states for two qubits. For example, in the three-qubit case the two GHZ states read I 1, I, 1) + IO, 0,O). By local transformation on the first qubit the following pair of states obtains IO, 1, 1) f / I, 0,O). Acting similarly on the second and third qubits provides a total of eight mutually orthogonal states that form a basis of the three-qubit Hilbert space. In this paper we did not address the question 01 unicity. However, our experience leads us to conjecture that the GHZ states and the states obtained from them by local transformations are the unique states that violate maximally the Bell inequality (5) and are the unique states satisfying independently the criteria of Sections 3.2,3.3 and 3.4. Hence, we conjecture that our four criteria are not only compatible, but equivalent characterization of maximally entangled states.
Acknowledgement Stimulating discussions with A. Ekert, B. Huttner, T. Mor, S. Popescu, A. Zeilinger and M. Zukowski are acknowledged. In particular we thank M. Zukowski for bringing Refs. [ 31 to our attention. H.B.-P. is supported by the Danish National Science Research Council (grant no. 9601645). This work profited also from support by the Swiss National Science Foundation.
Appendix A. Symmetric
states of n qubits
In this Appendix we summarize some useful properties and notations for symmetric states. Let ]j, n) denote the sum of all product states with j 11) and n-j /O).Forexample,/1,3)r/1,0,0)+/0,l,0)+ IO, 0, 1). Hence 1j, n) is a 2” component vector repre-
6
N. Gisin. H. Bechmann-Pasquinucci/Physics
senting a n-qubit are given by
state. The norms and inner products
_
n! (n - j) !j! .
2 Ik, m) @Ij - k,
Fn,+ F,i,
-Fn-,, 4
&+I - a;,,, 2
m - n) .
(A.21
I
a”,+1 +
It is also relatively easy to change from one basis to another. For example, if 11)X E IO), + 1l)z and IO), E IO), - II),, then
c,f_2kc;re
2
c
Accordingly read
)
the GHZ states expressed
4
Hence, assuming m-t- 1.
(A.4)
(n+l)P
12k-t l,n),.
(A.5)
k=O Similarly, the GHZ states expressed in y-basis ( I l), z
lljz + i(o),,lo),= lo),+ ill),) read n), = k(ik
Fnt+I +
(B.3) -
4
FL,+* F’
n-(lllfl)
( 10) holds for m, it also holds for
References
k=O
In,
n-Cm+l)
in the x-bases
IO,n), + In,n), = c 1%n),,
IO,n)Z f
FM~~+I)
(A.3)
nl2
c
(B.2)
(I3.4)
c~_2k_lc~~eI I& n),.
l&n): - In,n), =
>
Frnan,+~ - F;,a;,+,F’
4
k=O
F’n-(n1+l)
(nl+l)
“‘+lFn-(nt+~)
(j-e-1)/2 -
a;,+, F,-
= F,,+I + FL,+1 Fn-(n,+~)
k=O
e=o
2
+ &$,,+I = Fm”k+i 4
+
I (i-e)/2
2 ( .‘c
+
(B.1)
,
F;_,,
F’n-(nt+l)
k=O
n
(
4 +
F,,, - F'
+ 7
Fm+ FL, GlL+l
(A.1)
The decomposition of symmetric states of n qubits on symmetric states of m-qubit states is straightforward,
Ij, n), =
=
n
(j+nlk, n) = S,j,kCy E Sj,k
lj, n) =
F
Letters A 246 (1998) l-6
f indk) Ik, n)r.
C.4.6)
k=O
Appendix B. Proof of formula (10) Let us prove it by induction on m. Defining Fl (a) E 2a, formula ( 10) holds for m = 1 and, recalling that FL: = Fn,, one has
[I] J.S. Bell, Physics I (1964) 195. [2] J.F. Clauser, M.A. Home, A. Shimony, R.A. Holt, Phys. Rev. Len 23 (1969) 880. [3] D.N. Klyshko, Phys. Lett. A 172 ( 1993) 399; A.V. Belinskii, D.N. Klyshko, Phys. Usp. 36 ( 1993) 653. [4] B.S. Cirel’son, Lett. Math. Phys. 4 (1980) 93. [5] N.D. Mermin, Phys. Rev. Lett. 65 (1990) 1838. [6] M. Ardehali, Phys. Rev. A 46 (1992) 5375. [7] G. Svetlichny, Phys. Rev. D 35 ( 1987) 3066. [S] M. iukowsky, D. Kaszlikowski, Phys. Rev. A 56 ( 1997) R1682. [9] D. Aharonov, private communication. [ 101 G. Brassard, T. Mor, Multi-particle entanglement via 2particle entanglement, preprint, Universite de Montreal, 1998. [I l] D.M. Greenberger, M. Home, A. Zeilinger, in: Bell’s Theorem, Quantum Theory and Conceptions of the Universe, ed. M. Kafatos (Kluwer. Dordrecht, 1989). [ 121 Th.M. Cover, J.A. Thomas, Elements of Information Theory. (Wiley, New York, 1991) [ 131 N. Linden, S. Popescu, quant-ph 9711016.
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 246 ( 1998) l-6
Bell inequality, Bell states and maximally entangled states for n qubits N. Gisin, H. Bechmann-Pasquinucci Group of Applied Physics, University of Geneva I21 I Geneva 4, Swilzerland
Received 26 May 1998; accepted for publication 29 June 1998 Communicated
by P.R. Holland
Abstract First, we present a Bell-type inequality for n qubits, assuming that M out of the n qubifs are independent. Quantum mechanics violates this inequality by a ratio that increases exponentially with m. Hence an experiment on n qubits violating this inequality sets a lower bound on the number m of entangled qubits. Next, we propose a definition of maximally entangled states of n qubits. For this purpose we study five different criteria. Four of these criteria are found compatible. For any number n of qubits, they determine an orthogonal basis consisting of maximally entangled states generalizing the Bell states. @ 1998 Elsevier Science B.V.
1. Introduction We consider n qubits, all spatially separated from each other. The purpose of this paper is to find characterizations of maximally entangled states of the n qubits. We assume that the n qubits are equivalent (there is no privileged qubit(s)), hence we restrict ourself to symmetric states. We test the following five criteria that could, a priori, characterize such symmetric maximally entangled (SME) states: ( 1) maximally entangled states violate Bell inequality [ l] maximally; (2) maximally entangled states are maximally fragile; (3) whenever m < n qubits are measured (in some appropriate basis), the outcomes determine a maximally entangled state of the m - n remaining qubits; (4) the mutual information of measurement outcomes is maximal; (5) all partial traces of maximally entangled states
are maximally mixed. The Bell inequality we use is presented in the next section and the above criteria are made more precise and analysed in Section 3. The result of our analysis is that the last one of the above criteria is not a good characterization. However, we find that all the other four properties are compatible and define SME states. By local transformation, the SME states generate a basis of the entire Hilbert space.
2. Bell-Klyshko
inequality for n qubits
In this section we briefly present a generalization to n qubits of the Bell-CHSH [ 21 inequality well known in the two-qubit case. This generalization was first presented by Klyshko and Belinskii [ 31. Let us start with a brief review of the well-known two-qubit case. Let a = f 1 and a’ = &I denote two possible outcomes of two possible measurements on
0375.9601/98/$ - see front matter @ 1998 Eisevier Science B.V. All rights reserved. P/150375-9601(98)00.516-7
N. Gisin. H. Bechmann-Pasquinucci/Physics
2
the first qubit and similarly b = f 1 and b’ = fl for the second qubit. Let us consider the following linear combination of joint results,
ietlers A 246 (1998) 1-6
again the linearity of the mean, one obtains the BellKlyshko inequality for n qubits, Eindep.(F,(at,...,a,))
F2 = ab + ab’ + a’b - a’b’ =(a+a’)b+(a_a’)b’<2.
(1)
The above inequality holds because either a = a’ and the second term vanishes or a = -a’ and the first term vanishes. In the first case the inequality follows because la + a’1 6 2 and lb1 6 1, and similarly in the second case. Assuming “local realism”, the a’s and b’s are independent and the probabilities of joint resuits are simply means of products. Using the linearity of the mean operator M and denoting &n&r, (a, b) = M( a sb) the expectation value of the products of the results of the experiments a and b, one obtains the traditional Bell-CHSH inequality [ 21, &dep. -
(a,
6) + &dep.
Eindep.
(a,
b’)
-I Eindep. (a’,
(a’, b’) 6 2.
(2)
The quantum mechanical description associates to the first measurement the Pauli matrix au with normalized three-dimensional vectors a, and similarly for the other measurements. The mean correlation in state JI is given by the quantum expectations: EaM( a, b) = (au @Ibu)+. Hence, the Bell-CHSH inequality involves the Bell operator, t32 =aa~ba+au~b’o+a’a~ba-a’a@b’u. (3) A straightforward computation shows that B: < 8. Accordingly, the largest eigenvalue of & is 24 and the Bell-CHSH inequality can be violated by quantum state by a maximal factor of fi [4]. The above brief presentation of the standard BellCHSH inequality, valid for two qubits, motivates the following generalization for n qubits, which is defined recursively. Let a, = f 1 and a; = f 1 denote two possible outcomes of two possible measurements on the nth qubit and define F,, = ;(a,, + aL)F,,_l + $(a, - aa)FL_, < 2, where the aj cisely CHSH
Similarly to the two-qubit operator for n qubits,
(5) case, we define the Bell
a, = B,_~~~(a,u+a~u)+~~_,~~(a,u-a~u). (6) An upper bound on the eigenvalues the following lemma,
of t3, is given by
z?,” 6 2”f’.
(7)
Again, the proof follows closely the one for the twoqubit case, a,2 = a,‘_, @3;<1 +a,a;j + @_J2
6)
<2.
+
c9 ;(I
[B,-I,E$_,I
- a,a;>
8 ii
< (1 + la, A a#2”
6 2”+l.
(8)
The maximally entangled states discussed in the next section saturate the above bound. Hence, the largest eigenvalue of f3, is 2(“+‘)j2 and the Bell-Klyshko inequality for n qubits can be violated by quantum states by a maximal factor of 2(n-1)/2. The maximal violation of inequality (5) is larger than the one presented by Mermin [ 51 for even numbers n of qubits and is larger than the inequality derived by Ardehali [ 61 for odd numbers of qubits. Inequality (5) is symmetric among the n qubits. For example, for three qubits it reads E( a, 6, c’) + E(a, b’, c) + E(a’, b, c) - E(a’, b’,c’) < 2. An interesting property of inequality (5) is that if one assumes that only m < n qubits have independent elements of reality, then, combining inequalities (4) and (7)) one obtains Em indep. qubits( Fn > < 2(n--n’f’)‘2. Indeed, applying recursively obtains (see Appendix B)
(9) the definition
(4)
one
(4)
F,!, denote the same expression F,, but with all and a$ exchanged. Inequality (4) holds for prethe same reason as the standard two-qubit Bellinequality, as presented below Eq. ( 1) . Using
Fn = $(F,-,
+ F;_,,,)F,,
+ ;(F,-,
- F;_,,)F;,. (10)
Hence, if the m last qubits are independent, then F,,, < 2 and Fh < 2, thus F,, f F,,_,. Finally, if
N. Gisin, H. Bechmann-Pasquinucci/Physics
the n - m first qubits are maximally entangled, then E nt tndep. qubits(Fn) = EQM(F,-,) < 2(n-n*+‘)‘2. Note that by symmetry the same result holds if any m qubits are independent. Accordingly, from an experimental measurement result of E( F,,) between 2(n-m+‘)/2 and 2(n-n’+2)/2 one can infer that at least m qubits are entangled [7,8]. As a first example, consider the following three-qubit mixed state with only twoqubit entanglement [9]: p = ~(Ps @ PT + Pt 8 Ps), where Ps denotes the singlet state of two qubits and PT = 1 T) (T 1the “up” state of a single qubit. Then the maximal violation of the generalized Bell-CHSH inequality (5) equals EQM( Fj) = ( 1 + fi). This value is reached for (Y= --LY’= y = -y’ = rrTr/4,,6 = n-, p’ = -7r/2, where (Yand cy’ are the angles defining the directions a and a’, respectively, in the xz plane (angle 0 corresponding to the z direction) and similarly for the four other angles. From this value one concludes rightly that the state p contains two-qubit entanglement. Finally, consider a second example of “twowise entanglement” [ 1o] : p = i (Ps @ Pt + PT 8 4 ), where PT denotes the triplet state 1 tl) + 1 Lt). The maximal violation of inequality (5) indicates again that p contains only two-qubit entanglement, though with the maximal value: EQM( F3) = 2& ( LY= 0, ayl = 7r/2, p = 3%-14, p’ = -3lr14, y = y’ = 0).
3. Maximally entangled states of n qubits In this section we consider several criteria to characterize symmetric maximally entangled (SME) states of n qubits. 3.1. Maximal
violation of Bell inequality
Maximally entangled states should maximally violate the Bell inequality. This criterion is not precise, because there are infinitely many versions of the Bell inequality. In this subsection we consider the Klyshko version presented in Section 2, because it is the natural generalization of the Bell-CHSH inequality which nicely characterizes the two-qubit Bell states. The generalized Bell-Klyshko inequality (5) for n qubits is maximally violated by the GHZ [ 111 state / 7 . . T) + 1 1 . _. 1) for aj regularly distributed in the xy plane (on the Poincare sphere) with angles ( j - I ) ( - 1 )nfl (7r/2n) with respect to the x-axis and
Letters A 246 (1998) 1-6
ai I a,j. More generally,
3
the state LYI1 . . 7) + /3 1
I . . 1) with real LYand p satisfying a2 + fi2 = 1 violate the ri-qubit inequality (5) by a factor ap2(“-‘)/‘. 3.2. Maximal entanglement
as maximal fragility
It seems natural to assume that entanglement is fragile. More specifically, if the qubits are subject to noise, acting independently on each of them, the more entangled states are more affected than the less entangled ones. Noise acting on a qubit can be modelled by a fluctuating Hamiltonian: H = /3u, where the three components of the vector /3 are independent Wiener processes (i.e. white noise). Accordingly, the density matrix of the n qubits follows the master equation n
Pt = C(u.jp,Ui
- 3p,),
(II)
./=I
where the Pauli matrices a, act on the jth qubit. Now, a state If, is fragile if under the above evolution it quickly drifts away, that is if 1(t,!,lp$ I$)1 = I Cy=, (Uj); - 3nl is large. Accordingly, a state $ is maximally fragile iff (Uj),$ = 0 for all j. This condition is equivalent to the one-qubit partial state being equal to the maximally mixed state, p1 - Trz..,(P$)
= l/2.
( 12)
Another type of fragility, natural for entanglement, is fragile under measurement interactions: if one or some qubits are measured (in the computational basis) on a maximally entangled state, then all the entanglement is destroyed, i.e. the measurement projects the state onto a product state. The GHZ states IO. . . 0) & 11 . . . 1) satisfy both of these two criteria of maximal fragility. Let us note that since maximally entangled states are maximally fragile, they are ideal candidates for ultrasensitive sensors: the drawback of fast decoherence can be turned into a potential application! 3.3. Distribution
of entangled stated
Another criterion for maximal entanglement could be the following. A state of n qubits is maximally entangled if any k < n holders of a qubit can distribute to the k-n other qubit holders a maximally entangled
4
N. Gisin, H. Bechmann-Pasquinucci/Physics
state. Note that this criterion applies recursively to larger and larger number of qubits, starting from the well-known Bell states for two qubits. At first this criterion may seem in contradiction with the one of Section 3.2. However, this is not the case, provided the measurement is not done in the computational basis (the “z-basis”), but in the “x-basis”. Indeed, using the formula of the appendix the GHZ states can be rewritten as (0, n), + In, n), = (IO, k)z + Ik, k),) @ (I&n
- k)z + In - k,n - k),)
Letters A 246 (1998) 1-6
are equally probable, and H(a2,. . _, a,lai) is minimal if the outcome al fully determines all the results a2 to a,,. The GHZ states do clearly satisfy this criterion. Note that for n > 3 they are the only symmetric states satisfying this criterion (for n = 2, the third symmetric Bell state does also satisfy this criterion). Note also the similarity between this mathematical criterion and the more physical “maximally fragile” criterion of Section 3.2: maximizing H( al ) is equivalent to condition ( 12) which represents maximal fragility under noise and minimizing H(a2,. . . , an/al ) is equivalent to maximal fragility under one-qubit measurement in the computational basis.
+ (IO, k)z - IkYkj,) @ (10, n - k)z - In - k, n - k)z)
(13)
k/2 = c 12j, k)X @ (10, n - k)L + In - k, n - k)z) j=O (k+1)/2 +
12j+I,k).~(lO,n-k),-In-k,n-k),).
c
j=O (14) Hence, if k qubits are measured in the x-basis and an even number of 1 are obtained, then the remaining n-k qubits are in the GHZ state (0, n- k)z + In - k, n-k),. Else, if an odd number of 1 is obtained, the n-k qubits are in the other GHZ state, IO, n - k), - In - k, n - k),. 3.4. Mutual information of measurement outcomes on maximally entangled states Let aj denote the results of simultaneous measurements of all qubits in the computational basis. A natural criterion for maximal entanglement is that the mutual information of the n random variables aj is maximum. Recall that Z({aj}) E xi H( aj) - H(al , . _. a,), where H is the entropy function H(x) 3 -M(log(p(x))). Using the symmetry among the aj and the chain rule [ 121 H(al,..., a,) = H(al) + H(a2 ,..., a,lal), one obtains I({aj})
= (n-l>H(al)-H(a2,...,a,lal).
(15)
Accordingly, the mutual information I( {ai}) is maximal if H(al) is maximal and H(az, . . _ , a,lal) is minimal. Finally, H( ai ) is maximal if both outcomes
3.5. Partial states of maximally entangle states are maximally mixed? In Section 3.2 we found that the one-qubit partial state of a maximally fragile state is the maximally mixed state. A more general criterion for maximal entanglement could be that all partial states are maximally mixed. Since we assume all through this article symmetric states, maximally mixed states differ from states represented by a multiple of the identity matrix: maximally mixed states are homogenous mixtures of all the n + 1 symmetric states. For example, the maximally mixed state of two qubits is the mixture of the three symmetric Bell states, all three with the same weight. Note that the density matrices representing mqubit and (n - m)-qubit partial states of an n-qubit state have the same spectra. Hence, only the partial states for m = 1, . . . , [n/2] can be maximally mixed (where [n/2] is the largest integer smaller or equal to n/2). Consequently, the m-qubit partial states of a SME state should, according to this criterion, have exactly m + 1 identical eigenvalues different from zero, forallm= l,... , [n/2]. Note that the conditions for different m’s are not independent. Indeed, if the partial state for m = [n/2] is maximally mixed, then all partial states for smaller m’s are also necessarily maximally mixed. We found states of n qubits satisfying this criterion for n = 2,3,4 and 6. However, no such state exists for n = 5, nor for n 2 7! Hence, in general, no state satisfying this criterion exists. This is no surprise, since the number of constraints increases with n much faster than the number of parameters defining symmetric states. This criterion is thus not suitable to
N. Gisin, H. Bechmann-Pasquinucci/Physics
characterize maximal entanglement. Indeed, if entanglement can be measured, then maximally entangled states should exist for any number of qubits. For completeness we nevertheless list some examples of states satisfying this criterion, Jr.?,*1 = 1093) * 1333) , Ykt2
= 103
$4.11
=
3) k I1*3) - 1233) T 13,3) 3
-31(X4)
= IO.
‘b6.kl $6.2
$63
-3)(X6) = hjo,
f
1576)
(18) ( 19)
>
(20)
+ 1296) + 1496) - 3166) , 6) I!=
(17)
+ 124)
+ )4,4) ,
4) + $1274)
= 11,6) =
* &II,4)
- 3(4,4) )
f &13,4) $4.2
(16)
$13,6) + &\6,6)
,
(21) (22)
where Ij, n) denotes the sum of all product states with j 11) and n - j IO) (see Appendix A) and where the first index of the $‘s indicates the number of qubits and the second index labels the different states with maximally mixed partial states.
4. Conclusion The Bell-Klyshko inequality for n qubits has been presented. Maximally entangled quantum states violate this inequality by a factor that grows exponentially with II. For even and odd numbers of qubits, this maximal violation is larger than for the inequality devised by Mermin [ 51 and by Ardehali [ 61, respectively. If only n - m qubits are assumed independent, then the same inequality leads to a higher bound. Hence, from an experimental test of the Bell-Klyshko inequality for n qubits, one can infer a lower bound on the number of entangled qubits. Maximally entangled symmetric states of n qubits were analysed according to five different criteria. The criterion that the partial states should be maximally mixed is found to be of limited value, since no such states exist, neither for n = 5 nor for n 2 7. However, all four other criteria are shown to be compatible. The conclusion is that the two GHZ states [ 1 I] are the maximally entangled symmetric states: they violate Bell inequality maximally, they are maximally
Letters A 246 (I 998) 1-6
5
fragile and maximize the mutual information. Moreover, depending on the measurement bases, m holders of qubits can either distribute maximally entangled qubits to their II - m colleagues or leave them with product states. Clearly all states obtained by local transformation of a maximally entangled state are equally valid maximally states [ 131, though not necessarily symmetric states. Starting from the GHZ state one thus obtains 2” linearly independent maximally entangled states. Hence, the maximally entangled states form an orthogonal basis for any number of qubits, like the wellknown Bell states for two qubits. For example, in the three-qubit case the two GHZ states read I 1, I, 1) + IO, 0,O). By local transformation on the first qubit the following pair of states obtains IO, 1, 1) f / I, 0,O). Acting similarly on the second and third qubits provides a total of eight mutually orthogonal states that form a basis of the three-qubit Hilbert space. In this paper we did not address the question 01 unicity. However, our experience leads us to conjecture that the GHZ states and the states obtained from them by local transformations are the unique states that violate maximally the Bell inequality (5) and are the unique states satisfying independently the criteria of Sections 3.2,3.3 and 3.4. Hence, we conjecture that our four criteria are not only compatible, but equivalent characterization of maximally entangled states.
Acknowledgement Stimulating discussions with A. Ekert, B. Huttner, T. Mor, S. Popescu, A. Zeilinger and M. Zukowski are acknowledged. In particular we thank M. Zukowski for bringing Refs. [ 31 to our attention. H.B.-P. is supported by the Danish National Science Research Council (grant no. 9601645). This work profited also from support by the Swiss National Science Foundation.
Appendix A. Symmetric
states of n qubits
In this Appendix we summarize some useful properties and notations for symmetric states. Let ]j, n) denote the sum of all product states with j 11) and n-j /O).Forexample,/1,3)r/1,0,0)+/0,l,0)+ IO, 0, 1). Hence 1j, n) is a 2” component vector repre-
6
N. Gisin. H. Bechmann-Pasquinucci/Physics
senting a n-qubit are given by
state. The norms and inner products
_
n! (n - j) !j! .
2 Ik, m) @Ij - k,
Fn,+ F,i,
-Fn-,, 4
&+I - a;,,, 2
m - n) .
(A.21
I
a”,+1 +
It is also relatively easy to change from one basis to another. For example, if 11)X E IO), + 1l)z and IO), E IO), - II),, then
c,f_2kc;re
2
c
Accordingly read
)
the GHZ states expressed
4
Hence, assuming m-t- 1.
(A.4)
(n+l)P
12k-t l,n),.
(A.5)
k=O Similarly, the GHZ states expressed in y-basis ( I l), z
lljz + i(o),,lo),= lo),+ ill),) read n), = k(ik
Fnt+I +
(B.3) -
4
FL,+* F’
n-(lllfl)
( 10) holds for m, it also holds for
References
k=O
In,
n-Cm+l)
in the x-bases
IO,n), + In,n), = c 1%n),,
IO,n)Z f
FM~~+I)
(A.3)
nl2
c
(B.2)
(I3.4)
c~_2k_lc~~eI I& n),.
l&n): - In,n), =
>
Frnan,+~ - F;,a;,+,F’
4
k=O
F’n-(n1+l)
(nl+l)
“‘+lFn-(nt+~)
(j-e-1)/2 -
a;,+, F,-
= F,,+I + FL,+1 Fn-(n,+~)
k=O
e=o
2
+ &$,,+I = Fm”k+i 4
+
I (i-e)/2
2 ( .‘c
+
(B.1)
,
F;_,,
F’n-(nt+l)
k=O
n
(
4 +
F,,, - F'
+ 7
Fm+ FL, GlL+l
(A.1)
The decomposition of symmetric states of n qubits on symmetric states of m-qubit states is straightforward,
Ij, n), =
=
n
(j+nlk, n) = S,j,kCy E Sj,k
lj, n) =
F
Letters A 246 (1998) l-6
f indk) Ik, n)r.
C.4.6)
k=O
Appendix B. Proof of formula (10) Let us prove it by induction on m. Defining Fl (a) E 2a, formula ( 10) holds for m = 1 and, recalling that FL: = Fn,, one has
[I] J.S. Bell, Physics I (1964) 195. [2] J.F. Clauser, M.A. Home, A. Shimony, R.A. Holt, Phys. Rev. Len 23 (1969) 880. [3] D.N. Klyshko, Phys. Lett. A 172 ( 1993) 399; A.V. Belinskii, D.N. Klyshko, Phys. Usp. 36 ( 1993) 653. [4] B.S. Cirel’son, Lett. Math. Phys. 4 (1980) 93. [5] N.D. Mermin, Phys. Rev. Lett. 65 (1990) 1838. [6] M. Ardehali, Phys. Rev. A 46 (1992) 5375. [7] G. Svetlichny, Phys. Rev. D 35 ( 1987) 3066. [S] M. iukowsky, D. Kaszlikowski, Phys. Rev. A 56 ( 1997) R1682. [9] D. Aharonov, private communication. [ 101 G. Brassard, T. Mor, Multi-particle entanglement via 2particle entanglement, preprint, Universite de Montreal, 1998. [I l] D.M. Greenberger, M. Home, A. Zeilinger, in: Bell’s Theorem, Quantum Theory and Conceptions of the Universe, ed. M. Kafatos (Kluwer. Dordrecht, 1989). [ 121 Th.M. Cover, J.A. Thomas, Elements of Information Theory. (Wiley, New York, 1991) [ 131 N. Linden, S. Popescu, quant-ph 9711016.