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On the problem of contextuality in macroscopic magnetization measurements
Physics Letters A 377 (2013) 2856–2859
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On the problem of contextuality in macroscopic magnetization measurements a, b ´ Akihito Soeda a , Paweł Kurzynski , Ravishankar Ramanathan a , Andrzej Grudka b , a, c Jayne Thompson , Dagomir Kaszlikowski a,d,∗ a
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´ n, Poland c ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, Melbourne, Victoria 3010, Australia d Department of Physics, National University of Singapore, 2 Science Drive 3, 117542 Singapore, Singapore b
a r t i c l e
i n f o
Article history: Received 14 June 2013 Received in revised form 23 August 2013 Accepted 26 August 2013 Available online 3 September 2013 Communicated by P.R. Holland
a b s t r a c t We show that sharp measurements of total magnetization cannot be used to reveal contextuality in macroscopic many-body systems of spins of arbitrary dimension. We decompose each such measurement into set of projectors corresponding to well-defined value of total magnetization. We then show that such sets of projectors are too restricted to construct Kochen–Specker sets. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Quantum theory makes statistical predictions about the outcomes of different measurements. If one tries to explain these predictions in terms of some hidden variable theory, then one necessarily encounters the problem of contextuality [1]. In simple terms, contextuality means that either the act of measurement does not simply reveal preexisting outcomes or these outcomes must depend on other compatible measurements. All the existing experimental tests of contextuality were performed on systems consisting of a few microscopic particles [2]. On the other hand, contextuality seems to be irrelevant in the macroscopic regime, although the microscopic constituents of macroscopic systems obey the laws of quantum theory. In this domain we do not measure the properties of individual particles, but only observe their collective attributes, e.g., one cannot measure the spins of individual particles but only their total spin. It is an open question whether feasible macroscopic measurements with the restrictions imposed from this constraint can still be used to construct tests of contextuality. We answer this question in the negative for sharp projective measurements of total magnetization. These results suggest that macroscopic many-body spin systems do not exhibit quantum contextuality; however they do not preclude future observation of contextual behavior in any other (macroscopic) physical system. Imagine a physical system on which a set of N measurements can be performed that are represented by some physical observ-
*
Corresponding author. Tel.: +65 6516 6758; fax: +65 6516 6897. E-mail address: [email protected] (D. Kaszlikowski).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.08.039
ables A 1 , A 2 , . . . , A N . Some of these observables are assumed to be jointly measurable, say A 1 and A 2 , yielding a set of outcomes (a1 , a2 ) according to the respective joint probability distribution p (a1 , a2 ). In non-contextual hidden variable theory, these joint probability distributions are required to be the marginals of a common joint probability distribution of all observables, i.e. p (a1 , a2 , . . . , a N ) [3]. The contextuality test probes the existence of such total joint probability distribution. In order to test contextuality, some observables should be measured in more than one context. Intuitively, contexts can be understood as follows. Consider three observables A , B and C , such that observable A can be measured together with an observable B or with C but B and C cannot be measured simultaneously. In other words, [ A , B ] = [ A , C ] = 0 and [ B , C ] = 0, here measurements of B and C are understood to provide two different contexts for the measurement of A. To show that quantum theory is a contextual theory, i.e., it does not allow for a joint probability distribution, one needs both commutativity and non-commutativity. Suppose this were not the case, two possibilities arise. One being that all observables A 1 , . . . , A N commute, in which case the joint probability distribution exists trivially and can be measured directly in the laboratory. On the other hand, if all observables do not commute, a joint probability distribution that recovers the measurable N marginals can be simply constructed as p (a1 , . . . , a N ) = i =1 p (ai ). Fig. 1 provides a more detailed explanation of contexts. Each ellipsoid denotes a set of mutually commuting observables. The observable Λ belongs to all three sets, other observables in different sets do not commute. The observable Λ can be measured in three different contexts, for instance it can be measured either with A k or with B 1 or with C 3 . We say that the three observables A k , B 1 , C 3 provide contexts for the measurement of Λ.
A. Soeda et al. / Physics Letters A 377 (2013) 2856–2859
Fig. 1. Pictorial representation of observables with context. See text for more details.
Any KS-contextuality test can be recast in terms of projectors which are the simplest observables that correspond to events. Indeed, the projectors may be interpreted as logical propositions to which one assigns outcomes 1 or 0 corresponding to whether the proposition is true (event occurs) or false (event does not occur), respectively. While contextuality tests using non-projective measurements have been considered [4], it is still controversial whether they properly refute the existence of hidden variable models [5], hence the measurements we consider here are restricted to be projective measurements. In this work, we are interested in the magnetization measurements, however note that any magnetization measurement is equivalent to a set of propositions (projectors) of the form: “Is the value of magnetization along direc equal to m?”. We know from quantum mechanics that the tion n set of values m is discrete and finite, hence the number of projectors into which the magnetization operator is decomposed is finite as well. A macroscopic sample of magnetic material is modeled as a system consisting of a macroscopic number of spins. While the original Stern–Gerlach type measurement of magnetization measures the magnetic dipole moment along a given direction, the generalized Stern–Gerlach measurement [6] measures multipole moments, but it requires complicated electromagnetic fields to do so. Therefore, in this Letter we focus on the problem whether one can construct a KS set using projectors corresponding to magnetization measurements along arbitrary directions. Interestingly, we find that it is not even possible to find two different contexts for any of these projectors, hence one cannot verify KS-contextuality with these measurements.
We assume that there are N particles of spin-s. Let us denote by the magnetization operator of the ith spin along direction n S ni = k=x, y ,z nk S ki , where S ki denotes the spin operator on the ith spin along the k-axis. The total magnetization operator along this direction S n is then given by N
I1 ⊗ I2 ⊗ · · · ⊗ Ii −1 ⊗ S ni ⊗ Ii +1 ⊗ · · · ⊗ I N ,
i =1
where Ii denotes the identity operator on the ith spin. The commutator of the total magnetization operators along two and n is different directions n
S n , S
n
=
N i =1
Fig. 2. Graphical representation of spin-1 (the smallest contextual system) states , n . Spin-1 can be represented as a vector of length along two different directions n √ 2 in three-dimensional space and has three values along any measurement direction, +1, 0, −1, therefore we have three arrows per direction. Due to the Heisenberg uncertainty only one coordinate of the spin vector can be determined, namely the one along the measurement direction. The other two coordinates remain undeter (or n ) the tip of mined hence for each measurement outcome along direction n the spin vector is spread over the rim of the corresponding circle. Red circles depict the projectors P−1 ( n), P0 ( n), P+1 ( n) and blue ones depict P−1 ( n ), P0 ( n ), P+1 ( n ). [Pm ( n), Pm ( n )] = 0 if and only if the two vectors are parallel. Similar results hold for systems composed of a large number of spins of any dimension. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
and n are parallel (or anti-parallel). which is zero if and only if n Therefore no total magnetization operator has more than two contexts. On the other hand, let us express the spectral decomposition of S n by S n = m mPm ( n). We investigate whether some of these projectors have more than two contexts and show that this is not = ± the case, namely, [Pm ( n), Pn ( n )] = 0 if and only if n n (Fig. 2). Without loss of generality we may assume that |m| |n|, n lies on the X Z -plane. We denote the defines the Z -axis, and n projectors corresponding to the magnetization measurement along the Z -axis with the eigenvalue m by Pm . The projectors of the sec˜ n and are related to Pm via ond measurement are denoted by P ˜ n = D(β)Pm D† (β), where β is the rotation operator D (β), i.e., P Euler angle characterizing the rotation about the Y -axis needed to (oriented along the Z -axis) to n (in the X Z -plane). take n Since we are dealing with N 1 particles, there are two different representations of Pm . The first one is given via the tensor product of spin states corresponding to different particles
Pm =
|k1 ⊗ · · · ⊗ |k N k1 | ⊗ · · · ⊗ k N | ,
k1 +...+k N =m
where we define |ki by S iz |ki = ki |ki . The second is given via a direct sum of spin states corresponding to different values of total angular momentum
2. Lack of contexts in macroscopic systems
S n =
2857
n
I1 ⊗ I2 ⊗ · · · ⊗ Ii −1 ⊗ S ni , S i
⊗ I i +1 ⊗ · · · ⊗ I N ,
Pm =
| j , λ j , m j , λ j , m|,
(1)
j ,λ j
where j runs over all values allowed by standard angular momenta addition rules and λ j specifies a particular irreducible module of the N-spin-s particle representation of SO(3) with total angular momentum denoted by j and Z component of the magnetization denoted by m. Note, that N 1 leads to a high degeneracy of Pm and as a result guarantees that the sum in Eq. (1) runs over many different values of j. The general form of the rotation matrix (often referred to as the Wigner d-matrix) [7] is D (β) =
j , λ j , m|. It follows that
P˜ n =
˜ ,m ˜ j ,λ j ,m
j ,λ j ˜ ˜ dm j ,λ j ,m,m ˜ ,m (β)| j , λ j , m ×
˜ j, λ j , m ˜ , dm˜ ,n (β) dm˜ ,n (β)| j , λ j , m j ,λ j
j ,λ j
(2)
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A. Soeda et al. / Physics Letters A 377 (2013) 2856–2859
(m−m ,m+m )
p j −m
(cos β) = j − m ! j + m !
and
[Pm , P˜ n ] =
˜ j ,λ j ,m
j ,λ j j ,λ j ˜| dm,n (β) dm˜ ,n (β) | j , λ j , m j , λ j , m
˜ j , λ j , m| . − | j, λ j , m
(3)
The above commutator equals zero if and only if all terms of the corresponding matrix vanish. Let us define
j ,λ Γk,k j = j , λ j , k|[Pm , P˜ n ] j , λ j , k .
(4)
In the above we do not consider off-diagonal terms corresponding to different j’s and λ’s, since they are trivially equal to zero. One easily finds
j ,λ j ,λ j ,λ j ,λ Γk,k j = dm,nj (β) δk,m dk ,nj (β) − δk ,m dk,n j (β) .
(5)
j ,λ j
The commutator vanishes if Γk,k = 0 for all allowed j , λ j , k, k . Since there is no dependency on λ j , from now on we omit this superscript. Below we show that there always exists a set of j , k, k j for which Γk,k = 0. j
First, let us note that Γk,k can be nonzero only for k = m and k = m, or for k = m and k = m. Without loss of generality, it is enough to show that there exist j and k = m for which j
j
dm,n (β) dk,n (β) = 0. Note that j and k must satisfy sN j |m|, |n|, |k|; moreover j is lower bounded by 0 or s when N is even or odd, respectively. At this stage let us write explicitly j dm m (β)
1/2 m−m ( j + m)!( j − m)! β = sin ( j + m )!( j − m )! 2 m+m β (m−m ,m+m ) × cos P j −m (cos β),
(6)
2
(α ,β)
where P n
(x) is the Jacobi polynomial [8]. j
We now find conditions under which dm ,m (β) is nonzero. Let us note that in case of two nonparallel magnetization directions (0 < β < π ) all trigonometric functions in the above forj
mula are nonzero. Therefore dm m (β) is nonzero if and only if (m−m ,m+m ) P j −m (cos β) is nonzero. The Jacobi polynomials satisfy the
following recursion relation, (α ,β)
f (n, α , β) P n
(α ,β)
( z) = g (n, α , β) P n−1 ( z) (α ,β)
− h(n, α , β) P n−2 ( z) ,
n = 2, 3, . . . , (7)
where f (n, α , β) = 2n(n + α +β)(2n + α +β − 2), g (n, α , β) = (2n + α +β − 1){(2n + α +β)(2n + α +β − 2)z + α 2 −β 2 }, and h(n, α , β) = 2(n + α − 1)(n + β − 1)(2n + α + β), and their generating function is given by ∞
(α ,β)
Pn
( z) w n = 2α +β R −1 (1 − w + R )−α (1 + w + R )−β ,
(8)
n =0
where R = (1 − 2zw + w 2 )1/2 [9]. These relations imply that, for any given (α ,β)
α and β , at least
(α ,β)
or P n+1 is nonzero. Otherwise, Eq. (7) implies that (α ,β) = 0 for any k n + 2, thus the generating function (8) must Pk be a finite polynomial of w, which is a contradiction. Therefore, for any m and n such that |m|, |n| sN − 1 and β = 0 (mod π ), sN sN −1 at least one of dm ,n (β) = 0 or dm,n (β) = 0 holds. Next we prove that for any j such that j |n| and β = 0 one of P n
j
(m−m ,m+m )
(mod π ), d± j,n (β) = 0. The Jacobi polynomial P j −m is given by
×
(cos β)
s
(−1) j −m−s (sin2 β2 ) j −m−s (cos2 β2 )s , s!( j − m − s)!(m + m + s)!( j − m − s)!
(9)
where s runs over all integers such that the factorials take nonnegative integers. The first factorial implies that s 0, while 0 s also holds when |m | = j from the second or fourth factorial. There(m−m ,m+m )
fore P j −m
(cos β) is nonzero for any β when |m | = j. From j
Eq. (6) we see that this means d± j ,n (β) = 0. We are now ready to identify an appropriate choice of j and m to prove our claim. If |m| = sN, j must equal sN and set k = −m. On the other hand, if |m| sN − 1, choose sN or sN − 1 for j, sN sN −1 where at least one of dm ,n (β) or dm,n (β) is guaranteed to be nonzero, and set k = j or k = − j, where one of these choices sat = ± isfies k = m. Therefore, [Pm ( n), Pn ( n )] = 0 if and only if n n . 3. Discussion and conclusions We investigated whether contextuality can be probed using projectors corresponding to macroscopic magnetization measurements. We showed that since each such projector appears in a single context, they cannot be used for KS-contextuality tests. To have measurements with multiple contexts for some projectors we must seek other macroscopic quantities; these however involve higher moments which require considerably more complicated non-linear phenomena [6]. This result on sharp projective measurements complements other work demonstrating classical physics can emerge in macroscopic systems if only coarse-grained measurements are available [10]. This manuscript’s detailed analysis of macroscopic many-body spin systems highlights problems with observing contextuality in the macroscopic domain, however these results do not preclude contextuality in other macroscopic physical systems. Interestingly, contexts are known to exist for any system of a dimension greater than two [1,12,11]. Hence, as far as the dimension is concerned, a macroscopic system is contextual. In a sense, the restrictions imposed on the set of measurements lead to a simplification of quantum theory. This prevents the testability of certain non-intuitive features such as contextuality. On the contrary, other features such as entanglement [13,14] are still preserved under the same restriction. Similar behavior was observed in a toy model of quantum theory proposed in [15]. An interesting open question is to investigate the complexity of measurements needed for contextuality to emerge. A further question could be to identify the exact trade-off between the size of the system and the complexity of the measurements that can be performed on it. Acknowledgements This work is supported by the National Research Foundation and Ministry of Education in Singapore. P.K. was supported by the Foundation for Polish Science. A.G. acknowledges the support of the Polish Ministry of Science and Higher Education grant No. IdP2011 000361. References [1] [2] [3] [4] [5] [6] [7]
S. Kochen, E.P. Specker, J. Math. Mech. 17 (1967) 59. J. Thompson, et al., arXiv:1304.1292, 2013. A. Fine, Phys. Rev. Lett. 48 (1982) 291. A. Cabello, Phys. Rev. Lett. 90 (2003) 190401. A. Grudka, P. Kurzynski, Phys. Rev. Lett. 100 (2008) 160401. A.R. Swift, R. Wright, J. Math. Phys. 21 (1980) 77. V. Devanathan, Angular Momentum Techniques in Quantum Mechanics, Kluwer Academic Publishers, 2002.
A. Soeda et al. / Physics Letters A 377 (2013) 2856–2859
[8] L.C. Biedenharn, J.D. Louck, Angular Momentum in Quantum Physics, Addison– Wesley, Canada, 1981. [9] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1970. [10] J. Kofler, C. Brukner, Phys. Rev. Lett. 99 (2007) 180403.
[11] [12] [13] [14] [15]
P. Badziag, et al., Phys. Rev. Lett. 103 (2009) 050401. A.M. Gleason, J. Math. Mech. 6 (1957) 885. R. Ramanathan, et al., Phys. Rev. Lett. 107 (2011) 060405. B. Julsgaard, et al., Nature (London) 413 (2001) 400. R.W. Spekkens, Phys. Rev. A 75 (2007) 032110.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On the problem of contextuality in macroscopic magnetization measurements a, b ´ Akihito Soeda a , Paweł Kurzynski , Ravishankar Ramanathan a , Andrzej Grudka b , a, c Jayne Thompson , Dagomir Kaszlikowski a,d,∗ a
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´ n, Poland c ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, Melbourne, Victoria 3010, Australia d Department of Physics, National University of Singapore, 2 Science Drive 3, 117542 Singapore, Singapore b
a r t i c l e
i n f o
Article history: Received 14 June 2013 Received in revised form 23 August 2013 Accepted 26 August 2013 Available online 3 September 2013 Communicated by P.R. Holland
a b s t r a c t We show that sharp measurements of total magnetization cannot be used to reveal contextuality in macroscopic many-body systems of spins of arbitrary dimension. We decompose each such measurement into set of projectors corresponding to well-defined value of total magnetization. We then show that such sets of projectors are too restricted to construct Kochen–Specker sets. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Quantum theory makes statistical predictions about the outcomes of different measurements. If one tries to explain these predictions in terms of some hidden variable theory, then one necessarily encounters the problem of contextuality [1]. In simple terms, contextuality means that either the act of measurement does not simply reveal preexisting outcomes or these outcomes must depend on other compatible measurements. All the existing experimental tests of contextuality were performed on systems consisting of a few microscopic particles [2]. On the other hand, contextuality seems to be irrelevant in the macroscopic regime, although the microscopic constituents of macroscopic systems obey the laws of quantum theory. In this domain we do not measure the properties of individual particles, but only observe their collective attributes, e.g., one cannot measure the spins of individual particles but only their total spin. It is an open question whether feasible macroscopic measurements with the restrictions imposed from this constraint can still be used to construct tests of contextuality. We answer this question in the negative for sharp projective measurements of total magnetization. These results suggest that macroscopic many-body spin systems do not exhibit quantum contextuality; however they do not preclude future observation of contextual behavior in any other (macroscopic) physical system. Imagine a physical system on which a set of N measurements can be performed that are represented by some physical observ-
*
Corresponding author. Tel.: +65 6516 6758; fax: +65 6516 6897. E-mail address: [email protected] (D. Kaszlikowski).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.08.039
ables A 1 , A 2 , . . . , A N . Some of these observables are assumed to be jointly measurable, say A 1 and A 2 , yielding a set of outcomes (a1 , a2 ) according to the respective joint probability distribution p (a1 , a2 ). In non-contextual hidden variable theory, these joint probability distributions are required to be the marginals of a common joint probability distribution of all observables, i.e. p (a1 , a2 , . . . , a N ) [3]. The contextuality test probes the existence of such total joint probability distribution. In order to test contextuality, some observables should be measured in more than one context. Intuitively, contexts can be understood as follows. Consider three observables A , B and C , such that observable A can be measured together with an observable B or with C but B and C cannot be measured simultaneously. In other words, [ A , B ] = [ A , C ] = 0 and [ B , C ] = 0, here measurements of B and C are understood to provide two different contexts for the measurement of A. To show that quantum theory is a contextual theory, i.e., it does not allow for a joint probability distribution, one needs both commutativity and non-commutativity. Suppose this were not the case, two possibilities arise. One being that all observables A 1 , . . . , A N commute, in which case the joint probability distribution exists trivially and can be measured directly in the laboratory. On the other hand, if all observables do not commute, a joint probability distribution that recovers the measurable N marginals can be simply constructed as p (a1 , . . . , a N ) = i =1 p (ai ). Fig. 1 provides a more detailed explanation of contexts. Each ellipsoid denotes a set of mutually commuting observables. The observable Λ belongs to all three sets, other observables in different sets do not commute. The observable Λ can be measured in three different contexts, for instance it can be measured either with A k or with B 1 or with C 3 . We say that the three observables A k , B 1 , C 3 provide contexts for the measurement of Λ.
A. Soeda et al. / Physics Letters A 377 (2013) 2856–2859
Fig. 1. Pictorial representation of observables with context. See text for more details.
Any KS-contextuality test can be recast in terms of projectors which are the simplest observables that correspond to events. Indeed, the projectors may be interpreted as logical propositions to which one assigns outcomes 1 or 0 corresponding to whether the proposition is true (event occurs) or false (event does not occur), respectively. While contextuality tests using non-projective measurements have been considered [4], it is still controversial whether they properly refute the existence of hidden variable models [5], hence the measurements we consider here are restricted to be projective measurements. In this work, we are interested in the magnetization measurements, however note that any magnetization measurement is equivalent to a set of propositions (projectors) of the form: “Is the value of magnetization along direc equal to m?”. We know from quantum mechanics that the tion n set of values m is discrete and finite, hence the number of projectors into which the magnetization operator is decomposed is finite as well. A macroscopic sample of magnetic material is modeled as a system consisting of a macroscopic number of spins. While the original Stern–Gerlach type measurement of magnetization measures the magnetic dipole moment along a given direction, the generalized Stern–Gerlach measurement [6] measures multipole moments, but it requires complicated electromagnetic fields to do so. Therefore, in this Letter we focus on the problem whether one can construct a KS set using projectors corresponding to magnetization measurements along arbitrary directions. Interestingly, we find that it is not even possible to find two different contexts for any of these projectors, hence one cannot verify KS-contextuality with these measurements.
We assume that there are N particles of spin-s. Let us denote by the magnetization operator of the ith spin along direction n S ni = k=x, y ,z nk S ki , where S ki denotes the spin operator on the ith spin along the k-axis. The total magnetization operator along this direction S n is then given by N
I1 ⊗ I2 ⊗ · · · ⊗ Ii −1 ⊗ S ni ⊗ Ii +1 ⊗ · · · ⊗ I N ,
i =1
where Ii denotes the identity operator on the ith spin. The commutator of the total magnetization operators along two and n is different directions n
S n , S
n
=
N i =1
Fig. 2. Graphical representation of spin-1 (the smallest contextual system) states , n . Spin-1 can be represented as a vector of length along two different directions n √ 2 in three-dimensional space and has three values along any measurement direction, +1, 0, −1, therefore we have three arrows per direction. Due to the Heisenberg uncertainty only one coordinate of the spin vector can be determined, namely the one along the measurement direction. The other two coordinates remain undeter (or n ) the tip of mined hence for each measurement outcome along direction n the spin vector is spread over the rim of the corresponding circle. Red circles depict the projectors P−1 ( n), P0 ( n), P+1 ( n) and blue ones depict P−1 ( n ), P0 ( n ), P+1 ( n ). [Pm ( n), Pm ( n )] = 0 if and only if the two vectors are parallel. Similar results hold for systems composed of a large number of spins of any dimension. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
and n are parallel (or anti-parallel). which is zero if and only if n Therefore no total magnetization operator has more than two contexts. On the other hand, let us express the spectral decomposition of S n by S n = m mPm ( n). We investigate whether some of these projectors have more than two contexts and show that this is not = ± the case, namely, [Pm ( n), Pn ( n )] = 0 if and only if n n (Fig. 2). Without loss of generality we may assume that |m| |n|, n lies on the X Z -plane. We denote the defines the Z -axis, and n projectors corresponding to the magnetization measurement along the Z -axis with the eigenvalue m by Pm . The projectors of the sec˜ n and are related to Pm via ond measurement are denoted by P ˜ n = D(β)Pm D† (β), where β is the rotation operator D (β), i.e., P Euler angle characterizing the rotation about the Y -axis needed to (oriented along the Z -axis) to n (in the X Z -plane). take n Since we are dealing with N 1 particles, there are two different representations of Pm . The first one is given via the tensor product of spin states corresponding to different particles
Pm =
|k1 ⊗ · · · ⊗ |k N k1 | ⊗ · · · ⊗ k N | ,
k1 +...+k N =m
where we define |ki by S iz |ki = ki |ki . The second is given via a direct sum of spin states corresponding to different values of total angular momentum
2. Lack of contexts in macroscopic systems
S n =
2857
n
I1 ⊗ I2 ⊗ · · · ⊗ Ii −1 ⊗ S ni , S i
⊗ I i +1 ⊗ · · · ⊗ I N ,
Pm =
| j , λ j , m j , λ j , m|,
(1)
j ,λ j
where j runs over all values allowed by standard angular momenta addition rules and λ j specifies a particular irreducible module of the N-spin-s particle representation of SO(3) with total angular momentum denoted by j and Z component of the magnetization denoted by m. Note, that N 1 leads to a high degeneracy of Pm and as a result guarantees that the sum in Eq. (1) runs over many different values of j. The general form of the rotation matrix (often referred to as the Wigner d-matrix) [7] is D (β) =
j , λ j , m|. It follows that
P˜ n =
˜ ,m ˜ j ,λ j ,m
j ,λ j ˜ ˜ dm j ,λ j ,m,m ˜ ,m (β)| j , λ j , m ×
˜ j, λ j , m ˜ , dm˜ ,n (β) dm˜ ,n (β)| j , λ j , m j ,λ j
j ,λ j
(2)
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A. Soeda et al. / Physics Letters A 377 (2013) 2856–2859
(m−m ,m+m )
p j −m
(cos β) = j − m ! j + m !
and
[Pm , P˜ n ] =
˜ j ,λ j ,m
j ,λ j j ,λ j ˜| dm,n (β) dm˜ ,n (β) | j , λ j , m j , λ j , m
˜ j , λ j , m| . − | j, λ j , m
(3)
The above commutator equals zero if and only if all terms of the corresponding matrix vanish. Let us define
j ,λ Γk,k j = j , λ j , k|[Pm , P˜ n ] j , λ j , k .
(4)
In the above we do not consider off-diagonal terms corresponding to different j’s and λ’s, since they are trivially equal to zero. One easily finds
j ,λ j ,λ j ,λ j ,λ Γk,k j = dm,nj (β) δk,m dk ,nj (β) − δk ,m dk,n j (β) .
(5)
j ,λ j
The commutator vanishes if Γk,k = 0 for all allowed j , λ j , k, k . Since there is no dependency on λ j , from now on we omit this superscript. Below we show that there always exists a set of j , k, k j for which Γk,k = 0. j
First, let us note that Γk,k can be nonzero only for k = m and k = m, or for k = m and k = m. Without loss of generality, it is enough to show that there exist j and k = m for which j
j
dm,n (β) dk,n (β) = 0. Note that j and k must satisfy sN j |m|, |n|, |k|; moreover j is lower bounded by 0 or s when N is even or odd, respectively. At this stage let us write explicitly j dm m (β)
1/2 m−m ( j + m)!( j − m)! β = sin ( j + m )!( j − m )! 2 m+m β (m−m ,m+m ) × cos P j −m (cos β),
(6)
2
(α ,β)
where P n
(x) is the Jacobi polynomial [8]. j
We now find conditions under which dm ,m (β) is nonzero. Let us note that in case of two nonparallel magnetization directions (0 < β < π ) all trigonometric functions in the above forj
mula are nonzero. Therefore dm m (β) is nonzero if and only if (m−m ,m+m ) P j −m (cos β) is nonzero. The Jacobi polynomials satisfy the
following recursion relation, (α ,β)
f (n, α , β) P n
(α ,β)
( z) = g (n, α , β) P n−1 ( z) (α ,β)
− h(n, α , β) P n−2 ( z) ,
n = 2, 3, . . . , (7)
where f (n, α , β) = 2n(n + α +β)(2n + α +β − 2), g (n, α , β) = (2n + α +β − 1){(2n + α +β)(2n + α +β − 2)z + α 2 −β 2 }, and h(n, α , β) = 2(n + α − 1)(n + β − 1)(2n + α + β), and their generating function is given by ∞
(α ,β)
Pn
( z) w n = 2α +β R −1 (1 − w + R )−α (1 + w + R )−β ,
(8)
n =0
where R = (1 − 2zw + w 2 )1/2 [9]. These relations imply that, for any given (α ,β)
α and β , at least
(α ,β)
or P n+1 is nonzero. Otherwise, Eq. (7) implies that (α ,β) = 0 for any k n + 2, thus the generating function (8) must Pk be a finite polynomial of w, which is a contradiction. Therefore, for any m and n such that |m|, |n| sN − 1 and β = 0 (mod π ), sN sN −1 at least one of dm ,n (β) = 0 or dm,n (β) = 0 holds. Next we prove that for any j such that j |n| and β = 0 one of P n
j
(m−m ,m+m )
(mod π ), d± j,n (β) = 0. The Jacobi polynomial P j −m is given by
×
(cos β)
s
(−1) j −m−s (sin2 β2 ) j −m−s (cos2 β2 )s , s!( j − m − s)!(m + m + s)!( j − m − s)!
(9)
where s runs over all integers such that the factorials take nonnegative integers. The first factorial implies that s 0, while 0 s also holds when |m | = j from the second or fourth factorial. There(m−m ,m+m )
fore P j −m
(cos β) is nonzero for any β when |m | = j. From j
Eq. (6) we see that this means d± j ,n (β) = 0. We are now ready to identify an appropriate choice of j and m to prove our claim. If |m| = sN, j must equal sN and set k = −m. On the other hand, if |m| sN − 1, choose sN or sN − 1 for j, sN sN −1 where at least one of dm ,n (β) or dm,n (β) is guaranteed to be nonzero, and set k = j or k = − j, where one of these choices sat = ± isfies k = m. Therefore, [Pm ( n), Pn ( n )] = 0 if and only if n n . 3. Discussion and conclusions We investigated whether contextuality can be probed using projectors corresponding to macroscopic magnetization measurements. We showed that since each such projector appears in a single context, they cannot be used for KS-contextuality tests. To have measurements with multiple contexts for some projectors we must seek other macroscopic quantities; these however involve higher moments which require considerably more complicated non-linear phenomena [6]. This result on sharp projective measurements complements other work demonstrating classical physics can emerge in macroscopic systems if only coarse-grained measurements are available [10]. This manuscript’s detailed analysis of macroscopic many-body spin systems highlights problems with observing contextuality in the macroscopic domain, however these results do not preclude contextuality in other macroscopic physical systems. Interestingly, contexts are known to exist for any system of a dimension greater than two [1,12,11]. Hence, as far as the dimension is concerned, a macroscopic system is contextual. In a sense, the restrictions imposed on the set of measurements lead to a simplification of quantum theory. This prevents the testability of certain non-intuitive features such as contextuality. On the contrary, other features such as entanglement [13,14] are still preserved under the same restriction. Similar behavior was observed in a toy model of quantum theory proposed in [15]. An interesting open question is to investigate the complexity of measurements needed for contextuality to emerge. A further question could be to identify the exact trade-off between the size of the system and the complexity of the measurements that can be performed on it. Acknowledgements This work is supported by the National Research Foundation and Ministry of Education in Singapore. P.K. was supported by the Foundation for Polish Science. A.G. acknowledges the support of the Polish Ministry of Science and Higher Education grant No. IdP2011 000361. References [1] [2] [3] [4] [5] [6] [7]
S. Kochen, E.P. Specker, J. Math. Mech. 17 (1967) 59. J. Thompson, et al., arXiv:1304.1292, 2013. A. Fine, Phys. Rev. Lett. 48 (1982) 291. A. Cabello, Phys. Rev. Lett. 90 (2003) 190401. A. Grudka, P. Kurzynski, Phys. Rev. Lett. 100 (2008) 160401. A.R. Swift, R. Wright, J. Math. Phys. 21 (1980) 77. V. Devanathan, Angular Momentum Techniques in Quantum Mechanics, Kluwer Academic Publishers, 2002.
A. Soeda et al. / Physics Letters A 377 (2013) 2856–2859
[8] L.C. Biedenharn, J.D. Louck, Angular Momentum in Quantum Physics, Addison– Wesley, Canada, 1981. [9] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1970. [10] J. Kofler, C. Brukner, Phys. Rev. Lett. 99 (2007) 180403.
[11] [12] [13] [14] [15]
P. Badziag, et al., Phys. Rev. Lett. 103 (2009) 050401. A.M. Gleason, J. Math. Mech. 6 (1957) 885. R. Ramanathan, et al., Phys. Rev. Lett. 107 (2011) 060405. B. Julsgaard, et al., Nature (London) 413 (2001) 400. R.W. Spekkens, Phys. Rev. A 75 (2007) 032110.
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