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Approximation of quantum assemblages
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Physics Letters A ••• (••••) ••••••
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Approximation of quantum assemblages Xiaojing Yan a,∗ , Zhi Yin b , Longsuo Li a a b
School of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Institute of Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150006, PR China
a r t i c l e
i n f o
Article history: Received 30 May 2019 Received in revised form 18 October 2019 Accepted 21 October 2019 Available online xxxx Communicated by M.G.A. Paris
a b s t r a c t In this paper, we derive a set of projectors on a large Hilbert space which can universally work for approximating quantum assemblages with binary inputs and outputs. The dimension of the Hilbert space depends on the accuracy of the approximation. © 2019 Elsevier B.V. All rights reserved.
Keywords: Quantum steering Quantum realization Quantum assemblages Approximation
1. Introduction The quantum steering was introduced by Schrödinger [1] and reformulated by Wiseman et al. [2] in 2007. Roughly speaking, in this scenario, Alice would like to steer Bob’s state by manipulating the measurements on her side. An assemblage σ is a set of unnormalized states {σxa } which Alice steers Bob into, with her measurement x and outcome a. Thus it is used to describe the correlation between Alice and Bob. It is a natural to ask whether a given assemblage can be produced by quantum mechanics (has a quantum realization) in finitedimensional Hilbert space, or whether it can be approximated at least. When Bob’s local dimension is finite, Schrödinger [1] and later Hughston-Jozsa-Wooters [3] gave a positive answer to this question. When Bob’s local dimension is infinite, the answer becomes negative (see Theorem 2.2). In fact, the question above is a reformulation of “Tsirelson’s problem” [4] in the quantum steering scenario, which asks the compatibility of describing the quantum composite systems by using a single Hilbert space or by a tensor product of two Hilbert spaces [5]. Tsirelson’s problem was solved by Navascués and Perez-Garcia with a negative answer [4]. It still remains open in the Bell scenario [6–10]. However, there is a special case, namely, when the inputs and outputs of Alice are binary. Navascués et al. showed that an assemblage with binary inputs and outputs can be approximated by a
*
Corresponding author. E-mail address: [email protected] (X. Yan).
https://doi.org/10.1016/j.physleta.2019.126082 0375-9601/© 2019 Elsevier B.V. All rights reserved.
sequence of assemblages that have quantum realization regardless of the dimension of Bob’s local Hilbert space [11]. To approximate a given assemblage, it is necessary to determine a sequence of quantum states and projective measurements on Alice’s side. In this paper, based on Navascués’ results, we construct a set of projectors on a large Hilbert space by sampling their principle angles uniformly in a circle. An intriguing result is that these fixed projectors can universally work for all assemblages (see Theorem 3.3). It shall be noted that the required dimension of the Hilbert space depends on the accuracy of the approximation. 2. Quantum assemblages and their realizations Definition of quantum assemblages An assemblage is an ensemble of unnormalized states. It was first introduced by Schrödinger [1] and revisited by Wiseman et al. [2]. For any fixed integers M , K , an assemblage on the Hilbert space H 2 can be mathematically defined a=0,..., K −1 [13,14] as a set of (unnormalized) states σ = {σxa }x=0,..., M −1 ⊂ a B ( H 2 ), such a σx does not depend on x and that the sum a Tr a σx = 12 for every x, where 12 denotes the identity of B ( H 2 ). Definition. [3]. We say that an assemblage σ = {σxa }x,a has a quantum realization if there exist a Hilbert space H 1 and a quantum state ρ ∈ H 1 ⊗ H 2 , such that
σxa = Tr1 [ρ ( P x,a ⊗ 12 )],
(2.1)
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2
where Tr1 is the partial trace for the first part of the tensor product, and { P x,a }x,a are projective measurements on H 1 such that a P x,a = 11 for every x.
where A x = P x,0 − P x,1 is a self-adjoint unitary operator of A1 . Hence we can say σ = {σx }x has a quantum realization if Equation (2.3) holds.
When dim( H 2 ) < ∞, Schrödinger (or called the HJW theorem [3]) proved that every assemblage has a quantum realization.
Theorem 2.2. There are a C ∗ -algebra A2 ⊂ B ( H 2 ) with dim( H 2 ) = ∞, and a dichotomic assemblage σ = {σx }x on A2 , such that σ does not have a quantum realization.
Theorem 2.1. [3,8]. Suppose dim( H 2 ) = d2 < ∞, then any assema=0,..., K −1 blage σ = {σxa }x=0,..., M −1 ⊂ B ( H 2 ) has a quantum realization. Moreover, the dimension of H 1 can be taken as d2 . C ∗ -algebra terminologies for quantum assemblages As mentioned in the introduction, one of our main results is that the HJW theorem doesn’t hold when dim( H 2 ) = ∞. To deal with the infinite dimension, we adopt use the C ∗ -algebra terminologies. We will follow the line of Fritz’s paper [8]. Recall that a C ∗ -algebra A ⊂ B ( H ) is a Banach ∗-algebra where the ∗ operation is compatible with its norm · as follows: x∗ x = xx∗ = x2 , x ∈ A. We have the following recipe for the quantum theory by using C ∗ algebra terminologies: 1. The algebra of observables of Alice (resp. Bob) is described by an unital C ∗ -algebra A1 ⊂ B ( H 1 ) (resp. A2 ⊂ B ( H 2 )). 2. A state ρ : A → C is a unital semidefinite functional on the algebra A. By the famous Riesz representation theorem, we have ρ (x) = Tr(ρ x), x ∈ A, where ρ is the usual quantum state in B ( H ). 3. The composite system is described by the minimal tensor product of A1 ⊗min A2 ⊂ B ( H 1 ⊗ H 2 ). We refer [8] and [12] for more about the tensor products. With the above recipe, we can define the notion of assemblages analogously by using the C ∗ -algebra language. For any fixed integers M , K , an assemblage can be defined [8] as a set of (unnormal a a=0,..., K −1 ized) states σ = {σxa (·)}x=0,..., M −1 on A2 , such that the sum a σx a does not depend on x and σ (1 ) = 1 for every x, where 1 2 2 is a x the identity of the algebra A2 . Definition. [8]. We say that an assemblage σ = {σxa (·)}x,a on A2 has a quantum realization if there are a unital C ∗ -algebra A1 ⊂ B ( H 1 ) and a state ρ on A1 ⊗min A2 , such that
σxa (·) = ρ [ P x,a ⊗ ·],
(2.2)
where { P x,a }x,a are projectors of A1 such that a P x,a = 11 for every x. Indeed, in the case of dim( H 2 ) < ∞, Equation (2.1) coincides with Equation (2.2). The HJW theorem fails in infinite dimensions Now, we are ready to present our main result in this section. To this end, we recall the notion of dichotomic assemblages [13–15]. Definition. For a given integer M, the dichotomic assemblage {σx (·)}x=0,..., M −1 on A2 is given by
σ=
σx = σx0 − σx1 ,
σ
a x (·)
=
Tr d
⊗E
Id + (−1)a U x DU x∗ 2
· ,
a = 0, 1 , x = 0, . . . , M − 1 , where E is the expectation respect to the probability space (, P ). Let
σx (·) = σx0 (·) − σx1 (·) =
Tr d
⊗ E[U x DU x∗ ·], x = 0, . . . , M − 1.
Obviously, H 2 = C d ⊗ L 2 (, P ), and it is easy to see that σ = {σx }x=0,..., M −1 is a dichotomic assemblage on A2 . Define a set of operators F = { F x }x=0,..., M −1 ⊂ A2 as the following:
F x = U x DU x∗ , x = 0, . . . , M − 1. It is easy to see that
σx ( F x ) = E
x=0,..., M −1
Tr[ D 2 ]
x=0,..., M −1
d
= M.
(2.4)
On the other hand, suppose that the dichotomic assemblage σ can be represented by Equation (2.3), i.e., there are a state ρ on A1 ⊗min A2 and operators A x ∈ A1 , x = 0, . . . , M − 1, such that
σx (·) = ρ [ A x ⊗ ·]. Then we have
x=0,..., M −1
σx ( F x ) =
ρ[ Ax ⊗ F x]
x=0,..., M −1
≤ A ⊗ F x x , x=0,..., M −1
(2.5)
where · is the operator norm on A1 ⊗min A2 . It is well known in the random matrix theory that (e.g. see [17,18])
√ Ax ⊗ F x ≤2 M
as d → ∞. Therefore, we conclude that Equation (2.4) will be violated as d → ∞, which completes our proof. 2
a=0,1
where {σxa }x=0,..., M is an assemblage on A2 for K = 2. a=0,1
Suppose {σxa }x=0,..., M −1 has a quantum realization, then it follows from Equation (2.2) that we have
σx (·) = ρ [ A x ⊗ ·],
Proof. Let M > 2, and suppose D is a d × d deterministic diagonal matrix, where the diagonal terms of D are either 1 or −1 and Tr[ D ] = 0. Let U x , x = 0, . . . , M be independent Haarrandom matrices in the group of unitary matrices U (d). Thus U x , x = 0, . . . , M are independent random variables of some probability space (, P ) [16]. Now, we consider the C ∗ -algebra A2 := Md ( L ∞ (, P )) which is commonly used in the random matrix theory. Define the following unnormalized states on A2 :
(2.3)
Remark. Due to the random matrix theory, we note that the local (in Bob’s side) observables F x = U x DU x∗ , x = 0, . . . , M − 1 in the observable algebra Md (C) (with a faithful state Tr ⊗ E) can be d described by random Haar unitaries u x , x = 0, . . . , M − 1 in some
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infinite-dimensional C ∗ -algebra A2 with a faithful state φ (thus the pair (A2 , φ) is called a C ∗ -probability space [16]) when d → ∞. Therefore, we can just assume Bob’s system is described by A2 , and write
σxa (·) = φ
1 + (−1)a u x 2
· ,
Definition. Suppose H is a Hilbert space given by n
H=
σx ( F x ) = M ,
σ has a quantum realization, then we have √ σx ( F x ) ≤ A x ⊗ u x ≤ 2 M , x x and if we assume
Q 0,0 = ⊕nk=1 Q 0k,0 , Q 0,1 = 1 − Q 0,0 ,
Q 1k,0 and
Remark. It is crucial that F x is random. Otherwise, the operator norm x A x ⊗ F x has an upper bounded M by the triangle inequality. On the other hand, the upper bound M is saturated by taking dim( H 1 ) = d, and F x = A x , x = 0, . . . , M − 1, i.e.
d i j =1 |ii j j | A x ⊗ A x ≥ Tr Ax ⊗ Ax x d x 1 Tr[1d ] = M , =
= =
P=
2
, Q =
2
1 − cos θ sin θ
2
1 − cos αk sin αk
1. A ∗x = A x , A 2x = 1, and
lim
n→∞
sin θ 1 + cos θ
1 2n
Tr( A i 1 A i 2 . . . A il ) = 0.
(3.3)
Proof. 1. It is clear that A ∗x = A x and A 2x = 1. Moreover,
Tr( A x ) = Tr( Q x,0 − Q x,1 ) =
n
Tr( Q xk,0 − Q xk,1 )
k =1
=
n
Tr(2Q xk,0 − 1) = 0, x = 0, 1.
k =1
It was shown by Navascués et al. [11] that the HJW theorem (asymptotically) holds when ( M , K ) = (2, 2) regardless of the dimension. Indeed, the correlation between Alice and Bob can always be (approximately) realized in finite-dimensional local systems. Therefore, for ( M , K ) = (2, 2) case, it suffices to consider the assemblages on finite-dimensional C ∗ -algebras, which of course have a quantum realization due to the HJW theorem. Motivated by [19], we construct a set of projectors such that all assemblages for ( M , K ) = (2, 2) can be approximated by the assemblages realized by those fixed projectors. In [19], Marciniak et al. constructed a fixed set of projectors that would universally work for all quantum correlation boxes. It shall be noted that our work differs from theirs. The key point of our work is the C-S decomposition of two projectors as follows[20]. Given a pair of d-dimensional projectors P and Q , there exists an orthonormal basis in which the two projectors are jointly block-diagonal. Moreover, the blocks can be either 1-dimensional, i.e. P and Q either have a 0 or a 1 in that block, or 2-dimensional, in which case they can be written in the form
1
− sin αk , 1 + cos αk sin αk , 1 + cos αk
αk = k2nπ ∈ 0, π2 .
3. Approximation of assemblages when ( M , K ) = (2, 2)
1
1 − cos αk − sin αk
2. If a sequence i 1 , . . . , il ∈ {0, 1} satisfies i 1 = i 2 , i 2 = i 3 , . . . , il−1 = il , then we have
x
− sin θ 1 + cos θ
2
Tr( A x ) = 0, x = 0, 1.
The statement of Theorem 2.2 gave as a negative answer to the Tsirelson problem in the context of quantum steering. Indeed, not all assemblages can be implemented by Equation (2.2). It was first studied by M. Navascués and D. Perez-Garcia [4] that there exists a steering protocol where the tensor and the commutation assumptions for the bipartite correlation are distinguishable. In their ∗ (∗ Z ). paper, they used A2 = C red M 2
1 − cos θ − sin θ
1
Proposition 3.1. Let A x = Q x,0 − Q x,1 , x = 0, 1, then we have the following properties:
where we have used the fact that A ∗x = A x and A 2x = 1d .
(3.2)
Q 1,0 = ⊕nk=1 Q 1k,0 , Q 1,1 = 1 − Q 1,0 ,
Q 0k,0
which yields a contradiction.
d
(3.1)
k =1
where
x
1
H k , H k C2,
then we define the (fixed) projectors acting on H by
where u ∗x = u x , u 2x = 1. Set F x = u x . By repeating the argument in the above proof. We have
3
The angle θ is called the principal angle between the subspaces.
.
2. For Equation (3.3), it is equivalent to show that for any given integer l
Tr(( A 0 A 1 )l A 0 ) = 0, lim
n→∞
1 2n
Tr(( A 0 A 1 )l ) = 0,
and
Tr(( A 1 A 0 )l A 1 ) = 0, lim
n→∞
1 2n
Tr(( A 1 A 0 )l ) = 0.
To this end, by direct calculation we have
( A 0 A 1 )l = ⊕nk=1
cos 2lαk sin 2lαk
− sin 2lαk cos 2lαk
.
Clearly
( A 0 A 1 )l A 0 = ⊕nk=1 Thus, we have
Tr(( A 0 A 1 )l A 0 ) = 0. Moreover,
0 − sin(2l + 1)αk
− sin(2l + 1)αk 0
.
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4
lim
n→∞
1 2n
Tr(( A 0 A 1 )l ) = lim
n→∞
n 1
2n
k =1
1 n
= lim
n→∞
n
Note that the principle angles θk are fixed and only depend on the given projective measurements. It is possible (for sufficiently large n) to find the associated integer k0 for every k, such that the angle between θk and αk0 is sufficiently small. Namely, we have θk − αk0 ≤ π /2n for all k. By
2 cos 2lαk
cos
klπ
k =1
n
d /2
identifying ⊕k =1 H k0 C d , we have the following estimate: 0
π =
cos lθ dθ = 0. 0
Similarly, we can show that
Tr(( A 0 A 1 )l A 0 ) = 0, lim
1
n→∞
2n
k
Tr P 0k ,0 Q 0,00 = cos2
θk − αk0 2
≥ cos2
π 4n
.
Similarly, we have
Tr(( A 0 A 1 )l ) = 0. 2
k
Tr P kx,a Q x,0a ≥ cos2
The above proposition reveals that A 0 and A 1 are free observables in the asymptotic sense (as n → ∞), see [18] for the concept of free observables. As an immediate corollary of Proposition and [18], we get the following result: Corollary 3.2. The dichotomic observables A x , x = 0, 1 defined in the above proposition can be used to saturate the Tsirelson bound of CHSH-Bell inequality.
π 4n
for all x, a = 0, 1.
2 ι : ⊕kd0/= H k0 ⊗ C d → H ⊗ C d , 1 we can find the desired state ρ by extending ρ naturally via the embedding ι. Thus σxa,ρ = Tr1 ρ ( Q x,a ⊗ 1) = Tr1 ρ ( Q x,a |⊕k H k0 ⊗ 1) .
Using the natural embedding
0
Definition. For any given quantum state ρ ∈ B ( H ) ⊗ Md (C), we define a quantum assemblage σρ = {σxa,ρ }x,a using the projectors given in Equation (3.2),
σxa,ρ = Tr1 ρ ( Q x,a ⊗ 1d ) , x, a = 0, 1.
(3.4)
We consider the distance d(·, ·) as follows
d(σ , σρ ) :=
Finally, we have
d(σ , σρ ) =
x,a
x,a
x,a
where · 1 is the Schatten 1 norm on Md (C). Our main result is that for any given quantum assemblage σ ⊂ Md (C), we can always find a state ρ ∈ B ( H ) ⊗ Md (C), such that σ is approximated by σρ with respect to the distance d(·, ·). More precisely, we have the following theorem: Theorem 3.3. Let 2n be the dimension of Hilbert space in formula (3.1). For any quantum assemblage σ = {σxa }x,a=0,1 ⊂ Md (C), there is a state ρ ∈ B ( H ) ⊗ Md (C), such that
(3.5)
n→∞
1
x,a
= Tr1 ρ (( P x,a − Q x,a |⊕k ≤ ρ [( P x,a − Q x,a |⊕k
σxa − σxa,ρ 1 ,
lim d(σ , σρ ) = 0,
a σx − σxa,ρ
0
by Hölder inequality
≤
( P x,a − Q x,a |⊕k x,a
≤ 4d 1 − cos2
π 4n
0
H
k0
H k0 ) ⊗ 1) 0
H k0 ) ⊗ 1]
1
1
) ⊗ 1 ρ 2 2
→ 0, as n → ∞,
where · 2 denotes the Schatten 2 norm on Md (C).
2
Remark. Unfortunately, our construction doesn’t work for ( M , K ) = (2, 2), since C-S decomposition doesn’t hold for more than 2 projectors.
where H and σρ are given by Equations (3.1) and (3.4) respectively. Proof. By the HJW theorem, there is a quantum state Md (C) ⊗ Md (C) and projectors P x,a ∈ Md (C) such that
ρ ∈
σxa = Tr1 ρ ( P x,a ⊗ 1d ) , x, a = 0, 1.
(3.6)
For simplicity, we can assume that d is an odd number. We apply the C-S decomposition to the pair ( P 0,0 , P 1,0 ), and further assume the blocks are all 2-dimensional. Thus, we have
⎧ ⎨ P 0,0 = ⊕d/2 P k , P 0,1 = 1 − P 0,0 , k=1 0,0 ⎩P
1,0
d/2
= ⊕k=1 P 1k ,0 , P 1,1 = 1 − P 1,0 ,
where
P 0k ,0 = P 1k ,0 =
1 2 1 2
1 − cos θk − sin θk 1 − cos θk sin θk
− sin θk , 1 + cos θk sin θk . 1 + cos θk
(3.7)
4. Conclusion In this paper, we demonstrated that for ( M , K ) = (2, 2), when the local dimension is infinite, the HJW theorem doesn’t hold. In other words, there exists an assemblage in infinite dimension which doesn’t have a quantum realization. Indeed, it is a reformulation of Tsirelson’s problem in the context of the quantum steering, which was disproved by Navascués and Perez-Garcia. For ( M , K ) = (2, 2), we provided an elementary method to approximate the assemblages coming from the finite-dimensional Hilbert spaces. The significance of our results is that the projectors constructed on a large Hilbert space universally work for all assemblages. Acknowledgements X-J. Yan, Z. Yin and L-S. Li are partially supported by NSFC No. 11771106.
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References [1] E. Schrödinger, P.A.M. Dirac, Probability relations between separated systems, Math. Proc. Camb. Philos. Soc. 32 (3) (1936) 446–452, https://doi.org/10.1017/ S0305004100019137. [2] H.M. Wiseman, S.J. Jones, A.C. Doherty, Steering, entanglement, nonlocality, and the EPR paradox, Phys. Rev. Lett. 98 (2007) 140402, https://doi.org/10.1103/ PhysRevLett.98.140402. [3] L.P. Hughston, R. Jozsa, W.K. Wootters, A complete classification of quantum ensembles having a given density matrix, Phys. Lett. A 183 (1) (1993) 14–18, https://doi.org/10.1016/0375-9601(93)90880-9. [4] M. Navascués, D. Perez-Garcia, Quantum steering and space like separation, Phys. Rev. Lett. 109 (16) (2012) 160405, https://doi.org/10.1103/physrevlett. 109.160405. [5] V.B. Scholz, R.F. Werner, Tsirelson’s problem, arXiv:0812.4305, 2008. [6] M. Junge, M. Navascués, C. Palazuelos, D. Perez-Garcia, V.B. Scholz, R.F. Werner, Connes’ embedding problem and Tsirelson’s problem, J. Math. Phys. 52 (1) (2011) 012102, https://doi.org/10.1063/1.3514538. [7] N. Ozawa, Tsirelson’s problem and asymptotically commuting unitary matrices, J. Math. Phys. 54 (3) (2013) 251–263, https://doi.org/10.1063/1.4795391. [8] T. Fritz, Tsirelson’s problem and Kirchberg’s conjecture, Rev. Math. Phys. 24 (05) (2012) 1250012, https://doi.org/10.1142/S0129055X12500122. [9] W. Slofstra, Tsirelson’s problem and an embedding theorem for groups arising from non-local games, arXiv:1606.03140, 2016. [10] S.J. Kim, V.I. Paulsen, C. Schafhauser, A synchronous game for binary constraint systems, J. Math. Phys. 59 (3) (2018) 032201, https://doi.org/10.1063/ 1.4996867.
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[11] M. Navascués, T. Cooney, D. Perez-Garcia, N. Villanueva, A physical approach to Tsirelson’s problem, Found. Phys. 42 (2012) 985–995, https://doi.org/10.1007/ s10701-012-9641-0. [12] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York, 1979. [13] M.F. Pusey, Negativity and steering: a stronger Peres conjecture, Phys. Rev. A 88 (2013) 032313, https://doi.org/10.1103/PhysRevA.88.032313. [14] Z. Yin, M. Marciniak, M. Horodecki, Operator space approach to steering inequality, J. Phys. A, Math. Theor. 48 (13) (2015) 135303, https://doi.org/10. 1088/1751-8113/48/13/135303. [15] E.G. Cavalcanti, S.J. Jones, H.M. Wiseman, M.D. Reid, Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox, Phys. Rev. A 80 (3) (2009) 84, https://doi.org/10.1103/PhysRevA.80.032112. [16] A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability, Cambridge University Press, 2006. [17] B. Collins, C. Male, The strong asymptotic freeness of Haar and deterministic matrices, Ann. Sci. Éc. Norm. Supér. 47 (1) (2014) 147–163, https://doi.org/10. 24033/asens.2211. [18] Z. Yin, A.W. Harrow, M. Horodecki, M. Marciniak, A. Rutkowski, Random and free observables saturate the Tsirelson bound for CHSH inequality, Phys. Rev. A 95 (2017) 32101, https://doi.org/10.1103/PhysRevA.95.032101. [19] M. Marciniak, M. Horodecki, Z. Yin, Bell and steering scenarios in terms of operator systems, arXiv:1412.0522, 2014. [20] R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Springer, New York, 1997.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Approximation of quantum assemblages Xiaojing Yan a,∗ , Zhi Yin b , Longsuo Li a a b
School of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Institute of Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150006, PR China
a r t i c l e
i n f o
Article history: Received 30 May 2019 Received in revised form 18 October 2019 Accepted 21 October 2019 Available online xxxx Communicated by M.G.A. Paris
a b s t r a c t In this paper, we derive a set of projectors on a large Hilbert space which can universally work for approximating quantum assemblages with binary inputs and outputs. The dimension of the Hilbert space depends on the accuracy of the approximation. © 2019 Elsevier B.V. All rights reserved.
Keywords: Quantum steering Quantum realization Quantum assemblages Approximation
1. Introduction The quantum steering was introduced by Schrödinger [1] and reformulated by Wiseman et al. [2] in 2007. Roughly speaking, in this scenario, Alice would like to steer Bob’s state by manipulating the measurements on her side. An assemblage σ is a set of unnormalized states {σxa } which Alice steers Bob into, with her measurement x and outcome a. Thus it is used to describe the correlation between Alice and Bob. It is a natural to ask whether a given assemblage can be produced by quantum mechanics (has a quantum realization) in finitedimensional Hilbert space, or whether it can be approximated at least. When Bob’s local dimension is finite, Schrödinger [1] and later Hughston-Jozsa-Wooters [3] gave a positive answer to this question. When Bob’s local dimension is infinite, the answer becomes negative (see Theorem 2.2). In fact, the question above is a reformulation of “Tsirelson’s problem” [4] in the quantum steering scenario, which asks the compatibility of describing the quantum composite systems by using a single Hilbert space or by a tensor product of two Hilbert spaces [5]. Tsirelson’s problem was solved by Navascués and Perez-Garcia with a negative answer [4]. It still remains open in the Bell scenario [6–10]. However, there is a special case, namely, when the inputs and outputs of Alice are binary. Navascués et al. showed that an assemblage with binary inputs and outputs can be approximated by a
*
Corresponding author. E-mail address: [email protected] (X. Yan).
https://doi.org/10.1016/j.physleta.2019.126082 0375-9601/© 2019 Elsevier B.V. All rights reserved.
sequence of assemblages that have quantum realization regardless of the dimension of Bob’s local Hilbert space [11]. To approximate a given assemblage, it is necessary to determine a sequence of quantum states and projective measurements on Alice’s side. In this paper, based on Navascués’ results, we construct a set of projectors on a large Hilbert space by sampling their principle angles uniformly in a circle. An intriguing result is that these fixed projectors can universally work for all assemblages (see Theorem 3.3). It shall be noted that the required dimension of the Hilbert space depends on the accuracy of the approximation. 2. Quantum assemblages and their realizations Definition of quantum assemblages An assemblage is an ensemble of unnormalized states. It was first introduced by Schrödinger [1] and revisited by Wiseman et al. [2]. For any fixed integers M , K , an assemblage on the Hilbert space H 2 can be mathematically defined a=0,..., K −1 [13,14] as a set of (unnormalized) states σ = {σxa }x=0,..., M −1 ⊂ a B ( H 2 ), such a σx does not depend on x and that the sum a Tr a σx = 12 for every x, where 12 denotes the identity of B ( H 2 ). Definition. [3]. We say that an assemblage σ = {σxa }x,a has a quantum realization if there exist a Hilbert space H 1 and a quantum state ρ ∈ H 1 ⊗ H 2 , such that
σxa = Tr1 [ρ ( P x,a ⊗ 12 )],
(2.1)
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where Tr1 is the partial trace for the first part of the tensor product, and { P x,a }x,a are projective measurements on H 1 such that a P x,a = 11 for every x.
where A x = P x,0 − P x,1 is a self-adjoint unitary operator of A1 . Hence we can say σ = {σx }x has a quantum realization if Equation (2.3) holds.
When dim( H 2 ) < ∞, Schrödinger (or called the HJW theorem [3]) proved that every assemblage has a quantum realization.
Theorem 2.2. There are a C ∗ -algebra A2 ⊂ B ( H 2 ) with dim( H 2 ) = ∞, and a dichotomic assemblage σ = {σx }x on A2 , such that σ does not have a quantum realization.
Theorem 2.1. [3,8]. Suppose dim( H 2 ) = d2 < ∞, then any assema=0,..., K −1 blage σ = {σxa }x=0,..., M −1 ⊂ B ( H 2 ) has a quantum realization. Moreover, the dimension of H 1 can be taken as d2 . C ∗ -algebra terminologies for quantum assemblages As mentioned in the introduction, one of our main results is that the HJW theorem doesn’t hold when dim( H 2 ) = ∞. To deal with the infinite dimension, we adopt use the C ∗ -algebra terminologies. We will follow the line of Fritz’s paper [8]. Recall that a C ∗ -algebra A ⊂ B ( H ) is a Banach ∗-algebra where the ∗ operation is compatible with its norm · as follows: x∗ x = xx∗ = x2 , x ∈ A. We have the following recipe for the quantum theory by using C ∗ algebra terminologies: 1. The algebra of observables of Alice (resp. Bob) is described by an unital C ∗ -algebra A1 ⊂ B ( H 1 ) (resp. A2 ⊂ B ( H 2 )). 2. A state ρ : A → C is a unital semidefinite functional on the algebra A. By the famous Riesz representation theorem, we have ρ (x) = Tr(ρ x), x ∈ A, where ρ is the usual quantum state in B ( H ). 3. The composite system is described by the minimal tensor product of A1 ⊗min A2 ⊂ B ( H 1 ⊗ H 2 ). We refer [8] and [12] for more about the tensor products. With the above recipe, we can define the notion of assemblages analogously by using the C ∗ -algebra language. For any fixed integers M , K , an assemblage can be defined [8] as a set of (unnormal a a=0,..., K −1 ized) states σ = {σxa (·)}x=0,..., M −1 on A2 , such that the sum a σx a does not depend on x and σ (1 ) = 1 for every x, where 1 2 2 is a x the identity of the algebra A2 . Definition. [8]. We say that an assemblage σ = {σxa (·)}x,a on A2 has a quantum realization if there are a unital C ∗ -algebra A1 ⊂ B ( H 1 ) and a state ρ on A1 ⊗min A2 , such that
σxa (·) = ρ [ P x,a ⊗ ·],
(2.2)
where { P x,a }x,a are projectors of A1 such that a P x,a = 11 for every x. Indeed, in the case of dim( H 2 ) < ∞, Equation (2.1) coincides with Equation (2.2). The HJW theorem fails in infinite dimensions Now, we are ready to present our main result in this section. To this end, we recall the notion of dichotomic assemblages [13–15]. Definition. For a given integer M, the dichotomic assemblage {σx (·)}x=0,..., M −1 on A2 is given by
σ=
σx = σx0 − σx1 ,
σ
a x (·)
=
Tr d
⊗E
Id + (−1)a U x DU x∗ 2
· ,
a = 0, 1 , x = 0, . . . , M − 1 , where E is the expectation respect to the probability space (, P ). Let
σx (·) = σx0 (·) − σx1 (·) =
Tr d
⊗ E[U x DU x∗ ·], x = 0, . . . , M − 1.
Obviously, H 2 = C d ⊗ L 2 (, P ), and it is easy to see that σ = {σx }x=0,..., M −1 is a dichotomic assemblage on A2 . Define a set of operators F = { F x }x=0,..., M −1 ⊂ A2 as the following:
F x = U x DU x∗ , x = 0, . . . , M − 1. It is easy to see that
σx ( F x ) = E
x=0,..., M −1
Tr[ D 2 ]
x=0,..., M −1
d
= M.
(2.4)
On the other hand, suppose that the dichotomic assemblage σ can be represented by Equation (2.3), i.e., there are a state ρ on A1 ⊗min A2 and operators A x ∈ A1 , x = 0, . . . , M − 1, such that
σx (·) = ρ [ A x ⊗ ·]. Then we have
x=0,..., M −1
σx ( F x ) =
ρ[ Ax ⊗ F x]
x=0,..., M −1
≤ A ⊗ F x x , x=0,..., M −1
(2.5)
where · is the operator norm on A1 ⊗min A2 . It is well known in the random matrix theory that (e.g. see [17,18])
√ Ax ⊗ F x ≤2 M
as d → ∞. Therefore, we conclude that Equation (2.4) will be violated as d → ∞, which completes our proof. 2
a=0,1
where {σxa }x=0,..., M is an assemblage on A2 for K = 2. a=0,1
Suppose {σxa }x=0,..., M −1 has a quantum realization, then it follows from Equation (2.2) that we have
σx (·) = ρ [ A x ⊗ ·],
Proof. Let M > 2, and suppose D is a d × d deterministic diagonal matrix, where the diagonal terms of D are either 1 or −1 and Tr[ D ] = 0. Let U x , x = 0, . . . , M be independent Haarrandom matrices in the group of unitary matrices U (d). Thus U x , x = 0, . . . , M are independent random variables of some probability space (, P ) [16]. Now, we consider the C ∗ -algebra A2 := Md ( L ∞ (, P )) which is commonly used in the random matrix theory. Define the following unnormalized states on A2 :
(2.3)
Remark. Due to the random matrix theory, we note that the local (in Bob’s side) observables F x = U x DU x∗ , x = 0, . . . , M − 1 in the observable algebra Md (C) (with a faithful state Tr ⊗ E) can be d described by random Haar unitaries u x , x = 0, . . . , M − 1 in some
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infinite-dimensional C ∗ -algebra A2 with a faithful state φ (thus the pair (A2 , φ) is called a C ∗ -probability space [16]) when d → ∞. Therefore, we can just assume Bob’s system is described by A2 , and write
σxa (·) = φ
1 + (−1)a u x 2
· ,
Definition. Suppose H is a Hilbert space given by n
H=
σx ( F x ) = M ,
σ has a quantum realization, then we have √ σx ( F x ) ≤ A x ⊗ u x ≤ 2 M , x x and if we assume
Q 0,0 = ⊕nk=1 Q 0k,0 , Q 0,1 = 1 − Q 0,0 ,
Q 1k,0 and
Remark. It is crucial that F x is random. Otherwise, the operator norm x A x ⊗ F x has an upper bounded M by the triangle inequality. On the other hand, the upper bound M is saturated by taking dim( H 1 ) = d, and F x = A x , x = 0, . . . , M − 1, i.e.
d i j =1 |ii j j | A x ⊗ A x ≥ Tr Ax ⊗ Ax x d x 1 Tr[1d ] = M , =
= =
P=
2
, Q =
2
1 − cos θ sin θ
2
1 − cos αk sin αk
1. A ∗x = A x , A 2x = 1, and
lim
n→∞
sin θ 1 + cos θ
1 2n
Tr( A i 1 A i 2 . . . A il ) = 0.
(3.3)
Proof. 1. It is clear that A ∗x = A x and A 2x = 1. Moreover,
Tr( A x ) = Tr( Q x,0 − Q x,1 ) =
n
Tr( Q xk,0 − Q xk,1 )
k =1
=
n
Tr(2Q xk,0 − 1) = 0, x = 0, 1.
k =1
It was shown by Navascués et al. [11] that the HJW theorem (asymptotically) holds when ( M , K ) = (2, 2) regardless of the dimension. Indeed, the correlation between Alice and Bob can always be (approximately) realized in finite-dimensional local systems. Therefore, for ( M , K ) = (2, 2) case, it suffices to consider the assemblages on finite-dimensional C ∗ -algebras, which of course have a quantum realization due to the HJW theorem. Motivated by [19], we construct a set of projectors such that all assemblages for ( M , K ) = (2, 2) can be approximated by the assemblages realized by those fixed projectors. In [19], Marciniak et al. constructed a fixed set of projectors that would universally work for all quantum correlation boxes. It shall be noted that our work differs from theirs. The key point of our work is the C-S decomposition of two projectors as follows[20]. Given a pair of d-dimensional projectors P and Q , there exists an orthonormal basis in which the two projectors are jointly block-diagonal. Moreover, the blocks can be either 1-dimensional, i.e. P and Q either have a 0 or a 1 in that block, or 2-dimensional, in which case they can be written in the form
1
− sin αk , 1 + cos αk sin αk , 1 + cos αk
αk = k2nπ ∈ 0, π2 .
3. Approximation of assemblages when ( M , K ) = (2, 2)
1
1 − cos αk − sin αk
2. If a sequence i 1 , . . . , il ∈ {0, 1} satisfies i 1 = i 2 , i 2 = i 3 , . . . , il−1 = il , then we have
x
− sin θ 1 + cos θ
2
Tr( A x ) = 0, x = 0, 1.
The statement of Theorem 2.2 gave as a negative answer to the Tsirelson problem in the context of quantum steering. Indeed, not all assemblages can be implemented by Equation (2.2). It was first studied by M. Navascués and D. Perez-Garcia [4] that there exists a steering protocol where the tensor and the commutation assumptions for the bipartite correlation are distinguishable. In their ∗ (∗ Z ). paper, they used A2 = C red M 2
1 − cos θ − sin θ
1
Proposition 3.1. Let A x = Q x,0 − Q x,1 , x = 0, 1, then we have the following properties:
where we have used the fact that A ∗x = A x and A 2x = 1d .
(3.2)
Q 1,0 = ⊕nk=1 Q 1k,0 , Q 1,1 = 1 − Q 1,0 ,
Q 0k,0
which yields a contradiction.
d
(3.1)
k =1
where
x
1
H k , H k C2,
then we define the (fixed) projectors acting on H by
where u ∗x = u x , u 2x = 1. Set F x = u x . By repeating the argument in the above proof. We have
3
The angle θ is called the principal angle between the subspaces.
.
2. For Equation (3.3), it is equivalent to show that for any given integer l
Tr(( A 0 A 1 )l A 0 ) = 0, lim
n→∞
1 2n
Tr(( A 0 A 1 )l ) = 0,
and
Tr(( A 1 A 0 )l A 1 ) = 0, lim
n→∞
1 2n
Tr(( A 1 A 0 )l ) = 0.
To this end, by direct calculation we have
( A 0 A 1 )l = ⊕nk=1
cos 2lαk sin 2lαk
− sin 2lαk cos 2lαk
.
Clearly
( A 0 A 1 )l A 0 = ⊕nk=1 Thus, we have
Tr(( A 0 A 1 )l A 0 ) = 0. Moreover,
0 − sin(2l + 1)αk
− sin(2l + 1)αk 0
.
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lim
n→∞
1 2n
Tr(( A 0 A 1 )l ) = lim
n→∞
n 1
2n
k =1
1 n
= lim
n→∞
n
Note that the principle angles θk are fixed and only depend on the given projective measurements. It is possible (for sufficiently large n) to find the associated integer k0 for every k, such that the angle between θk and αk0 is sufficiently small. Namely, we have θk − αk0 ≤ π /2n for all k. By
2 cos 2lαk
cos
klπ
k =1
n
d /2
identifying ⊕k =1 H k0 C d , we have the following estimate: 0
π =
cos lθ dθ = 0. 0
Similarly, we can show that
Tr(( A 0 A 1 )l A 0 ) = 0, lim
1
n→∞
2n
k
Tr P 0k ,0 Q 0,00 = cos2
θk − αk0 2
≥ cos2
π 4n
.
Similarly, we have
Tr(( A 0 A 1 )l ) = 0. 2
k
Tr P kx,a Q x,0a ≥ cos2
The above proposition reveals that A 0 and A 1 are free observables in the asymptotic sense (as n → ∞), see [18] for the concept of free observables. As an immediate corollary of Proposition and [18], we get the following result: Corollary 3.2. The dichotomic observables A x , x = 0, 1 defined in the above proposition can be used to saturate the Tsirelson bound of CHSH-Bell inequality.
π 4n
for all x, a = 0, 1.
2 ι : ⊕kd0/= H k0 ⊗ C d → H ⊗ C d , 1 we can find the desired state ρ by extending ρ naturally via the embedding ι. Thus σxa,ρ = Tr1 ρ ( Q x,a ⊗ 1) = Tr1 ρ ( Q x,a |⊕k H k0 ⊗ 1) .
Using the natural embedding
0
Definition. For any given quantum state ρ ∈ B ( H ) ⊗ Md (C), we define a quantum assemblage σρ = {σxa,ρ }x,a using the projectors given in Equation (3.2),
σxa,ρ = Tr1 ρ ( Q x,a ⊗ 1d ) , x, a = 0, 1.
(3.4)
We consider the distance d(·, ·) as follows
d(σ , σρ ) :=
Finally, we have
d(σ , σρ ) =
x,a
x,a
x,a
where · 1 is the Schatten 1 norm on Md (C). Our main result is that for any given quantum assemblage σ ⊂ Md (C), we can always find a state ρ ∈ B ( H ) ⊗ Md (C), such that σ is approximated by σρ with respect to the distance d(·, ·). More precisely, we have the following theorem: Theorem 3.3. Let 2n be the dimension of Hilbert space in formula (3.1). For any quantum assemblage σ = {σxa }x,a=0,1 ⊂ Md (C), there is a state ρ ∈ B ( H ) ⊗ Md (C), such that
(3.5)
n→∞
1
x,a
= Tr1 ρ (( P x,a − Q x,a |⊕k ≤ ρ [( P x,a − Q x,a |⊕k
σxa − σxa,ρ 1 ,
lim d(σ , σρ ) = 0,
a σx − σxa,ρ
0
by Hölder inequality
≤
( P x,a − Q x,a |⊕k x,a
≤ 4d 1 − cos2
π 4n
0
H
k0
H k0 ) ⊗ 1) 0
H k0 ) ⊗ 1]
1
1
) ⊗ 1 ρ 2 2
→ 0, as n → ∞,
where · 2 denotes the Schatten 2 norm on Md (C).
2
Remark. Unfortunately, our construction doesn’t work for ( M , K ) = (2, 2), since C-S decomposition doesn’t hold for more than 2 projectors.
where H and σρ are given by Equations (3.1) and (3.4) respectively. Proof. By the HJW theorem, there is a quantum state Md (C) ⊗ Md (C) and projectors P x,a ∈ Md (C) such that
ρ ∈
σxa = Tr1 ρ ( P x,a ⊗ 1d ) , x, a = 0, 1.
(3.6)
For simplicity, we can assume that d is an odd number. We apply the C-S decomposition to the pair ( P 0,0 , P 1,0 ), and further assume the blocks are all 2-dimensional. Thus, we have
⎧ ⎨ P 0,0 = ⊕d/2 P k , P 0,1 = 1 − P 0,0 , k=1 0,0 ⎩P
1,0
d/2
= ⊕k=1 P 1k ,0 , P 1,1 = 1 − P 1,0 ,
where
P 0k ,0 = P 1k ,0 =
1 2 1 2
1 − cos θk − sin θk 1 − cos θk sin θk
− sin θk , 1 + cos θk sin θk . 1 + cos θk
(3.7)
4. Conclusion In this paper, we demonstrated that for ( M , K ) = (2, 2), when the local dimension is infinite, the HJW theorem doesn’t hold. In other words, there exists an assemblage in infinite dimension which doesn’t have a quantum realization. Indeed, it is a reformulation of Tsirelson’s problem in the context of the quantum steering, which was disproved by Navascués and Perez-Garcia. For ( M , K ) = (2, 2), we provided an elementary method to approximate the assemblages coming from the finite-dimensional Hilbert spaces. The significance of our results is that the projectors constructed on a large Hilbert space universally work for all assemblages. Acknowledgements X-J. Yan, Z. Yin and L-S. Li are partially supported by NSFC No. 11771106.
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