aa r X i v : . [ qu a n t - ph ] F e b Zero-Correlation Entanglement
Toru Ohira ∗ Graduate School of Mathematics, Nagoya University, Nagoya, JapanFebruary 25, 2020
Abstract
We consider a quantum entangled state for two particles, each par-ticle having two basis states, which includes an entangled pair of spin1/2 particles. We show that, for any quantum entangled state vectors ofsuch systems, one can always find a pair of observable operators X , Y with zero correlations ( h ψ | X Y | ψ i − h ψ | X | ψ i h ψ | Y | ψ i = 0). At thesame time, if we consider the analogous classical system of a “classicallyentangled” (statistically non-independent) pair of random variables tak-ing two values, one can never have zero correlations (zero covariance, E [ XY ] − E [ X ] E [ Y ] = 0). We provide a general proof to illustrate thedifferent nature of entanglements in classical and quantum theories. Entanglement is considered a key concept in understanding quantum phenom-ena. In understanding entanglement, correlations based on expectation values ofquantum observable operators for multi-particle systems have been investigatedboth theoretically and experimentally (e.g.,[1, 2, 3, 4, 5, 6, 7, 8]). For exam-ple, among the research efforts for obtaining conditions for the separability ofdensity matrices(e.g.,[9, 10, 11, 12, 13]), a recent work by Fujikawa et. al.[13]has associated saparability with zero correlations and analyzed experimentalresults.In this paper, we also study correlations to investigate entangled quantumsystems. Our approach, however, differs from previous works that associatedzero correlations with separability. Rather, entanglements are connected withzero correlations. In particular, we consider a system of two quantum particles,each taking two distinct states (2 × ∗ The author is also affiliated with Future Value Creation Research Center, Graduate Schoolof Informatics, Nagoya University, and with Mathematical Science Team, RIKEN Center forAdvanced Intelligence Project.
1s well as providing a general proof on the quantum system, we also notethat for analogous classical systems consisting of two stochastic variables, eachtaking on one of two possible values, there cannot be zero correlation unless thevariables are statistically independent. We may regard the statistical depen-dence of classical variables as analogous to quantum entanglement. Thus, zerocorrelation in the latter case but not the former provides yet another illustrationof the difference between classical and quantum probability theories.
We consider two random variables X and Y , such that they both take only twodistinct finite values ( x , x ) and ( y , y ). The joint probability distribution forthese variables is denoted as P ( X : Y ), and given by P ( X = x i : Y = y j ) = p ij , ( i, j ∈ { , } )The probability distributions P ( X ) for X and P ( Y ) for Y can be derived fromabove as. P ( X = x i ) = p i + p i , P ( Y = y j ) = p j + p j . By the requirement that both X , and Y take only two values, p ( x ) + p ( x ) = p ( y ) + p ( y ) = 1 . When the joint probability of X and Y can be decomposed as P ( X : Y ) = P ( X ) P ( Y ) , these stochastic variables are called statistically independent. If such decom-position is not possible, X and Y are not statistically independent (“classicallyentangled”).Expectation values are defined as E [ X ] = X i p ( x i ) x i , E [ Y ] = X i p ( y i ) y i , E [ XY ] = X i,j p ( x i : y j ) x i y j . From these we define the covariance of X and Y as Cov [ X, Y ] ≡ E [ XY ] − E [ X ] E [ Y ] . We define zero correlation to be when this covariance is zero or equivalently, E [ XY ] = E [ X ] E [ Y ] . We later mention in the main theorem that for this classical 2 × .2 Quantum case We consider two quantum particles A and B . The total normalized state vector | ψ i is given by | ψ i = X i,j ω i,j | a i i ⊗ | b j i , where | a i and | b i describe the state of particle A and B respectively, and ω i,j arequantum amplitudes given by complex scalars. This system is called a separable(product) state when the total state vector can be decomposed as a product ofeach normalized state of A and B , | ψ i = | ψ A i ⊗ | ψ B i where | ψ A i = X i ν i | a i i | ψ B i = X j τ j | b j i . There is a correspondence between the classical notion of statistical indepen-dence and this separability of state vectors. State vectors which are not separa-ble are called (quantum) entangled state vectors. Thus, quantum entanglementcorresponds to “classical entanglement,” i.e. lack of statical independence oftwo stochastic variables, as presented in the previous subsection.We now present the expectation values for the quantum system. For this,two operators X and Y are defined as X = Q A ⊗ B and Y = A ⊗ R B . Here, Q A and R B are quantum observable operators for A and B respectively, and is the identity operator.With these operators, we consider the relations between the expectationvalues of h ψ | X Y | ψ i and h ψ | X | ψ i h ψ | Y | ψ i . For separable states in general, itis straightforward to show that they are always equal, that is, zero correlated.We ask the same question for the case of entangled states by limiting our-selves to the case that particles A and B take only two distinctive states. In thiscase the most general quantum state vector is given as follows, with ( i ) = − | ψ i = α | a i ⊗ | b i + βe iφ | a i ⊗ | b i + γe iκ | a i ⊗ | b i + δe iλ | a i ⊗ | b i (1)where α, β, γ, δ, φ, κ , and λ are real-valued parameters with α + β + γ + δ = 1and 0 ≤ φ, κ, λ ≤ π .We note that this state vector is separable when αδ = βγ and λ = ( φ + κ ) mod 2 π , but is quantum entangled otherwise.Also, the most general observable operators are given by Hermitian matricesin the basis of | a (1 , i and | b (1 , i respectively, Q A = (cid:20) Q + qe is qe − is Q − (cid:21) , R B = (cid:20) R + re iv re − iv R − (cid:21) (2)where Q ± , R ± , q, r, s , and v are real-valued parameters with 0 ≤ s, v ≤ π .3ith the above setup, we will show in the next section that even for anyentangled state vector in the form above, we can always find a pair of Q A , R B that achieves zero correlation, h ψ | X Y | ψ i = h ψ | X | ψ i h ψ | Y | ψ i with X = Q A ⊗ B and Y = A ⊗ R B . With the set up in the previous section for a dual particle system, we discuss therelation between entanglements and zero-correlations. For non-entangled sys-tems, both classical (statistically independent) and quantum (separable) casescommonly lead to zero-correlations. For entangled systems, however, there is aclear difference between the classical and quantum systems, which is reiteratedin the following theorem.
Theorem
Classical Case:For any pair of random variables X and Y each taking two distinct finitevalues (for any values of ( p ij , x i , y j ), i, j ∈ { , } as set up above) that are notstatistically independent (“classically entangled”), they can NEVER be zerocorrelated.Quantum Case:For any quantum-entangled pure state for a dual particle system each takingtwo distinct states as set up above, one can ALWAYS find a pair of observableoperators X = Q A ⊗ B and Y = A ⊗ R B that are zero correlated. Proof
Classical Case:We have established the equivalence of statistical independence and zerocorrelation for such statistical variables X and Y in the previous work[14]. Thestatement here follows immediately. (In passing, we note that the classical2 × | ψ i in Eq. (1),we can always find a pair of observable operators X , Y such that the followingzero-correlation relation holds. h ψ | X Y | ψ i = h ψ | X | ψ i h ψ | Y | ψ i (3)with X = Q A ⊗ B and Y = A ⊗ R B .After tedious calculations, the above statement translates to the following:4iven any set of real-valued parameters as in Eq. (1) – α, β, γ, δ, φ, κ, λ with α + β + γ + δ = 1 and 0 ≤ φ, κ, λ ≤ π – one can find a set of real-valuedparameters Q ± , R ± , q, r, s, v with 0 ≤ s, v ≤ π such that the following holds. α Q + R + + β Q + R − + γ Q − R + + δ Q − R − + 2 αβ cos( φ + v ) Q + r + 2 αγ cos( κ + s ) qR + + 2 βδ cos( λ − φ + s ) qR − + 2 γδ cos( λ − κ + v ) Q − r + 2 αδ cos( λ + s + v ) qr + 2 βγ cos( κ − φ + s − v ) qr = [( α + β ) Q + + ( γ + δ ) Q − + 2 αγ cos( κ + s ) q + 2 βδ cos( λ − φ + s ) q ] × [( α + γ ) R + + ( β + δ ) R − + 2 αβ cos( φ + v ) r + 2 γδ cos( λ − κ + v ) r ](4)Finding the general solution, i.e., all possible sets of parameters Q ± , R ± , q, r, s, v ,is difficult. We can, however, find a set of parameters for which the above state-ment holds. For this we first set the phase parameters as s = 12 ( − κ + φ − λ ) , v = 12 ( κ − φ − λ ) (5)Further, if we define ξ = cos( ( λ − φ − κ )), Q ± = Q ± ǫ and R ± = R ± η , (4)can be simplified as follows:2( αδ − βγ )( αδ + βγ ) ǫη + ( αδ + βγ ) qr = 2( αδ − βγ )( αβ − γδ ) ξqη + 2( αδ − βγ )( αγ − βδ ) ξrǫ + 2( αγ + βδ )( αβ + γδ ) ξ qr (6)(Note, Q , R , s, v do not appear in Eq. (6).)Our aim now is to find ǫ, η, q , and r to satisfy Eq. (6) given any set of α, β, γ, δ, ξ with α + β + γ + δ = 1 and − ≤ ξ ≤
1. Let us also impose theconditions ǫ + q = 0, and η + r = 0, so that the observable operators havetwo distinct eigenvalues.We first note that for the separable case ( αδ = βγ and ξ = 1) Eq. (6) holdsfor any set of ǫ, η, q, r as expected by the fact that the separability of the statevector | ψ i entails the zero correlation.Even for the entangled case, one can find the desired set of parameters byexplicit constructions. We do this by considering different cases which, takenaltogether, comprise all possible values of α, β, γ, δ , and ξ . αδ − βγ = 0 and ξ = 0 αδ + βγ = 0( ǫ, η, q, r ) = ( ǫ, η, q = 0 , r = 0).No constraints on ǫ, η other than ǫ + q = 0, and η + r = 0. (Hereafter, thesame convention is used: unless specified, no constraints other than ǫ + q = 0,and η + r = 0.) 5 .1.2 αδ + βγ = 0This case needs to be considered by further classifications.(i) αβ − γδ = 0 and αγ − βδ = 0We can take either of the following two parameter settings. • ( ǫ, η, q, r ) = ( ǫ = 0 , η = 0 , q = 0 , r = 0) such that ( αδ + βγ ) ǫ = qξ ( αβ − γδ ) • ( ǫ, η, q, r ) = ( ǫ = 0 , η = 0 , q = 0 , r = 0) such that ( αδ + βγ ) η = rξ ( αγ − βδ )(ii) αβ − γδ = 0 and αγ − βδ = 0( ǫ, η, q, r ) = ( ǫ = 0 , η = 0 , q = 0 , r = 0)(iii) αβ − γδ = 0 and αγ − βδ = 0( ǫ, η, q, r ) = ( ǫ = 0 , η = 0 , q = 0 , r = 0)(iv) αβ − γδ = 0 and αγ − βδ = 0There are three possibilities:(a) ( α = 0 , β = 0 , γ = 0 , δ = 0)( ǫ, η, q, r ) such that 2 βγǫη = qr (b) ( α = 0 , β = 0 , γ = 0 , δ = 0)( ǫ, η, q, r ) such that − αδǫη = qr (c) ( α = δ = 0 , β = γ = 0 , α = β )When 16 α β ξ − = 0:( ǫ, η, q, r ) such that 2( α − β ) ǫη = (16 α β ξ − qr When 16 α β ξ − ǫ, η, q, r ) such that ǫη = 0 αδ − βγ = 0 and ξ = 0 αδ + βγ = 0No constraints on ( ǫ, η, q, r ) αδ + βγ = 0( ǫ, η, q, r ) such that 2( αδ − βγ ) ǫη = ( αδ + βγ ) qr αδ − βγ = 0 and ξ = 0 βγ = 0No constraints on ( ǫ, η, q, r ) 6 .3.2 βγ = 0( ǫ, η, q, r ) such that qr = 0 αδ − βγ = 0 and ξ = 0 ξ = 1No constraints on ( ǫ, η, q, r ) (This is the separable case.) ξ = 1(i) βγ − ξ ( αγ + βδ )( αβ + γδ ) = 0No constraints on ( ǫ, η, q, r )(ii) βγ − ξ ( αγ + βδ )( αβ + γδ ) = 0( ǫ, η, q, r ) such that qr = 0Q.E.D. A set of Bell state vectors is a representative example of entangled state vectorsfor the 2 × | Φ ± i = 1 √ | a i ⊗ | b i ± | a i ⊗ | b i ) ≡ √ (cid:20) (cid:21) A ⊗ (cid:20) (cid:21) B ± (cid:20) (cid:21) A ⊗ (cid:20) (cid:21) B ) , (7) | Ψ ± i = 1 √ | a i ⊗ | b i ± | a i ⊗ | b i ) ≡ √ (cid:20) (cid:21) A ⊗ (cid:20) (cid:21) B ± (cid:20) (cid:21) A ⊗ (cid:20) (cid:21) B ) . (8)When we apply our theorem to these state vectors, we obtain that the fol-lowing set of two observable operators leads to zero correlation. • The Bell state vectors in Eq. (7) correspond to the classification . . (iv-b): − αδǫη = qr . • The Bell state vectors in Eq. (8) correspond to the classification . . (iv-a):2 βγǫη = qr .For a very simple example, with q = r = m ( = 0), we can have the followingpairs of operators yielding zero correlations for the Bell states:7 or {| Φ + i , | Ψ − i} : Q A = (cid:20) Q − m mm Q + m (cid:21) = Q + m ( σ x − σ z ) R B = (cid:20) R + m mm R − m (cid:21) = R + m ( σ x + σ z ) F or {| Φ − i , | Ψ + i} : Q A = (cid:20) Q + m mm Q − m (cid:21) = Q + m ( σ x + σ z ) R B = (cid:20) R + m mm R − m (cid:21) = R + m ( σ x + σ z )In the above, we have used the identity operator and the Pauli matricies σ x = (cid:20) (cid:21) , σ z = (cid:20) − (cid:21) . From a broader perspective, the theorem presented here is only one example ofthe intricate relations among quantum and classical probability concepts. The“quantum pigeonhole effect” recently proposed by Aharonov et. al. [17] shedseven more light on the deeper characteristics of quantum entanglements by cre-ating correlations from separable product states, which are normally consideredas non-entangled.Classically the pigeonhole principle states that if we have a number of pigeonsto be placed in a smaller number of holes, at least one hole must contain multiplepigeons. The analogous system is considered in quantum mechanics with threetwo-state quantum particles (pigeons), each in a superposition of two (hole)states (a quantum 3 × .2 Other issues We would like to return to our theorem and discuss a couple of points.We again note that our derivation of a zero–correlation condition is only onepossibility, and other choices are possible. For example, two options are derivedin classification . . (i). Exploring and categorizing different types of solutionsis left for the future.In general, one does not have a priori knowledge of the quantum state vector,which makes the construction of observable operators with zero-correlationsdifficult. On the other hand, if one can infer or conjecture the quantum stateof a 2 × × × Acknowledgments
The author would like to thank Philip M. Pearle, Professor Emeritus of HamiltonCollege, for his comments and encouragement. This work was supported byfunding from Ohagi Hospital, Hashimoto, Wakayama, Japan, and by Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of ScienceNos.19H01201 and 18H04443.
Appendix
After the original version of this paper is placed in the arXiv, I received acomment from Dr. Shuming Cheng that my result can be extended to theentangled mixed states of the 2 × × AB = 14 ( A ⊗ B + ~a · ~σ ⊗ B + A ⊗ ~b · ~σ + X ij F ij σ i ⊗ σ j ) (9)where is the 2 × ~a,~b are vectors consists of 3 real numbers( · is the inner product), and F ij are real number elements of a 3 × F ,and ~σ = ( σ x , σ y , σ z ) is a vector with the Pauli matrices. σ x = (cid:20) (cid:21) , σ y = (cid:20) − ii (cid:21) , σ z = (cid:20) − (cid:21) . Now, the two quantum observable operators Q A , R B can be also expressedusing the Pauli matrices up to a scale as Q A = 12 ( A + ~x · ~σ ) , R B = 12 ( B + ~y · ~σ ) (10)where ~x, ~y are three dimensional real vectors.Then, for operators X = Q A ⊗ B and Y = A ⊗ R B , we can calculate theexpectation values as follows; hX Yi = Tr AB [ ρ AB X Y ] = 14 (1 + ~a · ~x + ~b · ~y + ~x · F · ~y ) , and hX i = 12 (1 + ~a · ~x ) , hYi = 12 (1 + ~b · ~y ) . This leads to hX Yi − hX ihYi = 14 ( ~x · F · ~y + ( ~a · ~x )( ~b · ~y )) = 14 ~x · ( F − ~a · ~b T ) · ~y, (11)where ~a · ~b T is the outer product of ~a,~b . Hence, the zero correlation conditionis given as ~x · ( F − ~a · ~b T ) · ~y = 0 . (12)Given any density matrix (9), the corresponding real 3 × F − ~a · ~b T ) isfixed. One can always find a pair of ( ~x, ~y ), i.e., two observable operators in (10),satisfying (12) as it is an orthogonality relation in the real three–dimensionalspace. References [1] J. S. Bell, Physics, , 195 (1964).[2] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt, Phys. Rev. Lett., ,880 (1969).[3] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett., , 460 (1981).104] A. J. Leggett, and A. Garg, Phys. Rev. Lett., , 857 (1985).[5] V. Vedral, M. B. Plenio, M. A. Pippin, and P. L. Knight, Phys. Rev. Lett., , 2257 (1997).[6] J. B. Altepeter, E. R. Jeffrey, P. G. Kwiat, S. Tanizilli, N. Gisin, and A.Acin, Phys. Rev. Lett., , 033601 (2005).[7] H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawahata, H. Kuboki, Y. Maeda,N. Matsui, C.Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda,A. Tamio, T. Uesaka, and K. Yako, Phys. Rev. Lett., , 150405 (2006).[8] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod.Phys., , 865 (2009).[9] R. F. Werner, Phys. Rev. A, , 4277 (1989).[10] N. Gisin, Phys. Lett. A, , 141 (1996).[11] A. Peres, Phys. Rev. Lett., , 1413 (1996).[12] K. Fujikawa, Phys. Rev. A, , 012315 (2009).[13] K. Fujikawa, C. H. Oh, K. Umetsu and S. Yu, Annals of Physics, , 248(2016).[14] T. Ohira, Prog. Theor. Exp. Phys., , 083A02 (2018).[15] W. Feller, An Introduction to Probability Theory and Its Applications , vol.1,(John Wiley & Sons, New York, NY, 1957).[16] L. Bain, and M. Engelhardt,
Introduction to Probability and MathematicalStatistics (2nd ed.) , pp. 185-186, (Duxbury Press, Belmont, CA, 1992). (Thisreference provides continuous variable cases: if one specifies joint probabil-ity distribution function as a bivariate normal distribution, the equivalencebetween statistical independence and zero–correlation holds.)[17] Y. Aharonov, F. Colombo, S. Popescu, I. Sabadini, D. C. Struppa and J.Tollaksen, Proc. Natl. Acad. Sci. USA, , 532 (2016).[18] M. Chen, C. Liu, Y. Luo, H. Huang, B. Wang, X. Wang, L. Li, N. Liu, andC. Lu and J. Pan, Proc. Natl. Acad. Sci. USA, , 1549 (2019).[19] H. Araki and M. M. Yanase,
Phys. Rev. , , 622 (1960).[20] M. M. Yanase, Phys. Rev. , , 666 (1961).[21] S. Popescu and L. Vaidman, Phys. Rev. A , , 4331 (1994).[22] K. Hess, K. Michielsen and H. De Raedt, Europhys. Lett. , , 60007 (2009).[23] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Phys. Rep.,525