Spin-momentum correlation in relativistic single particle quantum states
aa r X i v : . [ qu a n t - ph ] A ug Spin-momentum correlation in relativisticsingle particle quantum states
M. A. Jafarizadeh a,b,c ∗ , M. Mahdian a † a Department of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz 51664, Iran. b Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Iran. c Research Institute for Fundamental Sciences, Tabriz 51664, Iran.
October 28, 2018 ∗ E-mail:[email protected] † E-mail:[email protected] pin-momentum correlation in relativistic single particle quantum states Abstract
This paper was concerned with the spin-momentum correlation in single-particlequantum states, which is described by the mixed states under Lorentz transformations.For convenience, instead of using the superposition of momenta we use only two momen-tum eigen states ( p and p ) that are perpendicular to the Lorentz boost direction. Con-sequently, in 2D momentum subspace we show that the entanglement of spin-momentumin the moving frame depends on the angle between them. Therefore, when spin andmomentum are perpendicular the measure of entanglement is not observer-dependentquantity in inertial frame. Likewise, we have calculated the measure of entanglement (byusing the concurrence) and has shown that entanglement decreases with respect to theincreasing of observer velocity. Finally, we argue that, Wigner rotation is induced byLorentz transformations can be realized as controlling operator. Keywords : Spin-momentum correlation, Relativistic entanglement, Quan-tum gate
PACS numbers: 03.67.Hk, 03.65.Ta
In two recent decades quantum entanglement has become as one of the most important re-sources in the rapidly growing field of quantum information processing with remarkable ap-plications on it [1], and was based on the fact that the existence of entangled states producesnonclassical phenomena. Therefore, specifying that a particular quantum state is entangled orseparable is important because if the quantum state be separable then its statistical propertiescan be explained entirely by classical statistics.Relativistic aspects of quantum mechanics have recently attracted much attention in thecontext of the theory of quantum information, especially on quantum entanglement[2, 3, 4,5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Peres et al.[6] have recently observed thatthe reduced spin density matrix of a single spin- particle is not a relativistic invariant, and pin-momentum correlation in relativistic single particle quantum states particles or the observer with respectto the laboratory. Alsing and Milburn studied the Lorentz invariance of entanglement andshowed that the entanglement fidelity of the bipartite state is preserved explicitly. Instead ofstate vector in the Hilbert space, they have used a 4-component Dirac spinor or a polarizationvector in favor of quantum field theory [11]. Ahn also calculated the degree of violation ofthe Bell’s inequality which is decreases with increasing of velocity of the observer [12]. Mostof the previous works were concerned with the pure states although authors in [13, 14, 15]have considered mixed quantum states that are described by superposition of momenta withGaussian distribution, where Lorentz transformation introduces a transfer of entanglementbetween different degrees of freedom. While the entanglement between spins and momentumsof particles may change, separately. However, the total entanglement of particle-particle isthe same in all inertial frames. Beside of the pervious works that are concerned of studyon the entanglement between quantum states of two particles, here we generalize this to thespin-momentum correlation of relativistic single-particle (by using the concurrence) and showthat the measure of entanglement depends on the angle between spin and momentum and itdecreases with increasing of velocity of the observer. Also it has been shown that the Wignerangle depends on momentum, so Wigner rotation behaves as a quantum gate or controllingoperators. Thus using this quantum gate the spin-momentum entanglement changes in theframework of special relativity.This paper is organized as follows: Sec. II, is devoted to single-particle relativistic quantumstates. In Sec. III, we calculate explicitly the spin-momentum entanglement of relativistic pin-momentum correlation in relativistic single particle quantum states Suppose we have a bipartite system with its quantum degrees of freedom distributed amongtwo parties A and B with Hilbert spaces H A and H B , respectively, (the standard Hilbert spaceof dimension d endowed with usual inner product denoted by h . i ). In this paper quantumstate is made up of a single-particle having two types of degrees of freedom : momentum p and spin σ . The former is a continuous variable with Hilbert space of infinite dimension butwe restrict ourselves here to 2D momentum subspace with two eigen-state p and p , while thelatter is a discrete one with Hilbert space of spin particle. The pure quantum state of such asystem can always be written as | ψ i = X i =1 − n X j = n c ij | p i i ⊗ | j i , (2.1)where | p i are two momentum eigen states of each particle and the kets | j i are the eigenstatesof spin operator. c ij ’s are complex coefficients such that P i,j | c ij | = 1 .A bipartite quantum mixed state is defined as a convex combination of bipartite pure states(2.1), i.e. ρ = X i =1 P i | ψ i ih ψ i | , (2.2)where P i ≥ P i P i = 1 . | ψ i i ( i = 1 , , ,
4) as four orthogonal maximal entangled Bell states( BD ) are belong to the product space H A ⊗ H B and in terms of momentum and spin statesare well-known as | ψ i = 1 √ | p i ⊗ | n i + | p i ⊗ | − n i ) , pin-momentum correlation in relativistic single particle quantum states | ψ i = 1 √ | p i ⊗ | n i − | p i ⊗ | − n i ) , | ψ i = 1 √ | p i ⊗ | n i + | p i ⊗ | − n i ) , | ψ i = 1 √ | p i ⊗ | n i − | p i ⊗ | − n i ) . (2.3)Here, | ± n i are the Bloch sphere representation of spin state (qubit) as | n i = cos ξ e iτ sin ξ , | − n i = sin ξ − e iτ cos ξ , (2.4)where ξ and τ are polar and azimuthal angles, respectively. particle quantum states We assumed that spin and momentum are in the yz-plane ( τ = π , ~p = (0 , p sin θ, p cos θ ))and the Lorentz boost is orthogonal to it. For an observer in another reference frame S ′ described by an arbitrary boost Λ in the x-direction, the transformed BD states are given by(see Appendix A) | ψ i i −→ U (Λ) | ψ i i , | Λ ψ i = 1 √ | Λ p i ⊗ cos ξ cos Ω ~p − i sin Ω ~p sin ζ i sin ξ cos Ω ~p + sin Ω ~p cos ζ + | Λ p i ⊗ sin ξ cos Ω ~p − i sin Ω ~p cos ζ − i cos ξ cos Ω ~p − sin Ω ~p sin ζ , | Λ ψ i = 1 √ {| Λ p i ⊗ cos ξ cos Ω ~p − i sin Ω ~p sin ζ i sin ξ cos Ω ~p + sin Ω ~p cos ζ −| Λ p i ⊗ sin ξ cos Ω ~p − i sin Ω ~p cos ζ − i cos ξ cos Ω ~p − sin Ω ~p sin ζ , pin-momentum correlation in relativistic single particle quantum states | Λ ψ i = 1 √ {| Λ p i ⊗ cos ξ cos Ω ~p + i sin Ω ~p sin ζ i sin ξ cos Ω ~p − sin Ω ~p cos ζ + | Λ p i ⊗ sin ξ cos Ω ~p + i sin Ω ~p cos ζ − i cos ξ cos Ω ~p + sin Ω ~p sin ζ , | Λ ψ i = 1 √ {| Λ p i ⊗ cos ξ cos Ω ~p + i sin Ω ~p sin ζ i sin ξ cos Ω ~p − sin Ω ~p cos ζ −| Λ p i ⊗ sin ξ cos Ω ~p + i sin Ω ~p cos ζ − i cos ξ cos Ω ~p + sin Ω ~p sin ζ , (2.5)where ζ = ( ξ − θ ) and {| Λ p i , | Λ p i} are two orthogonal momentum eigen-state after Lorentztransformation.The BD density matrix (2.2), which describes the state of the single-particle at non-relativisticframe, is exchanged to the density matrix ρ ′ after Lorentz transformation, i.e. ρ −→ U (Λ) ρ,ρ ′ = U (Λ) ρ = X i =1 P i | Λ ψ i ih Λ ψ i | . (2.6)It can be calculate that | ψ i i will be orthogonal after Lorentz transformation, i. e. h Λ ψ i | Λ ψ j i = δ ij . We know that a system is entangle when its density matrix cannot be written as a convex sumof product states. For a pure state, dividing the system into two subsystems, A and B , allows pin-momentum correlation in relativistic single particle quantum states We show that by the von Neumann entropy the entanglement for a pure state in the Schmidtform [25] is not invariant after Lorentz transformation, and depend on the angles between spinand momentum. We introduce the following pure state | ψ i = p λ | n i ⊗ | p i + p λ | − n i ⊗ | p i , (3.7)where λ + λ = 1.We take the trace over the momentum eigen states and we obtain the following reduced spindensity matrix ρ ′ = T r Λ p , Λ p ( | Λ ψ ih Λ ψ | ) , with the following two different eigenvalues η = 12 { λ + λ − q λ + λ + λ λ (cos 2 ϕ − ϕ cos (Ω p − Ω p ) − } ,η = 12 { λ + λ + q λ + λ + λ λ (cos 2 ϕ − ϕ cos (Ω p − Ω p ) − } , (3.8)where ϕ is the angle between the spin and momentum ( ϕ = ξ − θ ). After some mathematical pin-momentum correlation in relativistic single particle quantum states E ( ρ ′ ) ≤ E ( ρ ) . (3.9)It shows that inequality (3.9) shows that when Lorentz boost and momentum are perpendic-ular, spin-momentum entanglement is decreases with increasing of velocity of the observer, aswell as when spin and momentum are perpendicular, i. e. ϕ = π ⇒ η = λ , η = λ , that show Lorentz transformation does not change the entanglement between them, i.e. E ( ρ ′ ) = E ( ρ ) . This subsection is devoted to calculate the concurrence of relativistic BD mixed state is givenin (2.6). By using the Appendix A, we obtain the following result: λ = 12 √ { q A + B − p C D } ,λ = 12 √ { q A + B + p C D } ,λ = 12 √ { q A + B − p C D } ,λ = 12 √ { q A + B + p C D } , where A = 3 P + 3 P − ( P + P ) cos 2 ϕ,B = 2 cos ϕ (2 P P + ( P − P ) cos ω ) ,C = ( P − P ) ( − ϕ − ϕ cos ω ) ,D = ( − (3 P + P )( P +3 P )+( P − P ) (cos 2 ϕ − ϕ cos ω )) , pin-momentum correlation in relativistic single particle quantum states λ i ’s are the square roots of the eigenvalues ρ ˜ ρ and ω = (Ω p + Ω p ) . First index ”1” in(A,B,C,D) corresponds to the ( P , P ) and the second index ”2” corresponds to the ( P , P ) , Therefore C ( ρ ′ ) = max { , λ − λ − λ − λ } , ( λ ≥ λ ≥ λ ≥ λ ) . (3.10)To see the behavior of concurrence with respect to the boost in x direction, after some calcu-lation we obtain the following results,( λ − λ ) = ( P − P ) (1 − cos ϕ sin ω , ( λ + λ ) = ( P + P ) − ( P − P ) cos ϕ sin ω . (3.11)Using by Eqs (3.11), we obtain( λ − λ ) − ( λ + λ ) = ( P − P ) r (1 − cos ϕ sin ω − r ( P + P ) − ( P − P ) cos ϕ sin ω , it is easy to see that( P + P ) − ( P − P ) cos ϕ sin ω ≥ ( P + P ) (1 − cos ϕ sin ω , so we have( λ − λ ) − ( λ + λ ) ≤ ( P − P ) r (1 − cos ϕ sin ω − ( P + P ) r (1 − cos ϕ sin ω P − P − P − P ) r (1 − cos ϕ sin ω ≤ ( P − P − P − P )therefore C ( ρ ′ ) ≤ C ( ρ ) . This shows that the spin-momentum correlation in single-particle mixed quantum state is pin-momentum correlation in relativistic single particle quantum states ϕ = π then the concurrence is not an observer-dependent quantity ininertial frame, namely C ( ρ ′ ) = C ( ρ ). We explain how the Lorentz transformations can be realized as quantum control gates. To dothis, we consider the pure sate of (3.7) under Lorentz transformations as U (Λ) | ψ i = p λ | Λ p i ⊗ W ( n , p ) | n i + p λ | Λ p i ⊗ W ( n , p ) | n i , (4.12)where W ( n i , p j ) is Wigner rotation and the spinors are rotated by the Wigner angles. Asa result, the Wigner rotation essentially behaves like a quantum control gate or controllingoperator with the control quantum sates {| p i , | p i} and target states ( | n i , | n i ). In order tobetter see the quantum control gate, we assume that the reference frame S ′ is described by anarbitrary Lorentz boost in the x-direction and momentum and spin are parallel in z-direction,i.e. ϕ = 0. Then the transformed states in 2 ⊗ | p i ⊗ | i → cos Ω p | Λ p i ⊗ | i + sin Ω p | Λ p i ⊗ | − i , | p i ⊗ | − i → − sin Ω p | Λ p i ⊗ | i + cos Ω p | Λ p i ⊗ | − i , | p i ⊗ | i → cos Ω p | Λ p i ⊗ | i + sin Ω p | Λ p i ⊗ | − i , | p i ⊗ | − i → − sin Ω p | Λ p i ⊗ | i + cos Ω p | Λ p i ⊗ | − i , pin-momentum correlation in relativistic single particle quantum states {| Λ p i| i , | Λ p i| − i , | Λ p i| i , | Λ p i| − i} is calculated as:Λ = cos Ω p sin Ω p − sin Ω p cos Ω p Ω p sin Ω p − sin Ω p cos Ω p , In the special case where (Ω p + Ω p ) = π , we obtaincos Ω p p , sin Ω p p p → Lim Ω p → Λ = − . (4.13)We know that, the Controled-Not (CNOT) gate is a two-qubit circuit that transforms targetqubit from its initial eigen-state to the opposite basis state iff the ’control’ qubit is in eigen-state | − i . Obviously, the quantum operation (4.13) flips the spin states, when the controlmomentum state is | p i , so the matrix representation (4.13) is similar to the Controlled-Not(CNOT) gate. This CNOT is a nonlocal operation because it can actually create a maximallyentangled state from a product state or vice versa. For instance, after applying the gate(4.13)onthe product state( | p i + | p i ) ⊗ | i , we obtain the following entangled state( | p i + | p i ) ⊗ | i → | Λ p i ⊗ | i + | Λ p i ⊗ | − i , (4.14)and for maximally-entangled Bell state1 √ | p i ⊗ | i + | p i ⊗ | − i ) → √ | Λ p i − | Λ p i ) ⊗ | i , (4.15)which is a separable state. pin-momentum correlation in relativistic single particle quantum states In this paper, we have considered spin-momentum correlation of massive single spin- particlequantum states which furnish an irreducible representation of the Poincare group. Insteadof the superposition of all momenta we have considered only two momenta p and p eigenstates. We have shown that the spin-momentum correlation of relativistic single spin- particlemixed state( when the momentum is perpendicular to the boost direction) is dependent on theangle between spin and momentum and when they are parallel the measure of entanglementdecreases with increasing of velocity of the observer. We have also shown that the Lorentztransformations can be realized as quantum control gates and they become like the CNOTgate in the limit where β → APPENDIX A
Wigner representation for spin- In Ref. [26], is shown that effect of an arbitrary Lorentz transformation Λ unitarily imple-mented as U (Λ) on single-particle states is U (Λ)( | p i ⊗ | σ i ) = s (Λ p ) p X σ ′ D σ ′ σ ( W (Λ , p ))( | Λ p i ⊗ | σ ′ i ) , (A-i)where W (Λ , p ) = L − (Λ p )Λ L ( p ) , (A-ii)is the Wigner rotation [7]. We will consider two reference frames in this work: one is the restframe S and the other is the moving frame S ′ in which a particle whose four-momentum p in Sis seen as boosted with the velocity ~v . By setting the boost and particle moving directions inthe rest frame to be ˆ v with ˆ e as the normal vector in the boost direction and ˆ p , respectively, pin-momentum correlation in relativistic single particle quantum states n = ˆ e × ˆ p , the Wigner representation for spin- particle is found as [12], D ( W (Λ , p ) = cos Ω ~p i sin Ω ~p ~σ. ˆ n ) , (A-iii)where cos Ω ~p α cosh δ + sinh α sinh δ (ˆ e. ˆ p ) q [ + cosh α cosh δ + sinh α sinh δ (ˆ e. ˆ p )] , (A-iv)sin Ω ~p n = sinh α sinh δ (ˆ e × ˆ p ) q [ + cosh α cosh δ + sinh α sinh δ (ˆ e. ˆ p )] , (A-v)and cosh α = γ = 1 p − β , cosh δ = E m , β = vc . APPENDIX B
Entanglement of formation
Let | ψ i = P Ni,j =1 a ij e i ⊗ e j , a ij ∈ C be an two-particle pure states with normalization P Ni,j =1 | a ij | = 1. For this pure state the entanglement of formation E is defined as theentropy of either of the two sub-Hilbert space, i. e. E ( | ψ i ) = − T r ( ρ log ρ ) = − T r ( ρ log ρ ) . (A-vi)where ρ (respectively, ρ ) is the partial trace of | ψ ih ψ | over the first (respectively, second)Hilbert space. A given density matrix ρ on H d ⊗ H d has pure-state decompositions of | ψ i i ofthe form (2.2) with probabilities P i , The entanglement of formation for the mixed state ρ is pin-momentum correlation in relativistic single particle quantum states ρ , i. e. E ( ρ ) = min X i P i E ( | ψ i i ) . (A-vii)In the case of n=2, (A-vi) can be written as E ( | ψ i ) | n =2 = H ( 1 + √ − C , (A-viii)where H ( x ) = − x log x − (1 − x ) log (1 − x ) is binary entropy and C is called concurrence.Thus calculation of (A-vii) can be reduced to calculate the corresponding minimum of C ( ρ ) = min Σ kb =1 p b C ( | ψ b i ) . Wootters in [20] has shown that for a 2-qubit system entanglement of formation of a mixedstate ρ can be defined as E ( ρ ) = H ( 1 + √ − C , (A-ix)by C ( ρ ) = max (0 , λ − λ − λ − λ ) , (A-x)where the λ i are the non-negative eigenvalues, in decreasing order, of the Hermitian matrix R ≡ q √ ρ ˜ ρ √ ρ, and ˜ ρ = ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) , where ρ ∗ is the complex conjugate of ρ when it is expressed in a fixed basis such as | ↑i , | ↓i ,and σ y is − ii on the same bases.In order to obtain the concurrence of BD states (2.2) we follow the method presented byWootters in [20]. We define subnormalized orthogonal eigenvectors | v i i as | v i i = p P i | ψ i i , h v i | v i i = P i δ ij , pin-momentum correlation in relativistic single particle quantum states | x i i as | x i i = Σ j =1 U ∗ ij | v i i for i = 1 , , , h x i | ˜ x j i = ( U τ U T ) ij = λ i δ ij , | ˜ x j i = σ y ⊗ σ y | x ∗ j i where τ ij = h v i | ˜ v j i is a symmetric but not necessarily Hermitian matrix. In construction of | x i i we have considered the fact that for any symmetric matrix τ one can always find a unitarymatrix U in such a way that λ i are real and non-negative, that is, they are the square roots ofeigenvalues of τ τ ∗ which are the same as the eigenvalues of R. Moreover one can always findU such that λ being the largest one. After some calculations we get the following values for λ i , λ = P , λ = P , λ = P , λ = P , and concurrence can be evaluated as C ( ρ ) = ( P − P − P − P ) . (A-xi) pin-momentum correlation in relativistic single particle quantum states References [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. , 777 (1935).[2] N.L.Harshman, Phys. Rev.A , 022312 (2005).[3] Daeho Lee and Ee Chang-Young, New Journal of Physics . , 022312 (2004).[4] Pawel Caban and Jakub Rembielinski, Phys. Rev.A , 012103 (2005).[5] Stephen D.Bartlett and Daniel R.Terno, Phys. Rev.A , 012302 (2005).[6] A.peres , P.F.Scudo and D.R.Terno, Phys. Rev. Lett , 230402 (2002).[7] E. P. Wigner, Ann. Math. , 149 (1939).[8] R.M.Gingrich and C.Adami,Phys.Rev.Lett. , 27 (2002);[9] H.Terashimo and M.Ueda , LANL e-print ,quant-ph/0204138.[10] M.Czachor, phys.Rev.A , 72 (1997).[11] P.M.Alsing and G.J.Milburn , LANL e-print ,quant-ph/0203051.[12] Doyeol Ahn, Hyuk-jae Lee, Young Hoon Moon, and Sung Woo Hwang, Phys. Rev.A ,012103 (2003).[13] M. A. Jafarizadeh and R. Sufiani , Phys. Rev.A , 012105 (2008).[14] L.lamata, M.A.Martin-Delgado, E.solano, Phys.Rev.Lett. , 250502 (2006).[15] L.lamata, Juan leon, David Salgado, Phys.Rev.A . , 052325 (2006).[16] Jian-Ming Cai, Zheng-Wei Zhou, Ye-Fei Yuan, and Guang-Can Guo, Phys. Rev.A ,042101 (2007). pin-momentum correlation in relativistic single particle quantum states , 052109 (2009).[18] Andr G. S. Landulfo and George E. A. Matsas, Phys. Rev. A , 032315 (2009).[19] C.H.Bennett, D.P.Divincenzo, J.A.Smolin, W.K.Wootters, Phys. Rev.A , 3824 (1935).[20] W.K.Wootters, Phys.Rev.Lett. , 2245 (1998).[21] Ping-Xing Chen, Lin-Mei Liang, Cheng-Zu Li, Ming-Qiu Huang, Phys. Lett.A , 175-177 (2002).[22] Arture Lozinski, Andreas Buchleitner, Karol Zyczkowski and Thomas Wellens, LANLe-print, quant-ph/0302144v1.[23] V.Vedral, M.B.Plenio, M.A.Rippin, P.L.Knight, Phys.Rev.Lett.53