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Quantum Theory of Computation and Relativistic Physics

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Quantum Theory of Computation and Relativistic Physics View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Quantum theory of computation and relativistic physics Alexander Yu. Vlasov∗ 197101, Mira Street 8, IRH St.{Petersburg, Russia Jan 1997 (Jul 1999) quant-ph/9701027 Title: Quantum Theory of Computation and Relativistic Physics Authors: Alexander Yu. Vlasov (FCR/IRH, St.-Petersburg, Russia) Comments: 6 pages, LaTeX2e, 2 columns, 4 PostScript figures are included by epsfig.sty; based on poster in Proc. PhysComp '96 Workshop (BU, Boston MA, 22-24 Nov 1996) pp. 332-333, a "no-go" result for bounded quantum networks; v3/4 -- typos corrected, minor changes. Report-no: RQC-VAY11/v4 In the e-print is discussed a few steps to introducing of "vocabulary" of relativistic physics in quantum theory of information and computation (QTI&C). The behavior of a few simple quantum systems those are used as models in QTI&C is tested by usual relativistic tools (transformation properties of wave vectors, etc.). Massless and charged massive particles with spin 1/2 are considered. Field theory is also discussed briefly. Abstract In the paper are described some steps for merger between relativistic quantum theory and theory of computation. The first step is consideration of transformation of qubit state due to rotation of coordinate system. The Lorentz transformation is considered after that. The some new properties of this transformation change usual model of qubit. The system of q2bit seems more fundamental relativistic model. It is shown also that such model as electron is really such q2bit system and for modelling of qubit is necessary to use massless particle like electron neutrino. The quantum field theory (QFT) is briefly discussed further. The wave vectors of interacted particle now described by some operator and it can produce some multiparticle (‘nonlinear’) effects. PACS numbers: 03.30.+p, 03.65.-w, 03.70.+k, 89.70.+c Keywords: Quantum, Computation, Relativistic ∗E-Mails: [email protected], [email protected], [email protected] 1 Introduction with complex coefficients. The state of quantum system is described as a ray in complex Hilbert space and for two- The paper describes some approaches to relativistic quan- state system it can be considered as complex projective . The main purpose of the work tum theory of computation space CP C .Eachray(c0,c1) is presented by ∼ ∪{∞} is to consider essentially new properties of quantum com- complex number ζ = c0/c1.The 0 corresponds to 0 and puters [1, 2, 3, 4, 5] due to relativistic phenomena rather the 1 to . There is correspondence|i [8] between the than some small corrections to nonrelativistic formulae. plane| iζ and∞ a sphere S due to stereographic projection ζ =(x iy)/(1 z) (see Fig. 1). Expressions for coordinate At first, in relativistic theory it is necessary to consider − − a qubit in different coordinate systems. In simplest case (x, y, z) on the unit sphere are: it may be 3D local rotations and SU(2) spinors. 2Re ζ c c +c c = = 0 1 1 0 For consideration of temporal coordinate it is necessary x 2 ζ +1 c0c0 +c1c1 to use Lorentz transformations and 4D spinors. The more |2Im| ζ i(c c c c ) = = 0 1 1 0 correct approach include full Poincare group and quantum y 2 − − (2) − ζ +1 c0c0 + c1c1 field theory. |ζ|2 1 c c c c z = | | − = 0 0 − 1 1 ζ 2 +1 c c +c c | | 0 0 1 1 2 Qubit Due to the equation Eq. (1) we can consider (X, Y, Z) instead: A quantum two-state system is often called quantum bit X = c0c1 + c1c0 or ‘qubit’ [6, 7]. Let us consider a particle with spin 1/2as Y =i(c c c c ) (3) 0 1− 1 0 a model of the qubit. The quantum state of the system is Z=c0c0 c1c1 ψ = c 0 + c 1 ,wherec and c are complex numbers − 0 | i 1 | i 0 1 The 0 and 1 map to opposite poles of the sphere. and the norm of ψ is: | i | i 2.1 Spatial rotation of coordinate system 2 2 2 ψ ψ∗ψ = c0 + c1 =1,c0,c1 C (1) || || ≡ | | | | ∈ A transformation of the state due to a spatial rotation of coordinate system is described by unitary matrix with determinant unity: False ab a = d, c = b (c1 =0) ψ0 = ψ, − (4) 1 cd ad bc= a 2 + b 2 =1 S − | | | | This is the group of unitary 2 2 matrices, SU(2). It × z corresponds to principle, that transformation of the wave x vector is described by some representation of a group of y coordinate transformation. The group SU(2) is represen- CP tation of the group of spatial rotations SO(3) in a space Im of 2D complex vectors. c 2 0 Due to 2–1 isomorphism SU(2) and SO(3), any rota- True Re × c1 tion corresponds to unitary matrix up to sign. We can see simple correspondence between any 1–gate and “pas- sive” transformation, i.e. transition to other coordinate Figure 1: Riemann sphere for qubit system. A state of a qubit can be described as a superposition The equations Eq. (3) can be used for demonstration of two logical states of usual bit (False, True or 0, 1) of relation between SO(3) and SU(2). If we apply some 1 unitary transformation Eq. (4) U :(c ,c ) (c0 ,c0 )then Linear transformations with determinant unity of a 0 1 → 0 1 (X, Y, Z) (X0,Y0,Z0). Unitary matrices do not change qubit correspond to Lorentz transformation of the vector the norm→ Eq. (1) and length of the vector: (T,X,Y,Z): 2 2 2 2 2 2 X + Y + Z =(c0 + c1 ) (5) | | | | ψ = Aψ;detA=1 Angles between vectors also do not change. Unitary trans- 0 V0 =2Aψ(Aψ)∗ =2Aψψ∗A∗ = AVA ∗ formations of a state of the qubit correspond to rotations 2 2 2 2 (8) det V0 = T 0 X0 Y 0 Z0 = of the sphere (Fig. 1). Two matrices: U and U produce − − − − =detV=T2 X2 Y2 Z2 the same rotation due to Eq. (3). − − − The transformations of a state of n–qubits due to spa- tial rotation can be described by unitary 22n matrices. t 3 The relativistic consideration of light cone a qubit ψ =1 jj jj 3.1 Lorentz transformation rotation (T,X,Y,Z) For Lorentz transformation of coordinate system there is similar isomorphism between the group SO(3, 1) and the group SL(2, C) of all complex 2 2 matrices with de- boost × terminant unity. The group SL(2, C) is isomorphic with x Lorentz group in the same way as the group SU(2) with group of 3D rotations [8]. The group SL(2, C)isarep- resentation of Lorentz group SO(3, 1) in a space of 2D complex vectors. On the other hand, we should not directly apply such y representation of relativistic group SL(2, C) to a qubit. Only the subgroup of unitary matrix saves the norm Figure 2: Null vector (T,X,Y,Z) Eq. (1). The expression Eq. (1) in relativistic theory is not invariant scalar, but temporal part of 4–vector. Sim- 1 ple relation between transformations of coordinate system Only if the matrix A is unitary, AVA ∗ = AVA − and and unitary matrices is broken here. Trace V i.e. the norm Eq. (6) does not change. Otherwise Let us denote: Eq. (6) should be considered as the ‘T –component’ of a 2 4–vector. T = ψ ψ∗ψ = c c + c c (6) || || ≡ 0 0 1 1 We can write1 , using equations Eq. (3), Eq. (6) : The relation between SL(2, C) and Lorentz group T + ZXiY c c c c Eq. (8) is valid not only for null vectors. Any vector is a V − =2 0 0 0 1 ≡ X+iYTZ c1c0 c1c1 sum of two null vectors and − 1 c0 A(V + U)A∗ = AVA ∗ + AUA∗. 2V= (c0 c1)=ψψ∗ (7) c1 The qubit is described by two-component complex vec- 2 2 2 2 det V = T X Y Z = tor or Weyl spinor. It corresponds to massless particle =2c c 2c c− 2c−c 2c−c =0 0 0 1 1− 1 0 0 1 with spin 1/2. Such particle always moves with the speed 1 of light. The equations Eq. (7) show a correspondence In the matrix notation ∗ is scalar and ∗ is 2 2matrix (with Dirac notation: and respectively). × between such spinor and 4D null vector (Fig. 2). This h j i j ih j 2 vector can be also rewritten by using Pauli matrices: t 01 0 i 10 σ = ,σ= ,σ= x 10 y i −0 z 0 1 − (9) V=T1+Xσx + Yσy +Zσz, 0 1 0 j = ψ∗ψ Vi = 2Tr(σiV)=Tr(σiψψ∗)=ψ∗σiψ; ψ∗γ ψ j σ= σ ,σ ,σ :(T, X, Y, Z )=(ψ∗ψ,ψ∗σψ) { x y z} { } light cone 3.2 Massive particle x Massive charged particle with spin 1/2 like an electron is described by two Weyl spinors and has four complex y components: ψ0 Figure 3: Massive particle ϕR 2 ψ1 ψ = ϕR,ϕL C ; ψ = (10) ϕL ∈ 0ψ2 1 ψ The representation is finite dimensional. B 3 C • It is possible to consider such massive@ particleA as two The representation is unitary in a definite norm. qubits: • It can be considered as some mathematical reasons for: ψ = c 00 + c 01 + c 10 + c 11 (11) 00| i 01| i 10| i 11| i Using of quantum field theory (QFT) instead of sys- The first index is similar to and for each ϕR,ϕL.
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