Lorentz covariant reduced - density - operator theory for relativistic quantum information processing

In this paper, we derived Lorentz covariant quantum Liouville equation for the density operator which describes the relativistic quantum information processing from Tomonaga-Schwinger equation and an exact formal solution for the reduced-density-operator is obtained using the projector operator technique and the functional calculus. When all the members of the family of the hypersurfaces become flat hyperplanes, it is shown that our results agree with those of non-relativistic case which is valid only in some specified reference frame. As an example, we derived the polarization of the vacuum in quantum electrodynamics up to the second order. The formulation presented in this work is general and could be applied to related fields such as quantum electrodynamics and relativistic statistical mechanics. e-mail:[email protected] e-mail:[email protected] e-mail:[email protected] 1 Recently, there has been growing interest in the relativistic formulation [1][7] of quantum operations for possible near future applications to relativistic quantum information processing such as teleportation [8], entanglement-enhanced communication [9], and quantum clock synchronization [10], [11]. In the non-relativistic case, the key element for studying quantum information processing is the density operator of a quantum register which is derived from the solution of a quantum Liouville equation (QLE) [12], [13] for the total system including an environment. The QLE is an integro-differential equation and it is in general nontrivial to obtain the solution of the form ρ E → ρ = Ê [ρ], (1) where ρ is the reduced density operator of the quantum register and Ê is the superoperator describing the evolution of ρ by the quantum information processing. In the previous works, we have employed a time-convolutionless reduced-density-operator formalism to model quantum devices [14] and noisy quantum channels [15], [16]. The first step toward the relativistic quantum information theory would be the formulation of Lorentz covariant QLE and the derivation of the reduced-density-operator which is a solution of the covariant QEL. The goal of this paper is to derive Lorentz covariant quantum Liouville equation which describes the relativistic quantum information processing and obtain a formal solution for the reduced-density-operator pertaining to the system (or electrons) part alone. It is well known that neither the non-relativistic Schrödinger equation nor the QLE is Lorentz covariant. As a result, it is expected that the usual non-relativistic definition of the reduced-density-operator and its functionals such as quantum entropy have no invariant meaning in special relativity. Another conceptual barrier for the relativistic treatment of quantum information processing is the difference of the role played by the wave fields and the state vectors in the quantum field theory. In non-relativistic quantum mechanics both the wave function and the state vector in Hilbert space give the probability amplitude which 2 can be used to define conserved positive probability densities or density matrix. On the other hands, in relativistic quantum field theory, covariant wave fields are not probability amplitude at all, but operators which create or destroy particles in spanned by states defined as containing definite numbers of particles or antiparticles in each normal mode [17]. The role of the fields is to make the interaction or S-matrix satisfy the Lorentz invariance and the cluster decomposition principle. The information of the particle states is contained in the state vectors of the Hilbert space spanned by states containing 0, 1, 2, · · · particles as in the case of non-relativistic quantum mechanics. So it seems like that one needs to obtain the covariant equation of motion for the state vector and derive the covariant QLE out of it. Some fifty years ago, Tomonaga [18] and Schwinger [19] derived a covariant equation of motion for the quantum state vector in terms of the functional derivative, known as Tomonaga-Schwinger (T-S) equation, i δΨ[σ] δσ(x) = Hint(x)Ψ[σ], (2) in the interaction picture. Here x is a space-time four-vector, σ is the spacelike hypersurface, Ψ[σ] is the state vector which is a functional of σ, Hint(x) = Hint[φα(x)] is the interaction Hamiltonian density which is a functional of quantum field φα[x], and δ δσ(x) is the Lorentz invariant functional derivative [20]. The functional derivative of Ψ[σ] is defined as δΨ[σ] δσ(x) = lim δω→0 Ψ[σ]−Ψ[σ] δω , (3) where δω is an infinitesimal four-dimensional volume between two hypersurfaces σ and σ. The formal solution of equation (2) is given by Ψ[σ] = U [σ, σ0]Ψ[σ0], (4) where the generalized transformational functional satisfies the T-S equation i δU [σ, σ0] δσ(x) = Hint(x)U [σ, σ0] (5)