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arXiv:quant-ph/0511067 · Physical Review A

Maps for Lorentz transformations of spin

Thomas F. Jordan, Anil Shaji, E. C. G. Sudarshan

Published Mar 13, 2006• 33 citations
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Abstract

Lorentz transformations of spin density matrices for a particle with positive mass and spin $1∕2$ are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are not independent, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit correlated with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverses are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.

DOI
10.1103/physreva.73.032104

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Lorentz transformations of spin density matrices for a particle with positive mass and spin 1∕21∕21∕2 are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are not independent, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit correlated with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverses are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.
Linear maps of matrices describing the evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that of completely positive maps. Only some minus signs are inserted in the operator-sum representation. Each map is the difference of two completely positive maps. The maps are first obtained as maps of mean values and then as maps of basis matrices. These forms also prove to be useful. An example for two entangled qubits is worked out in detail. The relation to earlier work is discussed.
Lorentz boosts applied to particles with spin and momentum degrees of freedom induce momentum-dependent rotations. As, in general, different particles have different momenta, the transformation of the whole state is not a representation of the rotation group. Here we identify the group that acts on a two-particle system and, for the case where the momenta of the particles are correlated, find invariant subspaces that have interesting properties for quantum information processes in relativistic scenarios. A basis of states for the study of transformations of spin states under Lorentz boosts is proposed, which is a good candidate for use in building quantum communication protocols in relativistic scenarios.
Simple examples are presented of Lorentz transformations that entangle the spins and momenta of two particles with positive mass and spin 1∕21∕21∕2. They apply to indistinguishable particles, produce maximal entanglement from finite Lorentz transformations of states for finite momenta, and describe entanglement of spins produced together with entanglement of momenta. From the entanglements considered, no sum of entanglements is found to be unchanged.
Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form [Formula: see text] of the Hilbert–Schmidt (HS) decomposition of the density matrix. For the generic case, in which Lorentz transformations diagonalize [Formula: see text] into [Formula: see text], we give relations between the [Formula: see text] and the [Formula: see text]. In particular, we consider two cases: (a) Two qubits density matrices with one pair of linear terms in the HS decomposition. (b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case (a) are analyzed and the non-generic property is related explicitly to Lorentz velocity [Formula: see text] which is not reachable physically.
This paper was concerned with the spin-momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigen states (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin-momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not observer-dependent quantity in inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and has shown that entanglement decreases with respect to the increasing of observer velocity. Finally, we argue that, Wigner rotation is induced by Lorentz transformations can be realized as controlling operator.
This paper is concerned with the spin–momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigenstates (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin and momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not an observer-dependent quantity in the inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and have shown that entanglement decreases with respect to the increase in observer velocity. Finally, we argue that Wigner rotation is induced by Lorentz transformations and can be realized as a controlling operator.