We have shown that the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has a pure quantum contribution that does not exist in the classical covariant spin operator. Based on this equivalence, reduced spin states can be clearly defined. We have shown that depending on the relative motion of an observer, the change in the entropy of a reduced spin density matrix sweeps through the whole range.
Relativistic effects affect nearly all notions of quantum information theory. The vacuum behaves as a noisy channel, even if the detectors are perfect. The standard definition of a reduced density matrix fails for photon polarization because the transversality condition behaves like a superselection rule. We can however define an effective reduced density matrix which corresponds to a restricted class of positive operator-valued measures. There are no pure photon qubits, and no exactly orthogonal qubit states. Reduced density matrices for the spin of massive particles are well-defined, but are not covariant under Lorentz transformations. The spin entropy is not a relativistic scalar and has no invariant meaning. The distinguishability of quantum signals and their entanglement depends on the relative motion of observers.
Lorentz transformation of the reduced helicity matrix for a massive spin 1/2 particle is first investigated in the framwork of relativistic quantum information theory. The corresponding entangled helicity entropy is calculated and has no invariant meaning as that of spin. The variation of the entropy as the relative velocity of motion, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of Lorentz group. The underlying physical explanations are also discussed.
Lorentz transformation of the reduced helicity density matrix for a massive spin- particle is investigated in the framework of relativistic quantum information theory for the first time. The corresponding helicity entropy is calculated, which shows no invariant meaning as that of spin. The variation of the helicity entropy with the relative speed of motion of inertial observers, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of the Lorentz group. This novel and odd behavior unique to the helicity may be readily detected by high-energy physics experiments. The underlying physical explanations are also discussed.
A foundational result in relativistic quantum information theory due to Peres, Scudo, and Terno, is that von Neumann entropy is not Lorentz invariant. Motivated by the "It from Qubit" paradigm, here we show that Lorentzian symmetries of quantum information emerge naturally in a pre-spacetime setting, without any reference to external variables such as position or momentum. In particular, we derive the natural action of the restricted Lorentz group
SO+(1,3) on the internal degrees of freedom of a single qubit from a simple, information-theoretic principle we refer to as preservation of linear entropy. It is then shown that the Lorentz invariance of the linear entropy of a relativistic qubit is a special case of a much more general phenomenon, namely, that any spectral invariant of an operator we term the '
W-matrix' is an
SL(2,C)⊗n invariant scalar. Consequently, the linear
n-partite quantum mutual information is shown to be an
SL(2,C)⊗n invariant for all
n-qubit states. Finally, we show that the correlation function associated with a pair of qubits in the singlet state yields the Minkowski metric on the space of qubit observables, whose symmetry group is the full Lorentz group
SO(1,3). In accordance with the "It from Qubit" paradigm, our results thus establish the natural emergence of relativistic spacetime structure from intrinsic properties of quantum information.
Variation of the von Neumann entropy by the Lorentz transformation is discussed. Taking the spin-singlet state in the center-of-mass frame, the von Neumann entropy in the laboratory frame is calculated from the reduced density matrix obtained by taking the trace over 4-momentum after the Lorentz transformation. As the model to discuss the EPR spin correlation, it is supposed that one parent particle splits into a superposition state of various pair states in various directions. Computing the von Neumann entropy and the Shannon entropy, we have shown a global behavior of the entropy to see a relativistic effect. We discuss also the superrelativistic limit, distinguishability between the two particles of the pair and so on.