Quantum and classical capacity boosted by a Lorentz transformation

In this paper we show that the quantum channel between two inertial observers who transmit quantum information by sending realistic photonic wave packets is a well-studied channel in quantum Shannon theory: the Pauli channel. The parameters of the Pauli channel and therefore its classical and quantum capacity depend on the magnitude of the Lorentz boost relating the two observers. The most striking consequence is that two inertial observers whose Pauli channel has initially zero quantum capacity can achieve nonzero quantum communication rates (reaching, in principle, its maximal value, equal to 1) by applying a boost in the right direction. This points to a fundamental connection between quantum channel capacities and special relativity.

Figure 1

The general wave packet can be thought of as a collection of $k$ vectors modulated by an envelope function. Each vector has a (complex) helicity space attached whose real projection can be visualized as an ${\mathbb{R}}^2$ plane perpendicular to the momentum vector (represented by two tangential planes). The amplitudes $a$ and $b$ of the real helicity vectors given by the projection of the dashed (blue) arrows on the helicity basis vectors [light (green) solid arrows] lying in the plane are identical for all $\mathbf{k}$'s, but because the momentum vectors point in different directions in momentum space, the helicity vectors point in different directions in ambient ${\mathbb{R}}^3$ space [see related discussion before Eqs. (2) and (4)]. To correct for this effect, we “unrotate” the polarization vectors by applying Eq. (4). This compensates the rotation of the $\mathbf{k}$ vector and is depicted in the plane perpendicular to the vector $k=k(\vartheta ,\varphi )$. But it works well only for small $\vartheta$. The unrotation becomes less effective as $\vartheta$ increases, and for $\vartheta \to \pi /2$ it is useless since the helicity vector points “downwards.” The important point is that, without the unrotation, reliable quantum communication is impossible [see Eq. (7)].

Figure 2

The classical capacity $C\left(\mathcal{P}\right)$ in bits per channel (upper dashed curve) and a lower bound on the quantum capacity $Q_{\uparrow }\left(\mathcal{P}\right)$ in bits per channel (lower curve) of a Pauli channel plotted as a function of $1/\Gamma$. The zero lower bound $Q_{\uparrow }\left(\mathcal{P}\right)$ becomes true zero quantum capacity below Cerf's bound where $c_{\downarrow }^0\ge 1/2$ (to the left of the vertical line).

Figure 3

The effect of a negative rapidity $\zeta$ in the $z$ direction (an approaching observer) on the quantum capacity's lower bound (in bits per channel) is illustrated for three initial wave packets. Two wave packets whose quantum capacity is 0 ($1/\Gamma =0.005$ and $1/\Gamma =0.05$; see Fig. 2) can be boosted to nonzero values. Already for a nonzero value of quantum capacity (illustrated as $1/\Gamma =0.3$) the boost further increases the communication rate.

Figure 4

Diagram describing the whole physical setup and the emergence of a relativistic Pauli channel $\mathcal{P}$ (note that ${\mathcal{P}}^{\prime }$ is $\mathcal{P}$ for $\zeta =0$). For a detailed description see the paragraph preceding Appendix pp1.