Information capacity of a single photon

Quantum states of light are the obvious choice for communicating quantum information. To date, encoding information into the polarization states of single photons has been widely used as these states form a natural closed two-state qubit. However, photons are able to encode much more—in principle, infinite—information via the continuous spatiotemporal degrees of freedom. Here we consider the information capacity of an optical quantum channel, such as an optical fiber, where a spectrally encoded single photon is the means of communication. We use the Holevo bound to calculate an upper bound on the channel capacity, and relate this to the spectral encoding basis and the spectral properties of the channel. Further, we derive analytic bounds on the capacity of such channels, and, in the case of a symmetric two-state encoding, calculate the exact capacity of the corresponding channel.

Figure 1

Example choice of encoding basis $\left\{{\psi }_i\left(\omega \right)\right\}$, where the basis states are not perfectly orthogonal, and an example channel spectral response function $\eta \left(\omega \right)$. The functions $\left\{{\psi }_i\left(\omega \right)\right\}$ are identical Gaussians of different means, separated in frequency by multiples of $\Delta \omega$. By letting the Gaussians have zero variance, one could achieve orthonormality and encode a qudit with an arbitrary number of levels. However, this would be physically unrealistic. In this example we have centered $\omega$ around 0, which is unphysical. Thus, $\omega$ should be interpreted as being relative to some central carrier frequency. All the integral overlaps defining the channel capacity are invariant under uniform translations in $\omega$, so the choice of center frequency does not affect the results. Other examples of encoding bases are time-bin encoding, which would yield the same results in the conjugate domain, or orbital angular momentum encoding.

Figure 2

Upper bound on the classical channel capacity (bits per transmitted photon) using Gaussian wave packets, displaced by multiples of $\Delta \omega$, where Alice employs uniform encoding, $p_i=1/N$. Left: Flat spectral response channel (efficiency $\eta$), with or without postselection. Middle: Gaussian spectral response channel (bandwidth $\sigma$), without postselection. Right: Gaussian spectral response channel, with postselection. We encode across $N=32$ basis states, a maximum of ${\log }_{2}N=5$ bits of information. In the limit of $\Delta \omega \gg 1$ (distinguishable photons), and $\sigma \gg 1$ or $\eta =1$ (a perfect channel), the upper bound on $\mathcal{C}$ asymptotes to the maximum achievable ${\log }_{2}N$. In this limit $\mathcal{C}$ saturates the bound.

Figure 3

Upper bound on the classical channel capacity (without postselection) as a function of the alphabet size $N$ and channel bandwidth $\sigma$. The red line shows the optimal $N$ for a given $\sigma$ based on the Holevo bound.

Figure 4

Maximum classical channel capacity for two-state Gaussian encoding as a function of the ratio $\lambda ={\sigma }_{\psi }/{\sigma }_{\eta }$.