On Simultaneous Information and Energy Transmission through Quantum Channels

TL;DR

It is shown that the capacity-power function for a quantum channel, for both unassisted and private protocol, is concave and also proved additivity for unentangled and uncorrelated ensembles of input signals, implying that regularized formulas for calculation do not need to be used.

Abstract

The optimal rate at which information can be sent through a quantum channel when the transmitted signal must simultaneously carry some minimum amount of energy is characterized. To do so, we introduce the quantum-classical analogue of the capacity-power function and generalize results in classical information theory for transmitting classical information through noisy channels. We show that the capacity-power function for a classical-quantum channel, for both unassisted and private protocol, is concave and also prove additivity for unentangled and uncorrelated ensembles of input signals for such channels. This implies we do not need regularized formulas for calculation. We show these properties also hold for all noiseless channels when we restrict the set of input states to be pure quantum states. For general channels, we find that the capacity-power function is piece-wise concave. We give an elegant visual proof for this supported by numerical simulations. We connect channel capacity and properties of random quantum states. In particular, we obtain analytical expressions for the capacity-power function for the case of noiseless channels using properties of random quantum states under an energy constraint and concentration phenomena in large Hilbert spaces.

Figures (17)

• Figure 1: Graphical summary: The effect of threshold energy on the information at the receiver: (a) The information is encoded in quantum states, which are sent through a physical channel. At the output, a detector captures the states and decodes the information. However, the detector is activated only after a certain threshold of energy $B$, carried by the output states. (b) A pictorial representation of the effect of an increase in $B$ on the decoded information. As the threshold of energy $B$ increases, information content at the receiver becomes poorer as shown by the increasingly mangled message at the receiver.
• Figure 2: Typical protocol for communication through a quantum channel with blocklength $n$.
• Figure 3: (a) Binary noiseless channel, (b) mutual information profile with respect to input distribution for noiseless channel (binary entropy function), (c) binary symmetric channel, (d) binary erasure channel
• Figure 4: Capacity-power function for three different channels when maximized over input states as well as probabilities. We can clearly see the function is not concave in general except for the noiseless channel (the black curve).
• Figure 5: Visual proof of Theorem \ref{['piece_wise_concavity']}: (a) Three curves of capacity-power function corresponding to three different sets of input signal states, $\mathcal{S}_1 = \{\rho_{1x} \}$, $\mathcal{S}_2 = \{\rho_{2x} \}$, $\mathcal{S}_3 = \{\rho_{3x} \}$ denoted by 1, 2, 3, respectively. The maximum of capacity-power function over all these sets of input signal states, shown in (b), is piecewise concave.

Theorems & Definitions (16)

• Theorem 1
• proof
• Corollary 1.1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof