Majorization Theory Approach to the Gaussian Channel Minimum Entropy Conjecture

A long-standing open problem in quantum information theory is to find the classical capacity of an optical communication link, modeled as a Gaussian bosonic channel. It has been conjectured that this capacity is achieved by a random coding of coherent states using an isotropic Gaussian distribution in phase space. We show that proving a Gaussian minimum entropy conjecture for a quantum-limited amplifier is actually sufficient to confirm this capacity conjecture, and we provide a strong argument towards this proof by exploiting a connection between quantum entanglement and majorization theory.

Figure 1

Any phase-insensitive Gaussian bosonic channel $\mathcal{M}$ is indistinguishable from a composed channel $\mathcal{A}\circ \mathcal{L}$, where $\mathcal{L}$ is a pure-loss channel and $\mathcal{A}$ a quantum-limited amplifier. The Stinespring dilation of $\mathcal{L}$ is a beam splitter of transmissivity $T$, while the amplifier $\mathcal{A}$ of gain $G$ becomes a two-mode squeezer of parameter $r$ ($G={cosh}^{2}r$) in which the input mode $A$ interacts with a vacuum environmental mode $E$.

Figure 2

Entanglement of the output state $|{\Psi }_{\lambda }^{(k)}⟩$ as a function of the squeezing parameter $r$. As explained in the text, the entanglement is monotonically increasing with $r$ for all Fock input states, while, for a fixed $r$, it monotonically increases with $k$. This behavior is in full agreement with the majorization relations (13, 15) proved in the text. The arrows in the figure indicate the majorization order.