Gaussian-only regenerative stations cannot act as quantum repeaters

Higher transmission loss diminishes the performance of optical communication—be it the rate at which classical or quantum data can be sent reliably, or the secure key generation rate of quantum key distribution (QKD). Loss compounds with distance—exponentially in an optical fiber, and inverse square with distance for a free-space channel. In order to boost classical communication rates over long distances, it is customary to introduce regenerative relays at intermediate points along the channel. It is therefore natural to speculate whether untended regenerative stations, such as phase-insensitive or phase-sensitive optical amplifiers, could serve as repeaters for long-distance QKD. The primary result of this paper rules out all bosonic Gaussian channels to be useful as QKD repeaters, which include phase-insensitive and phase-sensitive amplifiers as special cases, for any QKD protocol. We also delineate the conditions under which a Gaussian relay renders a lossy channel entanglement breaking, which in turn makes the channel useless for QKD.

Figure 1

(a) Any $n$-mode Gaussian channel ${\mathcal{N}}_G$ sandwiched between two pure-loss channel segments ${\mathcal{A}}_{{\eta }_1}^{\otimes n}$ and ${\mathcal{A}}_{{\eta }_2}^{\otimes n}$, respectively, can be decomposed into a single lossy channel ${\mathcal{A}}_{\eta }^{\otimes n}$ sandwiched by a pair of Gaussian channels, ${\mathcal{N}}_G^1$ and ${\mathcal{N}}_G^2$. The net loss in the channel is the sum (in dB) of the losses of the two individual lossy segments, i.e., $\eta ={\eta }_1{\eta }_2$ and the Gaussian channel at the receiver end ${\mathcal{N}}_G^2$ is a Gaussian unitary map. (b) Using this transformation recursively, one can “push” a collection of general Gaussian center stations interspersed through a lossy channel (b.1) to a single Gaussian operation at the input, and a single Gaussian operation at the output (b.2) of the entire loss accumulated in the center.

Figure 2

A general single-mode non-EB Gaussian channel ${\mathcal{N}}_G$ is unitary equivalent to a Gaussian channel $\mathcal{N}$, which can be one of two forms, a phase-insensitive channel (PIC), ${\mathcal{A}}_g^N$, or a phase-sensitive additive noise channel (ANC) ${\mathcal{I}}^{\epsilon }$. For both cases of the center stations sandwiched between two lossy segments ${\mathcal{A}}_{{\eta }_1}$ and ${\mathcal{A}}_{{\eta }_2}$, the total channel action ${\Phi }_0\equiv {\mathcal{A}}_{{\eta }_2}\circ {\mathcal{N}}_G\circ {\mathcal{A}}_{{\eta }_1}$ is shown to be unitary equivalent to a PIC channel ${\mathcal{A}}_{{\eta }_s}^{N_s}$ as in (b)1.c and (c)2.c. Green-shaded boxes denote single-mode unitary (reversible) operations, whereas red-shaded boxes denote (in-general irreversible) actions of a single-mode quantum channel—a trace-preserving completely positive map.

Figure 3

(a) A sequence of PSAs $S_i$ with $i=1,2,...,k$ connected by the lossy segments of transmission ${\eta }_i$ with $i=1,2,...,k+1$. The total loss is given by $\eta ={\prod }_{i=1}^{k+1}{\eta }_i$. (b) The PSA-loss chain ${\Phi }_0$ can be transformed to the standard form of a PIC ${\Phi }_s$ by using squeezing unitary operations $W$ and $S_0$. (c) The standard form ${\Phi }_s$ is decomposed into a thermal noise channel ${\mathcal{A}}_1^{N/\eta }$ and a transmission-$\eta$ pure lossy segment. Then, the origin channel ${\Phi }_0$ can be simulated by adding unitary operators to cancel out the unitary operators in (b) at the input end and output end. This turns ${\Phi }_0$ into the form with original loss sandwiched by the operation ${\mathcal{N}}_G^1$ and the optput-end operation ${\mathcal{N}}_G^1$—an explicit example of our main result explained in Fig. 1.