Energy-constrained two-way assisted private and quantum capacities of quantum channels

With the rapid growth of quantum technologies, knowing the fundamental characteristics of quantum systems and protocols is essential for their effective implementation. A particular communication setting that has received increased focus is related to quantum key distribution and distributed quantum computation. In this setting, a quantum channel connects a sender to a receiver, and their goal is to distill either a secret key or entanglement, along with the help of arbitrary local operations and classical communication (LOCC). In this work, we establish a general theory of energy-constrained, LOCC-assisted private and quantum capacities of quantum channels, which are the maximum rates at which an LOCC-assisted quantum channel can reliably establish a secret key or entanglement, respectively, subject to an energy constraint on the channel input states. We prove that the energy-constrained squashed entanglement of a channel is an upper bound on these capacities. We also explicitly prove that a thermal state maximizes a relaxation of the squashed entanglement of all phase-insensitive, single-mode input bosonic Gaussian channels, generalizing results from prior work. After doing so, we prove that a variation of the method introduced by Goodenough et al. [New J. Phys. 18, 063005 (2016)] leads to improved upper bounds on the energy-constrained secret-key-agreement capacity of a bosonic thermal channel. We then consider a multipartite setting and prove that two known multipartite generalizations of the squashed entanglement are in fact equal. We finally show that the energy-constrained, multipartite squashed entanglement plays a role in bounding the energy-constrained LOCC-assisted private and quantum capacity regions of quantum broadcast channels.

Figure 1

A secret-key-agreement protocol begins with Alice and Bob preparing a separable state of systems $A_1^{\prime }{A}_1{B}_1^{\prime }$ using LOCC. Alice then feeds the $A_1$ system into the first channel in order to generate the $B_1$ system. After repeating this procedure $n$ times, with rounds of LOCC interleaved between every channel use, Alice and Bob perform a final round of LOCC, which yields the key systems $K_A$ and $K_B$.

Figure 2

Alice and Bob alternate rounds of LOCC and channel uses, just as in Fig. 1. Each channel use is now purified, which yields outputs to Eve, the environment. Classical data are also collected by Eve from the LOCC. Eve's squashing channels are also purified and depicted above for the $n\mathrm{th}$ channel use.

Figure 3

A depiction of the isometric channel ${\mathcal{W}}_{A\to B{E}_1^{\prime }{E}_2^{\prime }{F}_1^{\prime }{F}_2^{\prime }}$ from Theorem 5. Note that this is the precise construction used in Ref. [7]. As stated in Theorem 5, the isometric channel ${\mathcal{W}}_{A\to B{E}_1^{\prime }{E}_2^{\prime }{F}_1^{\prime }{F}_2^{\prime }}$ is equal to $({\mathcal{V}}_{E_2\to E_2^{\prime }{F}_2^{\prime }}^{\mathcal{A}}\circ {\mathcal{U}}_{B_1\to B{E}_2}^{{\mathcal{A}}_{\mathcal{G},0}})\circ ({\mathcal{V}}_{E_1\to E_1^{\prime }{F}_1^{\prime }}^{\mathcal{L}}\circ {\mathcal{U}}_{A\to B_1{E}_1}^{{\mathcal{L}}_{T,0}})$. The modes labeled “env1” and “env2” are respective environmental modes for the isometric channels ${\mathcal{U}}^{{\mathcal{L}}_{T,0}}$ (top left) and ${\mathcal{U}}^{{\mathcal{A}}_{\mathcal{G},0}}$ (top right) and are prepared in the pure vacuum state. The other isometric channels ${\mathcal{V}}_{E_1\to E_1^{\prime }{F}_1^{\prime }}^{\mathcal{L}}$ (bottom left) and ${\mathcal{V}}_{E_2\to E_2^{\prime }{F}_2^{\prime }}^{\mathcal{A}}$ (bottom right) are chosen here to be 50:50 beam splitters, following [7]. The modes $F_1$ and $F_2$ are also prepared in the pure vacuum state. Given this setup, Theorem 5 states that, among all possible input states with mean photon number $\le N_S$, the thermal state $\theta \left(N_S\right)$ maximizes the entropy function $H\left(B\right|E_1^{\prime }{E}_2^{\prime })+H\left(B\right|F_1^{\prime }{F}_2^{\prime })$.

Figure 4

A depiction of the isometric channel ${\mathcal{W}}_{A\to B{E}_1^{\prime }{E}_2^{\prime }{F}_1^{\prime }{F}_2^{\prime }}$ needed for the bound in Proposition 2. This construction swaps the top-left beam splitter and top-right two-mode squeezer from Fig. 3 and corresponds to the channel decomposition in Eq. (173). This construction leads to an improvement of the bound from [7].

Figure 5

Comparison of the “DSW18 bound” from (177) with prior bounds from [7, 8, 11], with $\eta \in [0.5,1], N_S=0.1$, and $N_B=1$. The plot shows that the bound in Eq. (177) converges to zero as the channel becomes entanglement breaking.

Figure 6

Comparison of the “DSW18 bound” from (177) with prior bounds from [7, 8, 11], with $N_S\in [0,1], \eta =0.1$, and $N_B\in \{3\times {10}^{-7},1\times {10}^{-3},0.1\}$ [respectively panels (a)–(c)]. The DSW18 bound from (177) is indistinguishable from the bound from [7] for small $N_B$, but then the bounds are very different for higher $N_B$. In (a), GEW16 is not visible because it overlaps with DSW18.