Resource theory of unextendibility and nonasymptotic quantum capacity

In this paper, we introduce the resource theory of unextendibility as a relaxation of the resource theory of entanglement. The free states in this resource theory are the $k$-extendible states, associated with the inability to extend quantum entanglement in a given quantum state to multiple parties. The free channels are $k$-extendible channels, which preserve the class of $k$-extendible states. We define several quantifiers of unextendibility by means of generalized divergences and establish their properties. By utilizing this resource theory, we obtain nonasymptotic upper bounds on the rate at which quantum communication or entanglement preservation is possible over a finite number of uses of an arbitrary quantum channel assisted by $k$-extendible channels at no cost. These bounds are significantly tighter than previously known bounds for both the depolarizing and erasure channels. Finally, we revisit the pretty strong converse for the quantum capacity of antidegradable channels and establish an upper bound on the nonasymptotic quantum capacity of these channels.

Figure 1

A visual depiction of the conditions for the channel ${\mathcal{M}}_{A{B}_1\cdots B_k\to A^{\prime }{B}_1^{\prime }\cdots B_k^{\prime }}$ to be a $k$ extension of ${\mathcal{N}}_{AB\to A^{\prime }{B}^{\prime }}$. (a) The extension channel ${\mathcal{M}}_{A{B}_1\cdots B_k\to A^{\prime }{B}_1^{\prime }\cdots B_k^{\prime }}$ should be permutation covariant with respect to Bob's systems. (b) The extension channel ${\mathcal{M}}_{A{B}_1\cdots B_k\to A^{\prime }{B}_1^{\prime }\cdots B_k^{\prime }}$ should reduce to the original channel ${\mathcal{N}}_{AB\to A^{\prime }{B}^{\prime }}$ when tracing out the output systems $B_2^{\prime }\cdots B_k^{\prime }$ of ${\mathcal{M}}_{A{B}_1\cdots B_k\to A^{\prime }{B}_1^{\prime }\cdots B_k^{\prime }}$.

Figure 2

Depiction of an entanglement transmission protocol assisted by a $k$-extendible postprocessing channel. The quantum channel $\mathcal{N}$ is used $n$ times, in conjunction with an encoding channel ${\mathcal{E}}_{A^{\prime }\to A^n}$ and a $k$-extendible postprocessing decoding channel ${\mathcal{K}}_{R{B}^n\to R\widehat{A}}$, in order to establish entanglement shared between Alice and Bob.

Figure 3

Depiction of a quantum communication protocol using a quantum channel $\mathcal{N}$ assisted by $k$-extendible channels before and after every channel use. The quantum channel $\mathcal{N}$ is used $n$ times, in conjunction with the assisting $k$-extendible channels, in order to establish entanglement shared between Alice and Bob.

Figure 4

Upper bounds on the number of qubits that can be reliably transmitted over a depolarizing channel with $p=0.1$ and $\varepsilon =0.05$. The red dashed line is the bound from Theorem t27. The green dashed-dotted and blue dotted lines are upper bounds from [4] and [3], respectively.

Figure 5

Upper bounds on the number of qubits that can be reliably transmitted over a depolarizing channel with $p=0.25$ and $\varepsilon =5\times {10}^{-5}$. The red dashed line is the bound from Theorem t27. The green dashed-dotted and blue dotted lines are upper bounds from [4] and [3], respectively.

Figure 6

Upper bounds on the number of qubits that can be reliably transmitted over an erasure channel with $p=0.35$ and $\varepsilon =0.05$. The red dashed line is the bound from Theorem t27. The green dashed-dotted line is an upper bound from [4].

Figure 7

Upper bounds on the number of qubits that can be reliably transmitted over an erasure channel with $p=0.49$ and $\varepsilon =0.05$. The red dashed line is the bound from Theorem t27. The green dashed-dotted line is an upper bound from [4].