Amortization does not enhance the max-Rains information of a quantum channel

Abstract

Given an entanglement measure E, the entanglement of a quantum channel is defined as the largest amount of entanglement E that can be generated from the channel, if the sender and receiver are not allowed to share a quantum state before using the channel. The amortized entanglement of a quantum channel is defined as the largest net amount of entanglement Ethat can be generated from the channel, if the sender and receiver are allowed to share an arbitrary state before using the channel. Our main technical result is that amortization does not enhance the entanglement of an arbitrary quantum channel, when entanglement is quantified by the max-Rains relative entropy. We prove this statement by employing semi-definite programming (SDP)duality and SDPformulations for the max-Rains relative entropy and a channel's max-Rains information, found recently in Wang et al (arXiv:1709. 00200). The main application of our result is a single-letter, strong converse, and efficiently computable upper bound on the capacity of a quantum channel for transmitting qubits when assisted by positive-partial-transpose preserving (PPT-P) channels between every use of the channel. As the class of local operations and classical communication (LOCC) is contained in PPT-P, our result establishes a benchmark for the LOCC-assisted quantum capacity of an arbitrary quantum channel, which is relevant in the context of distributed quantum computation and quantum key distribution.In this paper, we are interested in placing upper bounds on the LOCC-assisted quantum capacity, and one way of simplifying the mathematics behind this task is to relax the class of free operations that the sender and receiver are allowed to perform between each channel use. With this in mind, we follow the approach of [7, 8]and relax the set LOCCto a larger class of operations known as PPT-preserving (PPT-P), standing for channels that are positive-partial transpose preserving. The resulting capacity is then known as the PPT-Passisted quantum capacity, and it is equal to the maximum rate at which qubits can be communicated reliably from a sender to a receiver, when they are allowed to use a PPT-P channel in between every use of the actual channel  . Figure 1 provides a visualization of such a PPT-P-assisted quantum communication protocol. Due to the containment LOCC⊂PPT-P [7, 8], the inequality    « « SB   ( ) Y J . 8 7 SB SB

Fundamental Limits on the Capacities of Bipartite Quantum Interactions

“…where R max (A; B) ρ := inf{λ : ρ AB ≤ 2 λ σ AB , σ AB ∈ PPT (A : B)} denotes the max-Rains information of the state ρ AB [34], with PPT (A : B) denoting the set of all positive semidefinite operators σ AB such that the trace norm T B (σ AB ) 1 ≤ 1 [35]. This observation was made in the case of point-to-point channels [36] and constitutes a contribution of our companion paper [30]. By successive application of the amortization relation in (3) to every use of N in an (n, M, ε)-protocol, it follows that (A : B), it follows by a data-processing argument that…”

Strong Converse Bound on the Two-Way Assisted Quantum Capacity

“…The main tool that we used is the formulation of the max-Rains relative entropy and max-Rains information as semi-definite programs [13]- [15]. We discuss in [22] how our strong converse result stands with respect to prior work on strong converses of quantum and private capacities. There, we also provide an elementary proof for the fact that amortization does not enhance a channel's max-relative entropy of entanglement -leading to a strong converse upper bound on the two-way assisted private capacity P ↔ (N ) of any quantum channel N (this latter result was first proven in [8] based on complex interpolation theory).…”

Defining quantum divergences via convex optimization

“…• For the Rains bound, C = PPT := {σ XY ∈ P(XY ) : σ Y XY 1 ≤ 1} and α = 1 [31], and the version with α = ∞ has also been studied in [4,38]. The notation Y denotes the partial transpose on system Y with respect to some fixed basis.…”