Optimal Interpolation of Entanglement Purification Protocols

TL;DR

This work tackles optimizing entanglement purification by interpolating between protocols to trade off rate and Bell fidelity. It develops a probabilistic interpolation framework that reduces to optimizing combinations of two protocols and provides closed-form expressions for fidelity at a target rate or rate at a target fidelity. The authors apply the method to DEJMPS, derive asymptotic rate bounds using entanglement measures such as the relative entropy of entanglement, and develop finite-size analysis tools (Markov-chain and iterative approaches) to bound output counts under fixed pool sizes. The results indicate that interpolating purification protocols can outperform any single protocol in networks where a specific rate and fidelity are required, with practical methods for bounding performance in finite resources.

Abstract

Bipartite entanglement purification is the conversion of copies of weakly entangled pairs shared between two separated parties into a smaller number of strongly entangled shared pairs using only local operations and classical communication. Choosing between different entanglement purification protocols generally involves weighing up a trade-off between the ratio of strongly entangled pairs produced to weakly entangled pairs consumed, which we call the rate of the protocol, and the degree of the entanglement of the strongly entangled pairs, typically measured by the fidelity of those pairs to maximally entangled states. By randomly choosing a protocol according to a probability distribution over a list of protocols for each pair we want to produce, we can achieve rates and fidelities not achieved by any of the original protocols. Here, we show how to choose this distribution to maximise the rate at which we produce qubit pairs with a given fidelity to a Bell state or, equivalently, to maximise the fidelity to a Bell state of the qubit pairs produced at a given rate. We investigate both the asymptotic case, where the number of initial pairs goes to infinity, and the finite-size regime, where protocols are restricted to a finite number of weakly entangled pairs.

Figures (7)

• Figure 1: The rates at which we can convert the output states of the depolarising channel acting on true ebits into approximate ebits with fidelity $0.9$ to a true ebit using the DEJMPS and interpolated DEJMPS protocols. The rate achievable by interpolating between different numbers of DEJMPS iterations is shown in blue. The rate achievable by simply applying the DEJMPS protocol iteratively without interpolation until we reach the target fidelity is shown in orange. The upper bound on the achievable rate shown in green is based on the relative entropy of entanglement (REE).
• Figure 2: Lower and upper bounds on the maximum number of output pairs per input pair at which the DEJMPS and interpolated DEJMPS protocols achieve the target fidelity of $0.9$ within a global infidelity of $10^{-7}$ against the number of initial pairs, generated by applying a depolarising channel to one qubit of our chosen Bell state. The Bell fidelities of the initial states are linearly spaced, ranging from $0.55$ in the top-left figure to $0.9$ in the bottom-right (at the top of each panel we show the depolarising probabilities corresponding to those fidelities). The lower and upper bounds for the DEJMPS protocol are in red and green, respectively, and for the interpolated DEJMPS protocol in yellow and magenta, respectively. The asymptotic rates of the DEJMPS and interpolated DEJMPS protocols are shown in blue and cyan, respectively.
• Figure 3: The same as Figure \ref{['fig:FiniteSizeDepolarising']}, except the maximum allowed infidelity is a per-pair infidelity of $10^{-7}$.
• Figure 4: An example of what performing $3$ iterations of the DEJMPS protocol might look like. The black lines represent the initial pairs. The purple lines represent pairs produced via $1$ iteration of the DEJMPS protocol. The green lines represent pairs produced via $2$ iterations of the DEJMPS protocol. The red line represents a pair produced via $3$ iterations of the DEJMPS protocol. The black crosses denote failed applications of the protocol.
• Figure 5: The same as Figure \ref{['fig:FiniteSizeDepolarising']}, except the initial states were generated by applying a dephasing channel to the chosen Bell state.