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arXiv:quant-ph/0004096 · Physical Review A

Enhanced quantum estimation via purification

Holger Mack, Dietmar G. Fischer, Matthias Freyberger

Published Sep 5, 2000• 9 citations
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Abstract

We analyze the estimation of a finite ensemble of quantum bits that have been sent through a depolarizing channel. Instead of using the depolarized qubits directly, we first apply a purification step and show that this improves the fidelity of subsequent quantum estimation. Even though we lose some qubits of our finite ensemble the information is concentrated in the remaining purified ones.

DOI
10.1103/physreva.62.042301

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We analyze the estimation of a finite ensemble of quantum bits that have been sent through a depolarizing channel. Instead of using the depolarized qubits directly, we first apply a purification step and show that this improves the fidelity of subsequent quantum estimation. Even though we lose some qubits of our finite ensemble the information is concentrated in the remaining purified ones.
Purification of mixed states in Quantum Mechanics, by which we mean the transformation into pure states, has been viewed as an {\it Operation} in the sense of Kraus et al and explicit {\it Kraus Operators} \cite{kra1,kra2,kra3} have been constructed for two seperate purification protocols. The first one, initially due to Schrodinger \cite{sch} and subsequently elaborated by Sudarshan et al \cite{sudar}, is based on the {\it preservation of probabilities}. We have constructed a second protocol here based on {\it optimization of fidelities}. Both purification protocols have been implemented on a single qubit in an attempt to improve the fidelity of the purified post measurement state of the qubit with the initial pure state. We have considered both {\it complete} and {\it partial} measurements and have established bounds and inequalities for various fidelities. We show that our purification protocol leads to better state reconstruction, most explicitly so, when partial measurements are made.
We present an approach to entanglement purification on complex quantum network architectures, that is, how a quantum network built from repeaters with limited processing fidelity can purify and distribute entanglement between users. In particular, we explore how noisy gates, depolarizing channels, and finite memory storage time influence the performance of entanglement purification in a quantum network. Finally, we apply the purification techniques we developed in the context of entanglement routing for different quantum network topologies in an effort to inform future design choices for quantum network configurations.
We consider the problem of purifying noisy qubit unitary channels. Given the ability to apply an unknown qubit unitary channel followed by depolarizing noise, we aim to construct a superchannel that purifies the noisy unitary back to the original unknown unitary. We first provide numerical evidence that sequential strategies can strictly outperform parallel strategies when the number of channel uses is finite, highlighting the fundamental distinction from state purification. We then provide a concrete U(2)\mathrm{U}(2)U(2)-covariant parallel protocol based on a novel entanglement-assisted quantum error-correcting code that suppresses the first-order noise strength as O(1/n)O(1/n)O(1/n) with nnn channel uses and show this scaling is asymptotically optimal in the low-noise regime, even when sequential strategies are allowed.
Quantum state purification is a process in which decoherence is partially reversed by using multiple copies of the input states that have been subject to the same decoherence effect. This thesis focuses on purifying the decoherence caused by the depolarizing channel. In the first half of the thesis, the purification problem is formally introduced and one efficient purification procedure featuring the swap-test is presented and analyzed. The rest of the thesis formulates the optimal purification problem as an optimization problem and applies it to qubit and qutrit purification.
Abstract We present a general-purpose quantum error correction primitive based on state purification via the SWAP test, which we refer to as purification quantum error correction (PQEC). This method operates on N noisy copies, requires minimally O(Mlog 2 N) data qubits to process the M-qubit inputs. In a similar way to standard QEC, the purification steps may be interleaved within a quantum algorithm to suppress the logical error rate. No postselection is performed and no knowledge of the state is required. We analyze its performance under a variety of error channels and find that PQEC is highly effective at boosting fidelity and reducing logical error rates, particularly for the depolarizing channel. Error thresholds for the local depolarizing channel are found to be 75% for any register size. For local dephasing, the error threshold is reduced to 50% but may be boosted using twirling.