Sufficient conditions and constraints for reversing general quantum errors

Gonzales, Alvin; Dilley, Daniel; Byrd, Mark

doi:10.1103/physreva.102.062415

Cited by 4 publications

(3 citation statements)

References 38 publications

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“…The known necessary and sufficient conditions for quantum error correction require that the error map is CP [21,22]. In the case of NCP errors, the necessary and sufficient conditions to correct a CP map can lead to codes that are not in the domain of the error map, which means that the process, as described, is not physical [23]. Thus, the results of the present paper may help to enable the correction of NCP error maps by transforming them via the ADM into CP maps.…”

Section: Introductionmentioning

confidence: 99%

Guaranteeing completely positive quantum evolution

Dilley¹,

Gonzales²,

Byrd³

2021

J. Phys. A: Math. Theor.

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In open quantum systems, it is known that if the system and environment are in a product state, the evolution of the system is given by a linear completely positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian maps, which also include non completely positive (NCP) maps. NCP maps can arise in evolutions such as non-Markovian evolution, where the CP divisibility of the map (writing the overall evolution as a composition of CP maps) usually fails. Positive but NCP maps are also useful as entanglement witnesses. In this paper, we focus on transforming an initial NCP map to a CP map through composition with the asymmetric depolarizing map. We use separate asymmetric depolarizing maps acting on the individual subsystems. Previous work have looked at structural physical approximation (SPA), which is a CP approximation of an NCP map using a mixture of the NCP map with a completely depolarizing map. We prove that the composition can always be made CP without completely depolarizing in any direction. It is possible to depolarize less in some directions. We give the general proof by using the Choi matrix and an isomorphism from a maximally entangled two qudit state to a set of qubits. We also give measures that describe the amount of disturbance the depolarization introduces to the original map. Given our measures, we show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map. Finally, we give some examples. For some measures and examples, completely depolarizing (while not necessary) in some directions can give a better approximation than keeping the depolarizing parameters bounded by the required depolarization if symmetric depolarization is used.

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Section: Introductionmentioning

confidence: 99%

Guaranteeing completely positive quantum evolution

Dilley¹,

Gonzales²,

Byrd³

2021

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The known necessary and sufficient conditions for quantum error correction require that the error map is CP [21,22]. In the case of NCP errors, these known CP necessary and sufficient conditions can lead to codes that are not in the domain of the error map, which means that the process as described is not physical [23]. Thus, the results of this paper may help with correcting for NCP error maps by transforming them via the asymmetric depolarizing map (ADM) into CP maps.…”

Section: Introductionmentioning

confidence: 99%

Guaranteeing Completely Positive Quantum Evolution

Dilley,

Gonzales,

Byrd

2021

Preprint

Self Cite

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In open quantum systems, it is known that if the system and environment are in a product state, the evolution of the system is given by a linear completely positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian maps, which also include non completely positive (NCP) maps. NCP maps can arise in evolutions such as non-Markovian evolution, where the CP divisibility of the map (writing the overall evolution as a composition of CP maps) usually fails. Positive but NCP maps are also useful as entanglement witnesses. In this paper, we focus on transforming NCP maps to CP maps through composition with the asymmetric depolarizing map. We use separate asymmetric depolarizing maps acting on the individual subsystems.Previous work have looked at structural physical approximation (SPA), which is a CP approximation of a NCP map using a mixture of the NCP map with a completely depolarizing map. We prove that the composition can always be made CP without completely depolarizing in any direction. It is often possible to depolarize less in some directions. We give the general proof by using the Choi matrix and an isomorphism from a maximally entangled two qudit state to qubits. We also give measures that describe the amount of disturbance the depolarization introduces to the original map. Given our measures, we show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map. Finally, we give some examples. For some measures and examples, completely depolarizing (while not necessary) in some directions can give a better approximation than keeping the depolarizing parameters bounded by the required depolarization if symmetric depolarization is used.

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“…But much of our theory relies on our assumed perfection of measurements and with the recent thermodynamic concerns, it is important to try to understand the practicality and implications. For example, initial system-environment correlations cause complications when defining a map for the system evolution [8][9][10][11][12] and can result in temporally correlated errors and/or non-Markovian evolutions that can imply restrictions on error correction methods [13][14][15][16][17][18]. So, does a non-perfectly projective measurement leave a state entangled, or otherwise correlated with other states?…”

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confidence: 99%

Entanglement Survives Most Measurements

Gonzales¹,

Dilley²,

Byrd³

2023

Preprint

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To prepare quantum states and extract information, it is often assumed that one can perform a perfectly projective measurement. Such measurements can achieve an uncorrelated system and environment. However, perfectly projective measurements can be difficult or impossible to perform. Using a sequence of weak measurements, we show that entanglement cannot be removed unless one of the measurement operators becomes perfectly projective through an extreme limiting process. Removing initial correlations and the scenario where measurement outcomes are not tracked are also discussed.

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Sufficient conditions and constraints for reversing general quantum errors

Cited by 4 publications

References 38 publications

Guaranteeing completely positive quantum evolution

Guaranteeing completely positive quantum evolution

Guaranteeing Completely Positive Quantum Evolution

Entanglement Survives Most Measurements

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