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arXiv:1707.06926 · Quantum

Gauge invariant information concerning quantum channels

Łukasz Rudnicki, Zbigniew Puchała, Karol Zyczkowski

Published Apr 11, 2018• 18 citations
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Abstract

Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.

DOI
10.22331/q-2018-04-11-60

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Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.
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A quantum holonomy reflects the curvature of some underlying structure of quantum-mechanical systems, such as that associated with quantum states. Here, we extend the notion of holonomy to families of quantum channels, i.e., trace-preserving completely positive maps. By the use of the Jamio\l{}kowski isomorphism, we show that the proposed channel holonomy is related to the Uhlmann holonomy. The general theory is illustrated for specific examples. We put forward a physical realization of the channel holonomy in terms of interferometry. This enables us to identify a gauge-invariant physical object that directly relates to the channel holonomy. Parallel transport condition and concomitant gauge structure are delineated in the case of smoothly parametrized families of channels. Finally, we point out that interferometer tests that have been carried out in the past to confirm the $4\ensuremath{\pi}$ rotation symmetry of the neutron spin can be viewed as early experimental realizations of the channel holonomy.
Noisy logic operations on a quantum computational register -- e.g., one or more qubits -- can be described by transfer matrices (a.k.a. CPTP maps or superoperators) that act linearly on the density matrix representing the register's quantum state. Collectively, these operations form a gate set. Gate sets have a gauge freedom; many gate sets that appear different actually predict the same experimental outcomes. A property of a gate set can be observable (and thus physically relevant) only if it is gauge-invariant. Unfortunately, no good gauge-invariant parameterizations of gate sets are known. We introduce the next best thing, a perturbative gauge-invariant parameterization of small Markovian errors in gate sets. We construct vector spaces of properties that are first-order gauge-invariant (FOGI). We show how to construct and understand FOGI properties, how to use them as coordinates to parameterize gate sets without gauge freedom, and how to extract approximately gauge-invariant error metrics.
Quantum superchannels are maps whose input and output are quantum channels. Rather than taking the domain to be the space of all linear maps we motivate and define superchannels on the operator system spanned by quantum channels. Extension theorems for completely positive maps allow us to apply the characterisation theorem for superchannels to this smaller set of maps. These extensions are non unique, showing two different superchannels act the same on all input quantum channels, and so this new definition on the smaller domain captures more precisely the action of superchannels as transformations between quantum channels. The non uniqueness can affect the auxilliary dimension needed for the characterisation as well as the tensor product of the superchannels.
Assume that are complex matrices. We give a computable criterion for existence of a common eigenvector of which generalize the result of D. Shemesh established for two matrices. We use this criterion to prove some necessary and sufficient condition for to have a common invariant subspace of dimension , , if every has pairwise different eigenvalues. Since the set of all matrices having multiple eigevalues has Lebesgue measure , the condition is sufficient in practical applications. Being motivated by quantum information theory, we give a flavour of such applications for irreducible completely positive superoperators.