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arXiv:1208.5121 · Physical Review A

Quantum channel detection

Chiara Macchiavello, Matteo A. C. Rossi

Published Oct 30, 2013• 24 citations
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Abstract

We present a method to detect properties of quantum channels, assuming that some a priori information about the form of the channel is available. The method is based on a correspondence with entanglement detection methods for multipartite density matrices based on witness operators. We first illustrate the method in the case of entanglement-breaking channels and nonseparable random unitary channels and show how it can be implemented experimentally by means of local measurements. We then study the detection of nonseparable maps and show that for pairs of systems of dimension higher than two the detection operators are not the same as in the random unitary case, highlighting a richer separability structure of quantum channels with respect to quantum states. Finally we consider the set of PPT maps, developing a technique to reveal NPT maps.

DOI
10.1103/physreva.88.042335

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We present a method to detect properties of quantum channels, assuming that some a priori information about the form of the channel is available. The method is based on a correspondence with entanglement detection methods for multipartite density matrices based on witness operators. We first illustrate the method in the case of entanglement-breaking channels and nonseparable random unitary channels and show how it can be implemented experimentally by means of local measurements. We then study the detection of nonseparable maps and show that for pairs of systems of dimension higher than two the detection operators are not the same as in the random unitary case, highlighting a richer separability structure of quantum channels with respect to quantum states. Finally we consider the set of PPT maps, developing a technique to reveal NPT maps.
We demonstrate experimentally the possibility of efficiently detecting properties of quantum channels and quantum gates. The experimentally realized quantum channel detection method has been recently proposed theoretically, and allows us to characterize the properties of quantum channels with a much smaller experimental effort than quantum process tomography. Here, the optimal detection scheme is first achieved for nonentanglement breaking channels of the depolarizing form and is based on the generation and detection of polarized entangled photons. We then demonstrate channel detection for nonseparable maps by considering the CNOT gate and employing two-photon hyperentangled states.
This thesis deals with methods of the detection of entanglement.After recalling some facts and definitions concerning entanglement and separability, we investigate two methods of the detection of entanglement.In the first part of this thesis we consider so-called entanglement witnesses, mainly in view of the detection of multipartite entanglement.Entanglement witnesses are observables for which a negative expectation value indicates entanglement.We first present a simple method to construct these witnesses.Since witnesses are nonlocal observables, they are not easy to measure in a real experiment.However, as we will show, one can circumvent this problem by decomposing the witness into several local observables which can be measured separately.We calculate the local decompositions for several interesting witnesses for two, three and four qubits.Local decompositions can be optimized in the number of measurement settings which are needed for an experimental implementation.We present a method to prove that a given local decomposition is optimal and discuss with this the optimality of our decompositions.Then we present another method of designing witnesses which are by construction measurable with local measurements.Finally, we shortly report on experiments where some of the witnesses derived in this part have been used to detect three-and four-partite entanglement of polarized photons.The second part of this thesis deals with separability criteria which are written in terms of uncertainty relations.There are two different formulations of uncertainty relations since one can measure the uncertainty of an observable by its variance as well as by entropic quantities.We show that both formulations are useful tools for the derivation of separability criteria for finite-dimensional systems and investigate the resulting criteria.Our results in this part exhibit also some more fundamental properties of entanglement: We show how known separability criteria for infinitedimensional systems can be translated to finite-dimensional systems.Furthermore, we prove that any entropic uncertainty relation on one part of the system gives rise to a separability criterion on the composite system.
We develop systematic frameworks for characterizing the entanglement properties of two-qubit channels beyond unitary settings. We introduce averaged local-unitary invariants, referred to as moments, obtained from Haar integrals over input states or unitaries. These moments provide computable descriptions of how a quantum channel can create, preserve, or destroy bipartite entanglement. We first show that second-order moments yield criteria for non-entangling and entanglement-breaking channels, which allow us to detect entanglement-creating and entanglement-preserving channels. We then demonstrate that higher-order moments can capture additional information and distinguish channels beyond second-order moments alone. Finally, we show that combinations of moments associated with different channel families improve the discrimination of locally inequivalent two-qubit unitaries.
For any $N\ensuremath{\bigotimes}N$-type bipartite system with even N>2N>2N>2, two-parameter continuous families of unital block-dephasing quantum channels and of weakly optimal entanglement witnesses are constructed from special forms of a parametrized positive map. The parameter domains of the families are complementary subsets of a convex set and this fact reveals unusual combination rules of quantum channels and entanglement witnesses. These rules make it possible to develop many physically implementable direct entanglement-detection methods that provide a unified framework to accomplish both information-processing and entanglement-detection tasks by means of smooth quantum channels, without any reference to entanglement witnesses.