Non-commutative graphs and quantum error correction for a two-mode quantum oscillator

Amosov, G. G.; Mokeev, A. S.; Pechen, Alexander

doi:10.1007/s11128-019-2554-5

Cited by 14 publications

(11 citation statements)

References 29 publications

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“…The operator systems can also be treated as "non-commutative" versions of graphs [41], and this point of view has applications to the theory of quantum error correction [42][43][44]. The presented theorems may be of interest to the study of dynamics on such graphs.…”

Section: Discussionmentioning

confidence: 99%

The Extension of Unital Completely Positive Semigroups on Operator Systems to Semigroups on $C^*$-algebras

Yashin

2022

Preprint

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The study of open quantum systems relies on the notion of unital completely positive semigroups on C * -algebras representing physical systems. The natural generalisation would be to consider the unital completely positive semigroups on operator systems. We show that any continuous unital completely positive semigroup on matricial system can be extended to a semigroup on a finite-dimensional C * -algebra, which is an injective envelope of the matricial system. In case the semigroup is invertible, this extension is unique.

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Section: Discussionmentioning

confidence: 99%

The Extension of Unital Completely Positive Semigroups on Operator Systems to Semigroups on $C^*$-algebras

Yashin

2022

Preprint

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“…Denote Π(G) the set of all probability distributions on G. Following to [15] we can define two quantum channels corresponding to POVM (8). One is the measurement channel Φ :…”

Section: Application To Quantum Channelsmentioning

confidence: 99%

“…At first, the task of their construction is of independent interest [2,3]. Then, they define measurement channels for which the task of calculating capacity is set [4,5] and noncommutative operator graphs that play an important role in the theory of quantum error correcting codes [6][7][8][9]. It should also be noted the use of operator-valued measures in quantum control problems [10,11].…”

Section: Introductionmentioning

confidence: 99%

On quantum channels generated by covariant positive operator-valued measures on a locally compact group

Amosov

2022

Preprint

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We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product of locally compact Abelian group G with its dual Ĝ. The method is based upon the Pontryagin duality allowing to establish an isometrical isomorphism between the space of Hilbert-Schmidt operators in L 2 (G) and the Hilbert space L 2 ( Ĝ × G). Any such a measure determines a pair of hybrid (containing classical and quantum parts) quantum channels consisting of the measurement channel and the channel transmitting an initial quantum state to the ensemble of quantum states on the group. It is shown that the second channel can be called a complementary channel to the measurement channel.

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“…Here we study the explicit example described in [8] in view of Section 4. For a Hamiltonian of a two-mode quantum oscillator…”

Section: Example: Two Mode Oscillatormentioning

confidence: 99%

“…In particular, if a non-commutative operator graph is linearly generated by a POVM covariant with respect to the action of a unitary-represented group it is possible to give the sufficient conditions of error-correcting code existence. Several examples of such graphs were constructed [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

On errors generated by unitary dynamics of bipartite quantum systems

Amosov,

Mokeev

2020

Preprint

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Given a quantum channel it is possible to define the noncommutative operator graph whose properties determine a possibility of errorfree transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group Z n and the additive group R are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.

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Non-commutative graphs and quantum error correction for a two-mode quantum oscillator

Cited by 14 publications

References 29 publications

The Extension of Unital Completely Positive Semigroups on Operator Systems to Semigroups on $C^*$-algebras

The Extension of Unital Completely Positive Semigroups on Operator Systems to Semigroups on $C^*$-algebras

On quantum channels generated by covariant positive operator-valued measures on a locally compact group

On errors generated by unitary dynamics of bipartite quantum systems

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