scholarly journals Corrigendum to “Completely-positive quantum operations generating thermostatistical states: A comparative study” [Physica A 409 (2014) 130–137]

2015 ◽  
Vol 422 ◽  
pp. 210
Author(s):  
Yasuyuki Matsuo ◽  
Sumiyoshi Abe
Keyword(s):  
Comparative Study ◽  
Completely Positive ◽  
Quantum Operations

scholarly journals Physical Implementability of Linear Maps and Its Application in Error Mitigation

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 600
Author(s):  
Jiaqing Jiang ◽  
Kun Wang ◽  
Xin Wang
Keyword(s):  
General Linear ◽  
Local Error ◽  
Decomposition Technique ◽  
Completely Positive ◽  
Linear Map ◽  
Quantum Operations ◽  
Error Mitigation ◽  
Linear Maps ◽  
Analytic Expressions ◽  
Sampling Cost

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

GEOMETRIC PHASES FOR MIXED STATES DURING UNITARY AND NON-UNITARY EVOLUTIONS

2003 ◽  
Vol 01 (01) ◽  
pp. 135-152 ◽  
Author(s):  
ARUN K. PATI
Keyword(s):  
Geometric Phase ◽  
Parallel Transport ◽  
Quantum Systems ◽  
Mixed States ◽  
Completely Positive Maps ◽  
Initial State ◽  
Unitary Evolution ◽  
Completely Positive ◽  
Quantum Operations ◽  
Positive Maps

Mixed states typically arise when quantum systems interact with the outside world. Evolution of open quantum systems in general are described by quantum operations which are represented by completely positive maps. We elucidate the notion of geometric phase for a quantum system described by a mixed state undergoing unitary evolution and non-unitary evolutions. We discuss parallel transport condition for mixed states both in the case of unitary maps and completely positive maps. We find that the relative phase shift of a system not only depends on the state of the system, but also depends on the initial state of the ancilla with which it might have interacted in the past. The geometric phase change during a sequence of quantum operations is shown to be non-additive in nature. This property can attribute a "memory" to a quantum channel. We explore these ideas and illustrate them with simple examples.

scholarly journals On Duality between Quantum Maps and Quantum States

2004 ◽  
Vol 11 (01) ◽  
pp. 3-42 ◽  
Author(s):  
Karol Życzkowski ◽  
Ingemar Bengtsson
Keyword(s):  
Probability Distributions ◽  
Dynamical Matrix ◽  
Density Matrices ◽  
Unitary Matrices ◽  
Completely Positive ◽  
Discrete Probability ◽  
Quantum Operations ◽  
Extended Space ◽  
Analogous Relation ◽  
Discrete Probability Distributions

We investigate the space of quantum operations, as well as the larger space of maps which are positive, but not completely positive. A constructive criterion for decomposability is presented. A certain class of unistochastic operations, determined by unitary matrices of extended dimensionality, is defined and analyzed. Using the concept of the dynamical matrix and the Jamiołkowski isomorphism we explore the relation between the set of quantum operations (dynamics) and the set of density matrices acting on an extended Hilbert space (kinematics). An analogous relation is established between the classical maps and an extended space of the discrete probability distributions.

scholarly journals Properties of local quantum operations with shared entanglement

2009 ◽  
Vol 9 (9&10) ◽  
pp. 739-764
Author(s):  
G. Gutoski
Keyword(s):  
Membership Problem ◽  
Noisy Channel ◽  
Complete Positivity ◽  
Linear Functionals ◽  
Completely Positive ◽  
Quantum Operations ◽  
Local Quantum ◽  
Natural Notion ◽  
No Signaling ◽  
Nonlocal Box

Multi-party local quantum operations with shared quantum entanglement or shared classical randomness are studied. The following facts are established: * There is a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. * The existence of the ball of local operations with shared randomness is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. * Local operations with shared entanglement are characterized in terms of linear functionals that are "completely'' positive on a certain cone K of separable Hermitian operators, under a natural notion of complete positivity appropriate to that cone. Local operations with shared randomness (but not entanglement) are also characterized in terms of linear functionals that are merely positive on that same cone K. * Existing characterizations of no-signaling operations are generalized to the multi-party setting and recast in terms of the Choi-Jamio\l kowski representation for quantum super-operators. It is noted that the standard nonlocal box is an example of a no-signaling operation that is separable, yet cannot be implemented by local operations with shared entanglement.

QUANTUM OPERATIONS INDUCED BY CLASSICAL RANDOMNESS: THE CASE OF QUBIT

2007 ◽  
Vol 05 (03) ◽  
pp. 397-408
Author(s):  
IOANNIS TSOHANTJIS
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Quantum Computation ◽  
Hopf Algebras ◽  
Quantum Dissipation ◽  
Density Matrices ◽  
Completely Positive ◽  
Quantum Operations ◽  
Unitary Transformations ◽  
Qubit States

The question of quantum computation gates and quantum dissipation maps generated by means of classical random walks is addressed here in operations involving qubit states. Classical variables determining the manifold of qubit state vectors and density matrices are left to perform a classical random walk formulated algebraically by means of Hopf algebras. It is shown that this induces quantum operations on the qubits and density matrices which are further identified with e.g. known unitary transformations and completely positive trace preserving maps, depending upon the kind of random walk chosen.

scholarly journals New encoding schemes for quantum authentication

2005 ◽  
Vol 5 (1) ◽  
pp. 1-12
Author(s):  
P. Garcia-Fernandez ◽  
E. Fernandez-Martinez ◽  
E. Perez ◽  
D.J. Santos
Keyword(s):  
Authentication Protocol ◽  
Completely Positive Maps ◽  
Authentication Protocols ◽  
Completely Positive ◽  
Quantum Operations ◽  
Positive Maps ◽  
General Quantum ◽  
Quantum Authentication

We study the potential of general quantum operations, Trace-Preserving Completely-Positive Maps (TPCPs), as encoding and decoding mechanisms in quantum authentication protocols. The study shows that these general operations do not offer significant advantage over unitary encodings. We also propose a practical authentication protocol based on the use of two successive unitary encodings.

scholarly journals Invertible quantum operations and perfect encryption of quantum states

2007 ◽  
Vol 7 (1&2) ◽  
pp. 103-110 ◽  
Author(s):  
A. Nayak ◽  
P. Sen
Keyword(s):  
Unitary Transformation ◽  
Quantum States ◽  
Quantum Channels ◽  
Quantum Operation ◽  
Straightforward Manner ◽  
Completely Positive ◽  
Quantum Bits ◽  
Quantum Operations ◽  
Self Testing

In this note, we characterize the form of an invertible quantum operation, i.e., a completely positive trace preserving linear transformation (a CPTP map) whose inverse is also a CPTP map. The precise form of such maps becomes important in contexts such as self-testing and encryption. We show that these maps correspond to applying a unitary transformation to the state along with an ancilla initialized to a fixed state, which may be mixed. The characterization of invertible quantum operations implies that one-way schemes for encrypting quantum states using a classical key may be slightly more general than the "private quantum channels'' studied by Ambainis, Mosca, Tapp and de Wolf {AmbainisMTW00}. Nonetheless, we show that their results, most notably a lower bound of 2n bits of key to encrypt n quantum bits, extend in a straightforward manner to the general case.

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