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.

EFFECTIVE METHODS IN INVESTIGATION OF IRREDUCIBLE QUANTUM OPERATIONS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260014 ◽  
Author(s):  
ANDRZEJ JAMIOŁKOWSKI
Keyword(s):  
Explicit Form ◽  
Quantum Channels ◽  
Quantum Operations ◽  
Kraus Representation

In this paper we discuss some constructive procedures which can be used in characterization of irreducible quantum operations and quantum channels. Our method makes use of an explicit form of a fixed operation in its Kraus representation. A generalization of the Shemesh criterion is discussed.

scholarly journals Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 522
Author(s):  
Bartosz Regula ◽  
Ryuji Takagi ◽  
Mile Gu
Keyword(s):  
Quantum Dynamics ◽  
Distance Measure ◽  
Quantum States ◽  
Quantum Channels ◽  
Linear Combinations ◽  
Operational Approach ◽  
Completely Positive ◽  
Error Mitigation ◽  
Linear Maps ◽  
The Cost

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

scholarly journals Entangling capacities and the geometry of quantum operations

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Jhih-Yuan Kao ◽  
Chung-Hsien Chou
Keyword(s):  
Quantum Information ◽  
Physical Significance ◽  
Quantum States ◽  
Important Class ◽  
Quantum Operation ◽  
Positive Partial Transpose ◽  
Quantum Operations ◽  
Partial Transpose ◽  
Ppt States

Abstract Quantum operations are the fundamental transformations on quantum states. In this work, we study the relation between entangling capacities of operations, geometry of operations, and positive partial transpose (PPT) states, which are an important class of states in quantum information. We show a method to calculate bounds for entangling capacity, the amount of entanglement that can be produced by a quantum operation, in terms of negativity, a measure of entanglement. The bounds of entangling capacity are found to be associated with how non-PPT (PPT preserving) an operation is. A length that quantifies both entangling capacity/entanglement and PPT-ness of an operation or state can be defined, establishing a geometry characterized by PPT-ness. The distance derived from the length bounds the relative entangling capability, endowing the geometry with more physical significance. We also demonstrate the equivalence of PPT-ness and separability for unitary operations.

scholarly journals Some Generalised Fixed Point Theorems Applied to Quantum Operations

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 759
Author(s):  
Umar Batsari Yusuf ◽  
Poom Kumam ◽  
Sikarin Yoo-Kong
Keyword(s):  
Fixed Point ◽  
Quantum State ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Quantum States ◽  
Contractive Condition ◽  
Quantum Operation ◽  
Quantum Operations ◽  
Finite Dimensional

In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea.

scholarly journals On Partially Trace Distance Preserving Maps and Reversible Quantum Channels

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Long Jian ◽  
Kan He ◽  
Qing Yuan ◽  
Fei Wang
Keyword(s):  
Quantum States ◽  
Quantum Channels ◽  
Trace Distance ◽  
Linear Maps ◽  
Unitary Transformations ◽  
Pure States ◽  
Positive Linear Maps

We give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices. Applying such results, we give a characterization of quantum channels leaving Helstrom's measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations.

INSIGHTS AND IMPLICATIONS FROM A BAYESIAN APPROACH TO QUANTUM INFORMATION

2005 ◽  
Vol 03 (01) ◽  
pp. 233-237 ◽  
Author(s):  
CHRISTOPHER A. FUCHS ◽  
MARCOS PÉREZ-SUÁREZ ◽  
DAVID J. SANTOS
Keyword(s):  
Quantum Theory ◽  
Quantum Information ◽  
Bayesian Approach ◽  
Stochastic Matrix ◽  
Quantum States ◽  
Structural Elements ◽  
Quantum Operation ◽  
Quantum Operations

Providing quantum theory, and thus quantum information, with some meaning, can only be the result of adopting a well-defined approach to the notion of probability. From a subjectivistic Bayesian approach to the latter, it follows that not only quantum states, but also quantum operations, are structural elements of a subjective nature. Furthermore, we provide a representation of any quantum operation as a stochastic matrix.

Characterization of very narrow quasi-one-dimensional quantum channels

1988 ◽  
Vol 37 (17) ◽  
pp. 10118-10124 ◽  
Author(s):  
K.-F. Berggren ◽  
G. Roos ◽  
H. van Houten
Keyword(s):  
Quantum Channels ◽  
One Dimensional

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.

Performance characterization of Pauli channels assisted by indefinite causal order and post-measurement

2020 ◽  
Vol 20 (15&16) ◽  
pp. 1261-1280
Author(s):  
Francisco Delgado ◽  
Carlos Cardoso-Isidoro
Keyword(s):  
Quantum Channel ◽  
Communication Channel ◽  
Quantum Communication ◽  
Quantum Channels ◽  
Causal Order ◽  
Combined Process ◽  
Performance Characterization ◽  
Output State ◽  
Transmission Of Information

Indefinite causal order has introduced disruptive procedures to improve the fidelity of quantum communication by introducing the superposition of { orders} on a set of quantum channels. It has been applied to several well characterized quantum channels as depolarizing, dephasing and teleportation. This work analyses the behavior of a parametric quantum channel for single qubits expressed in the form of Pauli channels. Combinatorics lets to obtain affordable formulas for the analysis of the output state of the channel when it goes through a certain imperfect quantum communication channel when it is deployed as a redundant application of it under indefinite causal order. In addition, the process exploits post-measurement on the associated control to select certain components of transmission. Then, the fidelity of such outputs is analysed to characterize the generic channel in terms of its parameters. As a result, we get notable enhancement in the transmission of information for well characterized channels due to the combined process: indefinite causal order plus post-measurement.

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