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Advances in physics theories and applications

Indeterminate Space-Time Quantum Mechanics, Geometrical Interpretation Of Quantum Information Compression, Hallgren's Efficient Quantum Algorithm For Solving Pell's Equation, Stronger No-Cloning Theorem, Quantum Template Matching, Perturbative And Nonpert

K.N.P. Kumar

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Bužek and M. Hillery analyze the possibility of (e) copying (that is, cloning) arbitrary states of (e) a quantum-mechanical spin-1/2 system. Authors show that there exists (eb) universal quantum-copying machine‘‘ (i.e., transformation) which approximately copies quantum-mechanical states such that (e) the quality of its output does not (e) depend on the input. Authors also examine a machine which combines a unitary transformation and (e&eb) a selective measurement to produce good copies of (e) states in the neighborhood of a particular state. Authors discuss the problem of (e) measurement of the output states. © 1996 The American Physical Society Received 5 February 1996 DOI:http://dx.doi.org/10.1103/PhysRevA.54.1844 Quantum copying: Beyond the no-cloning theorem V. Bužek and M. Hillery Phys. Rev. A 54, 1844 – Published 1 September 1996 STOP Authors derive a necessary and sufficient condition for two pure states, each entangled in two remote systems, to be clonable by the sequential access to the two systems. The result shows (eb) to what extent... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-496948-501458.pdf

Related papers

Quantum copying: Beyond the no-cloning theorem

Vladimír Bužek, Mark S. Hillery

We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system. We show that there exists a ``universal quantum-copying machine'' (i.e., transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. \textcopyright{} 1996 The American Physical Society.

arXiv:quant-ph/9607018•Physical Review A•Sep 1, 1996•1,124 citations

A quantum-copying machine for equatorial qubits

Heng Fan, Xiang‐Bin Wang, Keiji Matsumoto

Bužek and Hillery proposed a universal quantum-copying machine (UQCM) (i.e., transformation) to analyze the possibility of cloning arbitrary states. The UQCM copies quantum-mechanical states with the quality of its output does not depend on the input. We propose a slightly different transformation to analyze a restricted set of input states. We impose the conditions (I) the density matrices of the two output states are the same, and that (II) the distance between input density operator and the output density operators is input state independent. Using Hilbert-Schmidt norm and Bures fidelity, we show that our transformation can achieves the bound of the fidelity.

arXiv:quant-ph/0012033•Dec 7, 2000•0 citations

Some bounds for quantum copying

Alexey E. Rastegin

We propose new optimality criterion for the estimation of state-dependent cloning. We call this measure the relative error because the one compares the errors in the copies with contiguous size taking into account the similarity of states to be copied. A copying transformation and dimension of state space are not specified. Only the unitarity of quantum mechanical transformations is used. The presented approach is based on the notion of the angle between two states. Firstly, several useful statements simply expressed in terms of angles are proved. Among them there are the spherical triangle inequality and the inequality establishing the upper bound on the modulus of difference between probability distributions generated by two any states for an arbitrary measurement. The tightest lower bound on the relative error is then obtained. Hillery and Buzek originally examined an approximate state-dependent copying and obtained the lower bound on the absolute error. We consider relationship between the size of error and the corresponding probability distributions and obtain the tightest lower bound on the absolute error. Thus, the proposed approach supplements and reinforces the results obtained by Hillery and Buzek. Finally, the basic findings of investigation for the relative error are discussed.

arXiv:quant-ph/0108014•Aug 3, 2001•1 citation

Two-qubit quantum cloning machine and quantum correlation broadcasting

Azam Kheirollahi, Hamidreza Mohammadi, Seyed Javad Akhtarshenas

Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: ``Is there an optimal quantum cloning machine?'' Buzek and Hillery answered this question and constructed their famous B-H quantum cloning machine. The B-H machine clones the state of an arbitrary single qubit in an optimal manner and hence it is universal. Generalizing this machine for a two-qubit system is straightforward, but during this procedure, except for product states, this machine loses its universality and becomes a state-dependent cloning machine. In this paper, we propose some classes of optimal universal local quantum state cloners for a particular class of two-qubit systems, more precisely, for a class of states with known Schmidt basis. We then extend our machine to the case that the Schmidt basis of the input state is deviated from the local computational basis of the machine. We show that more local quantum coherence existing in the input state corresponds to less fidelity between the input and output states. Also we present two classes of a state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord, defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is, in general, very fragile during the broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.

arXiv:1604.04995•Physical Review A•Nov 9, 2016•1 citation

Quantum cloning, Bell's inequality and teleportation

Satyabrata Adhikari, Nirman Ganguly, Indranil Chakrabarty +1 more

We analyse the possibility of using the two-qubit output state from the Buzek–Hillery quantum copying machine (not necessarily a universal quantum cloning machine) as a teleportation channel. We show that there is a range of values of the machine parameter ξ for which the two-qubit output state is entangled and violates the Bell-CHSH inequality and for a different range it remains entangled but does not violate the Bell-CHSH inequality. Further, we observe that for certain values of the machine parameter the two-qubit mixed state can be used as a teleportation channel. The use of the output state from the Buzek–Hillery cloning machine as a teleportation channel provides an additional appeal to the cloning machine and motivation for our present work.

arXiv:0712.1404•Journal of Physics A Mathematical and Theoretical•Sep 15, 2008•13 citations

Quantum Cloning and Deletion in Quantum Information Theory

Satyabrata Adhikari

Quantum mechanics put restriction on performing some task which we can do classically. One such restriction is that we cannot copy an arbitrary quantum state. This is known as No-cloning theorem. Although quantum mechanics forbid us to construct a perfect cloner, there is no restriction to construct an imperfect cloner. Therefore, we have constructed and studied a special kind of quantum cloning machine named as Hybrid quantum cloning machine which is nothing but a different combination of pre-existing quantum cloning machines. We showed that this hybrid quantum cloning machine sometime gives better quality copy than the pre-existing quantum cloners. Quantum entanglement is a very astonishing property in a multi-partite system and it is a very useful ingredient of quantum information theory. Therefore broadcasting (cloning) of entanglement is a very important subject to study. In this regard we have studied the broadcasting of entanglement using state dependent quantum cloning machine. Further, we have investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners. Linearity of quantum mechanics put another restriction on deletion of an arbitrary quantum state i.e. an arbitrary quantum state against a copy cannot be perfectly deleted. Like cloning machine one can construct an approximate deletion machine also. Therefore, we have constructed an universal deletion machine in the sense that the fidelity of deletion does not depend on the input state and showed that the fidelity of deletion is 3/4 . Further we showed that the fidelity of deletion depends on the blank state at the output and it can be improved from the value 3/4 for certain blank state. Moreover, we have studied the case where two cloned copies are given at the input of the deletion machine.

arXiv:0902.1622•Feb 10, 2009•2 citations
Bužek and M. Hillery analyze the possibility of (e) copying (that is, cloning) arbitrary states of (e) a quantum-mechanical spin-1/2 system. Authors show that there exists (eb) universal quantum-copying machine‘‘ (i.e., transformation) which approximately copies quantum-mechanical states such that (e) the quality of its output does not (e) depend on the input. Authors also examine a machine which combines a unitary transformation and (e&eb) a selective measurement to produce good copies of (e) states in the neighborhood of a particular state. Authors discuss the problem of (e) measurement of the output states. © 1996 The American Physical Society Received 5 February 1996 DOI:http://dx.doi.org/10.1103/PhysRevA.54.1844 Quantum copying: Beyond the no-cloning theorem V. Bužek and M. Hillery Phys. Rev. A 54, 1844 – Published 1 September 1996 STOP Authors derive a necessary and sufficient condition for two pure states, each entangled in two remote systems, to be clonable by the sequential access to the two systems. The result shows (eb) to what extent... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-496948-501458.pdf
We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system. We show that there exists a ``universal quantum-copying machine'' (i.e., transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. \textcopyright{} 1996 The American Physical Society.
Bužek and Hillery proposed a universal quantum-copying machine (UQCM) (i.e., transformation) to analyze the possibility of cloning arbitrary states. The UQCM copies quantum-mechanical states with the quality of its output does not depend on the input. We propose a slightly different transformation to analyze a restricted set of input states. We impose the conditions (I) the density matrices of the two output states are the same, and that (II) the distance between input density operator and the output density operators is input state independent. Using Hilbert-Schmidt norm and Bures fidelity, we show that our transformation can achieves the bound of the fidelity.
We propose new optimality criterion for the estimation of state-dependent cloning. We call this measure the relative error because the one compares the errors in the copies with contiguous size taking into account the similarity of states to be copied. A copying transformation and dimension of state space are not specified. Only the unitarity of quantum mechanical transformations is used. The presented approach is based on the notion of the angle between two states. Firstly, several useful statements simply expressed in terms of angles are proved. Among them there are the spherical triangle inequality and the inequality establishing the upper bound on the modulus of difference between probability distributions generated by two any states for an arbitrary measurement. The tightest lower bound on the relative error is then obtained. Hillery and Buzek originally examined an approximate state-dependent copying and obtained the lower bound on the absolute error. We consider relationship between the size of error and the corresponding probability distributions and obtain the tightest lower bound on the absolute error. Thus, the proposed approach supplements and reinforces the results obtained by Hillery and Buzek. Finally, the basic findings of investigation for the relative error are discussed.
Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: ``Is there an optimal quantum cloning machine?'' Buzek and Hillery answered this question and constructed their famous B-H quantum cloning machine. The B-H machine clones the state of an arbitrary single qubit in an optimal manner and hence it is universal. Generalizing this machine for a two-qubit system is straightforward, but during this procedure, except for product states, this machine loses its universality and becomes a state-dependent cloning machine. In this paper, we propose some classes of optimal universal local quantum state cloners for a particular class of two-qubit systems, more precisely, for a class of states with known Schmidt basis. We then extend our machine to the case that the Schmidt basis of the input state is deviated from the local computational basis of the machine. We show that more local quantum coherence existing in the input state corresponds to less fidelity between the input and output states. Also we present two classes of a state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord, defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is, in general, very fragile during the broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.
We analyse the possibility of using the two-qubit output state from the Buzek–Hillery quantum copying machine (not necessarily a universal quantum cloning machine) as a teleportation channel. We show that there is a range of values of the machine parameter ξ for which the two-qubit output state is entangled and violates the Bell-CHSH inequality and for a different range it remains entangled but does not violate the Bell-CHSH inequality. Further, we observe that for certain values of the machine parameter the two-qubit mixed state can be used as a teleportation channel. The use of the output state from the Buzek–Hillery cloning machine as a teleportation channel provides an additional appeal to the cloning machine and motivation for our present work.
Quantum mechanics put restriction on performing some task which we can do classically. One such restriction is that we cannot copy an arbitrary quantum state. This is known as No-cloning theorem. Although quantum mechanics forbid us to construct a perfect cloner, there is no restriction to construct an imperfect cloner. Therefore, we have constructed and studied a special kind of quantum cloning machine named as Hybrid quantum cloning machine which is nothing but a different combination of pre-existing quantum cloning machines. We showed that this hybrid quantum cloning machine sometime gives better quality copy than the pre-existing quantum cloners. Quantum entanglement is a very astonishing property in a multi-partite system and it is a very useful ingredient of quantum information theory. Therefore broadcasting (cloning) of entanglement is a very important subject to study. In this regard we have studied the broadcasting of entanglement using state dependent quantum cloning machine. Further, we have investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners. Linearity of quantum mechanics put another restriction on deletion of an arbitrary quantum state i.e. an arbitrary quantum state against a copy cannot be perfectly deleted. Like cloning machine one can construct an approximate deletion machine also. Therefore, we have constructed an universal deletion machine in the sense that the fidelity of deletion does not depend on the input state and showed that the fidelity of deletion is 3/4 . Further we showed that the fidelity of deletion depends on the blank state at the output and it can be improved from the value 3/4 for certain blank state. Moreover, we have studied the case where two cloned copies are given at the input of the deletion machine.