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110
Quantum Error Correction Via Codes Over GF(4)
, 1997
"... The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on s ..."
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Cited by 236 (19 self)
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The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
Computational Complexity  A Modern Approach
, 2009
"... Not to be reproduced or distributed without the authors ’ permissioniiTo our wives — Silvia and RavitivAbout this book Computational complexity theory has developed rapidly in the past three decades. The list of surprising and fundamental results proved since 1990 alone could fill a book: these incl ..."
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Cited by 151 (2 self)
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Not to be reproduced or distributed without the authors ’ permissioniiTo our wives — Silvia and RavitivAbout this book Computational complexity theory has developed rapidly in the past three decades. The list of surprising and fundamental results proved since 1990 alone could fill a book: these include new probabilistic definitions of classical complexity classes (IP = PSPACE and the PCP Theorems) and their implications for the field of approximation algorithms; Shor’s algorithm to factor integers using a quantum computer; an understanding of why current approaches to the famous P versus NP will not be successful; a theory of derandomization and pseudorandomness based upon computational hardness; and beautiful constructions of pseudorandom objects such as extractors and expanders. This book aims to describe such recent achievements of complexity theory in the context of more classical results. It is intended to both serve as a textbook and as a reference for selfstudy. This means it must simultaneously cater to many audiences, and it is carefully designed with that goal. We assume essentially no computational background and very minimal mathematical background, which we review in Appendix A. We have also provided a web site for this book at
A Theory of Quantum ErrorCorrecting Codes
 Phys. Rev. A
, 1996
"... Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. ..."
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Cited by 74 (7 self)
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Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of errorcorrecting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate w...
Distinguishability and Accessible Information in Quantum Theory
, 1996
"... my mother Geraldine, who has been with me every day of my life and to my brother Mike, who gave me a frontier to look toward ii ACKNOWLEDGEMENTS No one deserves more thanks for the success of this work than my advisor and friend Carlton Caves. Carl is the model of the American work ethic applied to ..."
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Cited by 50 (5 self)
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my mother Geraldine, who has been with me every day of my life and to my brother Mike, who gave me a frontier to look toward ii ACKNOWLEDGEMENTS No one deserves more thanks for the success of this work than my advisor and friend Carlton Caves. Carl is the model of the American work ethic applied to physical thought. The opportunity to watch him in action has fashioned my way of thought, both in the scientific and the secular. He has been a valued teacher, and I hope my three years in Albuquerque have left me with even a few of his qualities. Special thanks go to Greg Comer, my philosophical companion. Greg’s influence on this dissertation was from a distance, but no less great because of that. Much of the viewpoint espoused here was worked out in conversation with him. I thank the home team, Howard Barnum, Sam Braunstein, Gary Herling, Richard Jozsa, Rüdiger Schack, and Ben Schumacher for patiently and critically listening to so many of my ideas—
On quantum fidelities and channel capacities
 IEEE Trans. Info. Theor
, 2000
"... Abstract — We show the equivalence of two different notions of quantum channel capacity: that which uses the entanglement fidelity as its criterion for success in transmission, and that which uses the minimum fidelity of pure states in a subspace of the input Hilbert space as its criterion. As a cor ..."
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Cited by 35 (2 self)
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Abstract — We show the equivalence of two different notions of quantum channel capacity: that which uses the entanglement fidelity as its criterion for success in transmission, and that which uses the minimum fidelity of pure states in a subspace of the input Hilbert space as its criterion. As a corollary, any source with entropy less than the capacity may be transmitted with high entanglement fidelity. We also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel. This enables us to simplify a known upper bound for the channel capacity. It also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity.
Computing the noncomputable
 Contemporary Physics
"... We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically non ..."
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Cited by 30 (7 self)
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We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantummechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not
Quantum Entanglement and Communication Complexity
 SIAM J. COMPUT
, 1998
"... We consider a variation of the communication complexity scenario, where the parties are supplied with an extra resource: particles in an entangled quantum state. We note that "quantum nonlocality" can be naturally expressed in the language of communication complexity. These are communication complex ..."
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Cited by 26 (6 self)
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We consider a variation of the communication complexity scenario, where the parties are supplied with an extra resource: particles in an entangled quantum state. We note that "quantum nonlocality" can be naturally expressed in the language of communication complexity. These are communication complexity problems where the "output" is embodied in the correlations between the outputs of the individual parties. Without entanglement, the parties must communicate to produce the required correlations; whereas, with entanglement, no communication is necessary to produce the correlations. In this sense, nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication. We show how to transform examples of nonlocality into more traditional communication complexity problems, where the output is explicitly determined by each individual party. The resulting problems require communication with or without entanglement, but the required communication is less when entanglement is available. All these results are a noteworthy contrast to the wellknown fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties.
On quantum statistical inference
 J. Roy. Statist. Soc. B
, 2001
"... [Read before The Royal Statistical Society at a meeting organized by the Research Section ..."
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Cited by 24 (5 self)
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[Read before The Royal Statistical Society at a meeting organized by the Research Section
Quantum Kolmogorov complexity based on classical descriptions
 IEEE Trans. Inform. Theory
, 2001
"... Abstract—We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity coincides with the classical Kolmogorov complexity on ..."
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Cited by 20 (1 self)
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Abstract—We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity coincides with the classical Kolmogorov complexity on the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approximated from above under certain conditions. With high probability a quantum object is incompressible. Upper and lower bounds of the quantum complexity of multiple copies of individual pure quantum states are derived and may shed some light on the nocloning properties of quantum states. In the quantum situation complexity is not subadditive. We discuss some relations with “nocloning ” and “approximate cloning ” properties. Keywords — Algorithmic information theory, quantum; classical descriptions of quantum states; information theory, quantum; Kolmogorov complexity, quantum; quantum cloning. I.