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43
On Duality between Quantum Maps and Quantum States
, 2004
"... 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 ..."
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Cited by 26 (0 self)
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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 Jamiolkowski 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.
Quantum channels, wavelets, dilations and representations of On
 Proc. Edinb. Math. Soc
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Upper Bounds on the Noise Threshold for Faulttolerant Quantum Computing
, 2008
"... We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary kqubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary onequbit gates that are essentially noisefree. We assume that the ..."
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We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary kqubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary onequbit gates that are essentially noisefree. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p> 1 − Θ(1 / √ k), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k = 2, our bound is p> 35.7%. Moreover, if the only allowed gate on more than one qubit is the twoqubit CNOT gate, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds. 1
Zyczkowski, Geometry of sets of quantum maps: a generic positive map acting on a highdimensional system is not completely positive
, 2008
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Parametrizing quantum states and channels
 Quantum Information Processing
, 2003
"... Abstract. This work describes one parametrization of quantum states and channels and several of its possible applications. This parametrization works in any dimension and there is an explicit algorithm which produces it. Included in the list of applications are a simple characterization of pure stat ..."
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Abstract. This work describes one parametrization of quantum states and channels and several of its possible applications. This parametrization works in any dimension and there is an explicit algorithm which produces it. Included in the list of applications are a simple characterization of pure states, an explicit formula for one additive entropic quantity which does not require knowledge of eigenvalues, and an algorithm which finds one Kraus operator representation for a quantum operation without recourse to eigenvalue and eigenvector calculations. 1.
Relative entropy and single qubit HolevoSchumacherWestmoreland channel capacity. quantph/0207128
 of Fields Institute Monographs. American Mathematical Society
, 2002
"... The relative entropy description of HolevoSchumacherWestmoreland (HSW) classical channel capacities is applied to single qubit quantum channels. A simple formula for the relative entropy of qubit density matrices in the Bloch sphere representation is derived. The formula is combined with the King ..."
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Cited by 5 (2 self)
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The relative entropy description of HolevoSchumacherWestmoreland (HSW) classical channel capacities is applied to single qubit quantum channels. A simple formula for the relative entropy of qubit density matrices in the Bloch sphere representation is derived. The formula is combined with the KingRuskaiSzarekWerner qubit channel ellipsoid picture to analyze several unital and nonunital qubit channels in detail. An alternate proof is presented that the optimal HSW signalling states for single qubit unital channels are those states with minimal channel output entropy. The derivation is based on symmetries of the relative entropy formula, and the KingRuskaiSzarekWerner qubit channel ellipsoid picture. A proof is given that the average output density matrix of any set of optimal HSW signalling states for a ( qubit or nonqubit) quantum
Comments on multiplicativity of maximal pnorms when p = 2
 Quantum Inf. Comput
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Lower Bounds on the Quantum Capacity and Highest Error Exponent of General Memoryless Channels
, 2002
"... Tradeoffs between the information rate and fidelity of quantum errorcorrecting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive linear map, where the dimension of the underlying H ..."
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Cited by 4 (1 self)
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Tradeoffs between the information rate and fidelity of quantum errorcorrecting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum errorcorrecting code of length n and rate R is proven to be lower bounded by 1 − exp[−nE(R) + o(n)] for some function E(R). The E(R) is positive below some threshold R0, a direct consequence of which is that R0 is a lower bound on the quantum capacity. This is an extension of the author’s previous result [M. Hamada, Phys. Rev. A, vol. 65, 052305, 2002; LANL ePrint, quantph/0109114, 2001]. While it states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.