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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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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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Rieffel “Quantum operations that cannot be implemented using a small mixed environment
 J. Math. Phys
, 2002
"... To implement any quantum operation (a.k.a. “superoperator ” or “CP map”) on a ddimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a d 2dimensional environment which is initialized in a fixed pure state. It has been suggested that a ddimen ..."
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To implement any quantum operation (a.k.a. “superoperator ” or “CP map”) on a ddimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a d 2dimensional environment which is initialized in a fixed pure state. It has been suggested that a ddimensional environment might be enough if we could initialize the environment in a mixed state of our choosing. In this note we show with elementary means that certain explicit quantum operations cannot be realized in this way. Our counterexamples map some pure states to pure states, giving strong and easily manageable conditions on the overall unitary transformation. Everything works in the more general setting of quantum operations from ddimensional to d ′dimensional spaces, so we place our counterexamples within this more general framework. 1 Quantum operations Quantum operations (see e.g. [3]) are also known as “superoperators”, “superscattering operators ” or “completely positive maps ” (“CP maps”). They can be viewed as a generalization of unitary transformations and are the most general transformations that can be applied to a quantum system in an unknown (possibly mixed) state. More precisely, quantum operations are the most general transformations that can be implemented deterministically, thus excluding operations which only succeed with a certain probability, like those depending on a measurement outcome. Under a quantum operation pure states are frequently mapped to mixed states. All quantum operations on a ddimensional system can be implemented as the partial trace of a unitary operator acting on the system together with an auxiliary system (the “environment”). The question is how small an environment suffices to implement all possible quantum operations on a ddimensional