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Atemporal diagrams for quantum circuits
 Phys. Rev. A
"... A system of diagrams is introduced that allows the representation of various elements of a quantum circuit, including measurements, in a form which makes no reference to time (hence “atemporal”). It can be used to relate quantum dynamical properties to those of entangled states (mapstate duality), ..."
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A system of diagrams is introduced that allows the representation of various elements of a quantum circuit, including measurements, in a form which makes no reference to time (hence “atemporal”). It can be used to relate quantum dynamical properties to those of entangled states (mapstate duality), and suggests useful analogies, such as the inverse of an entangled ket. Diagrams clarify the role of channel kets, transition operators, dynamical operators (matrices), and Kraus rank for noisy quantum channels. Positive (semidefinite) operators are represented by diagrams with a symmetry that aids in understanding their connection with completely positive maps. The diagrams are used to analyze standard teleportation and dense coding, and for a careful study of unambiguous (conclusive) teleportation. A simple diagrammatic argument shows that a Kraus rank of 3 is impossible for a onequbit channel modeled using a onequbit environment in a mixed state.
Introducing categories to the practicing physicist. In: What is Category Theory
 Advanced Studies in Mathematics and Logic 30, pp.45–74, Polimetrica Publishing
, 2006
"... We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra o ..."
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We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra of practicing physics. We will not provide rigorous definitions or anything resembling a coherent mathematical theory, but we will take the reader for a journey introducing concepts which are part of category theory in a manner that the physicist will recognize them. 1 Why? Why would a physicist care about category theory, why would he want to know about it, why would he want to show off with it? There could be many reasons. For example, you might find John Baez’s webside one of the coolest in the world. Or you might be fascinated by Chris Isham’s and Lee Smolin’s ideas on the use of topos theory in Quantum Gravity. Also the connections between knot theory, braided categories, and sophisticated mathematical physics such as quantum groups and topological quantum field theory might lure you. Or, if you are also into pure mathematics, you might just appreciate category theory due to its incredible unifying power of mathematical structures and constructions. But there is a far more onthenose reason which is never mentioned. Namely, a category is the exact mathematical structure of practicing physics! What do I mean here by a practicing physics? Consider a physical system of type A (e.g. a qubit, or two qubits, or an electron, or classical measurement data) and perform an operation f on it (e.g. perform a measurement on it) which results in a system possibly of a different type B (e.g. the system together with classical data which encodes the measurement outcome, or, just classical data in the case that the measurement destroyed the system). So typically we have
Facial structures for various notions of positivity and applications to the theory of entanglement
, 2012
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Entanglement of random subspaces via the Hastings bound
, 2009
"... Recently Hastings [16] proved the existence of random unitary channels which violate the additivity conjecture. In this paper we use Hastings’ method to derive new bounds for the entanglement of random subspaces of bipartite systems. As an application we use these bounds to prove the existence of no ..."
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Recently Hastings [16] proved the existence of random unitary channels which violate the additivity conjecture. In this paper we use Hastings’ method to derive new bounds for the entanglement of random subspaces of bipartite systems. As an application we use these bounds to prove the existence of nonunital channels which violate additivity of minimal output entropy.
Conditional density operators and the subjectivity of quantum operations
 Presented at Foundations of Probability and Physics–4
, 2006
"... Abstract. Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical a ..."
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Abstract. Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the update maps corresponding to a complete projective measurement, must be subjective, the situation is not so clear in other cases. Here, it is argued that all trace preserving completely positive maps, including unitary operators, should be regarded as subjective, in the same sense as a classical conditional probability distribution. The argument is based on a reworking of the ChoiJamiołkowski isomorphism in terms of “conditional ” density operators and trace preserving completely positive maps, which mimics the relationship between conditional probabilities and stochastic maps in classical probability.
Types of Quantum Information
, 2007
"... Quantum, in contrast to classical, information theory, allows for different incompatible types (or species) of information which cannot be combined with each other. Distinguishing these incompatible types is useful in understanding the role of the two classical bits in teleportation (or one bit in o ..."
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Cited by 4 (1 self)
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Quantum, in contrast to classical, information theory, allows for different incompatible types (or species) of information which cannot be combined with each other. Distinguishing these incompatible types is useful in understanding the role of the two classical bits in teleportation (or one bit in onebit teleportation), for discussing decoherence in informationtheoretic terms, and for giving a proper definition, in quantum terms, of “classical information.” Various examples (some updating earlier work) are given of theorems which relate different incompatible kinds of information, and thus have no counterparts in classical information theory.
Entanglement Distillation  A Discourse on Bound Entanglement in Quantum Information Theory
, 2006
"... In recent years entanglement has been recognised as a useful resource in quantum information and computation. This applies primarily to pure state entanglement which is, due to interaction with the environment, rarely available. Decoherence provides the main motivation for the study of entanglement ..."
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In recent years entanglement has been recognised as a useful resource in quantum information and computation. This applies primarily to pure state entanglement which is, due to interaction with the environment, rarely available. Decoherence provides the main motivation for the study of entanglement distillation. A remarkable effect in the context of distillation is the existence of bound entangled states, states from which no pure state entanglement can be distilled. The concept of entanglement distillation also relates to a canonical way of theoretically quantifying mixed state entanglement. This thesis is, apart from a review chapter on distillation, mainly a theoretical study of bound entanglement and the two major open problems in their classification. The first of these is the classification of PPT bound entanglement (separability problem). After having reviewed known tools we study in detail the multipartite permutation criteria, for which we present new results in their classification. We solve an open problem on the existence of certain PPT states. The Schmidt number of a quantum state is a largely unvalued concept, we analyse it in detail and introduce the Schmidt robustness. The notion of Schmidt number is exploited in the study of the second
yczkowski, Entropic characterization of quantum operations
 International Journal of Quantum Information
, 2011
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Deterministic and Unambiguous Dense Coding
, 2005
"... Optimal dense coding using a partiallyentangled pure state of Schmidt rank ¯ D and a noiseless quantum channel of dimension D is studied both in the deterministic case where at most Ld messages can be transmitted with perfect fidelity, and in the unambiguous case where when the protocol succeeds (p ..."
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Optimal dense coding using a partiallyentangled pure state of Schmidt rank ¯ D and a noiseless quantum channel of dimension D is studied both in the deterministic case where at most Ld messages can be transmitted with perfect fidelity, and in the unambiguous case where when the protocol succeeds (probability τx) Bob knows for sure that Alice sent message x, and when it fails (probability 1 − τx) he knows it has failed. Alice is allowed any singleshot (one use) encoding procedure, and Bob any singleshot measurement. For ¯ D ≤ D a bound is obtained for Ld in terms of the largest Schmidt coefficient of the entangled state, and is compared with published results by Mozes et al. For ¯D> D it is shown that Ld is strictly less than D 2 unless ¯ D is an integer multiple of D, in which case uniform (maximal) entanglement is not needed to achieve the optimal protocol. The unambiguous case is studied for ¯ D ≤ D, assuming τx> 0 for a set of ¯ DD messages, and a bound is obtained for the average 〈1/τ〉. A bound on the average 〈τ〉 requires an additional assumption of encoding by isometries (unitaries when ¯ D = D) that are orthogonal for different messages. Both bounds are saturated when τx is a constant independent of x, by a protocol based on oneshot entanglement concentration. Whether unitary (isometric) encoding suffices for optimal protocols remains a major unanswered question, both for our work and for previous studies of dense coding using partiallyentangled states, including noisy (mixed) states. I