Semicausal operations are semilocalizable

Eggeling, Tilo; Schlingemann, Dirk; Werner, Reinhard

doi:10.1209/epl/i2002-00579-4

Cited by 107 publications

(127 citation statements)

References 15 publications

(23 reference statements)

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“…Since the maps ρ → I ⊗ ρ, S * and I ⊗ σ → σ are all CP, we have σ = N * (ρ), where N * is a CP trace preserving map from states on K in to states on H in . Remarkably, the same mathematical structure of Lemma 2 characterizes semi-causal quantum operations [7], i.e. operations on bipartite systems that allow signaling from system A to system B but not viceversa.…”

Section: Proofmentioning

confidence: 92%

Transforming quantum operations: Quantum supermaps

Chiribella¹,

D’Ariano²,

Perinotti³

2008

Europhys. Lett.

328

489

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We introduce the concept of quantum supermap, describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and measurements, quantum supermaps describe all possible transformations between elementary quantum objects (quantum systems as well as quantum devices). After giving the axiomatic definition of supermap, we prove a realization theorem, which shows that any supermap can be physically implemented as a simple quantum circuit. Applications to quantum programming, cloning, discrimination, estimation, information-disturbance trade-off, and tomography of channels are outlined.PACS numbers: 03.65.Ta, 03.67.-aThe input-output description of any quantum device is provided by the quantum operation of Kraus [1], which yields the most general probabilistic evolution of a quantum state. Precisely, the output state ρ out is given by the quantum operation E applied to the input state ρ in as followswhere p(E|ρ in ) is the probability of E occurring on state ρ in , when E is one of a set of alternative transformations, such as in a quantum measurement. Owing to its physical meaning, a quantum operation E must be a linear, trace non-increasing, completely positive (CP) map (see, e.g.[2]). The most general form of such a map is known as Kraus formwhere the operators E j satisfy the bound1. Tracepreserving maps, i.e. those achieving the bound, are a particular kind of quantum operations: they occur deterministically and are referred to as quantum channels.In general it is convenient to consider two different input and output Hilbert spaces H in and H out , respectively. In this way, the concept of quantum operation can be used to treat also quantum states, effects, and measurements, which describe the properties of elementary quantum objects such as quantum systems and measuring devices. Indeed, states can be described as quantum operations with one-dimensional H in , i.e. with Kraus operators E j given by ket-vectors √ p j |ψ j ∈ H out , thus yielding the output state ρ out = E(1) = j p j |ψ j ψ j |.A quantum effect 0 ≤ P ≤ I [3] corresponds instead to a quantum operation with one-dimensional H out , i.e. with Kraus operators given by bra-vectorsMore generally, any quantum measurement can be viewed as a particular quantum operation, namely as a quantum-to-classical channel [4]. Channels, states, effects, and measurements are all special cases of quantum operations. What about then considering maps between quantum operations themselves? They would describe the most general kind of transformations between elementary quantum objects. For example a programmable channel [5] would be a map of this type, with a quantum state at the input and a channel at the output. Or else, a device that optimally clones a set of unknown unitary gates would be a map from channels to channels. We will call such a general class of quantum maps quantum supermaps, as they transform CP maps (sometimes referred to as superoper...

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Section: Proofmentioning

confidence: 92%

Transforming quantum operations: Quantum supermaps

Chiribella¹,

D’Ariano²,

Perinotti³

2008

Europhys. Lett.

328

489

View full text Add to dashboard Cite

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“…[58] Beckman, Gottesmann, Nielsen, and Preskill introduced the notions of semicausal and semilocalizable quantum channel for the purpose of studying the constraints on quantum dynamics of bipartite systems imposed by relativistic causality. Subsequently, Eggeling, Schlingemann, and Werner [59] proved the equivalence between semicausality and semilocalizability (see also Ref. [60] for an extensive discussion on the topic).…”

Section: Causally Ordered Channels and Channels With Memorymentioning

confidence: 99%

Probabilistic theories with purification

2010

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We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamio lkowski isomorphism in quantum theory. Such an isomorphism allows one to prove most of the basic features of quantum theory, like e.g. existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.

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“…It is easily proved that localizable operators are causal. It was shown that semicausal operators are always semilocalizable (Eggeling, Schlingemann, and Werner, 2002). However, there are causal operations that are not localizable (Beckman et al, 2001).…”

Section: E the No-communication Theoremmentioning

confidence: 99%

Quantum information and relativity theory

2004

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Quantum mechanics, information theory, and relativity theory are the basic foundations of theoretical physics. The acquisition of information from a quantum system is the interface of classical and quantum physics. Essential tools for its description are Kraus matrices and positive operator valued measures (POVMs). Special relativity imposes severe restrictions on the transfer of information between distant systems. Quantum entropy is not a Lorentz covariant concept. Lorentz transformations of reduced density matrices for entangled systems may not be completely positive maps. Quantum field theory, which is necessary for a consistent description of interactions, implies a fundamental trade-off between detector reliability and localizability. General relativity produces new, counterintuitive effects, in particular when black holes (or more generally, event horizons) are involved. Most of the current concepts in quantum information theory may then require a reassessment.

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Semicausal operations are semilocalizable

Cited by 107 publications

References 15 publications

Transforming quantum operations: Quantum supermaps

Transforming quantum operations: Quantum supermaps

Probabilistic theories with purification

Quantum information and relativity theory

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