Generalized channels: channels for convex subsets of the state space

Abstract

Let $K$ be a convex subset of the state space of a finite dimensional $C^{\ast }$-algebra. We study the properties of channels on $K$, which are defined as affine maps from $K$ into the state space of another algebra, extending to completely positive maps on the subspace generated by $K$. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on $K$. Moreover, if $K$ contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process POVMs.

37 pages, published version. Theorems 3 and 4 were replaced by Theorem 3, which was proved by using Arveson's extension theorem

References in corpus (6)

• Quantum Circuits Architecture
• Transforming quantum operations: quantum supermaps
• Process POVM: A mathematical framework for the description of process tomography experiments
• Optimal cloning of unitary transformations
• Quantum learning algorithms for quantum measurements
• Extremal quantum protocols

Cited by in corpus (27)

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• Comparison of Quantum Channels by Superchannels
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• Theoretical framework for Higher-Order Quantum Theory
• Dilations, inclusions of matrix convex sets, and completely positive maps
• Extending quantum operations
• Incompatible measurements in a class of general probabilistic theories
• Conditions on the existence of maximally incompatible two-outcome measurements in General Probabilistic Theory
• Witnessing incompatibility of quantum channels
• A necessary condition for incompatibility of observables in general probabilistic theories
• Base norms and discrimination of generalized quantum channels
• Asymptotics of quantum channels
• A prototype of quantum von Neumann architecture
• Correlation Measure Equivalence in Dynamic Causal Structures of Quantum Gravity
• Theory of Quantum Gravity Information Processing
• Normal completely positive maps on the space of quantum operations
• Interpolating between positive and completely positive maps: a new hierarchy of entangled states
• Strategies for single-shot discrimination of process matrices
• Extremality conditions for generalized channels
• Quantum circuit simulation of superchannels
• Interpolation for completely positive maps: numerical solutions
• Extremal generalized quantum measurements
• On the convex structure of process POVMs
• An Interpolation Problem for Completely Positive Maps on Matrix Algebras: Solvability and Parametrisation
• On the Alberti-Uhlmann Condition for Unital Channels
• On the simulation of quantum multimeters