Let K be a convex subset of the state space of a finite-dimensional C*-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 positive operator valued measures (POVMs).
Let K be a convex subset of the state space of a finite-dimensional C*-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 positive operator valued measures (POVMs).
Supermaps between quantum channels (completely positive trace-preserving (CPTP) maps of matrix algebras) were introduced in [Chiribella et al., EPL 83(3) (2008)]. In this work we generalise to supermaps between channels of any type; by channels we mean CPTP maps of finite-dimensional C*-algebras. Channels include POVMs, quantum instruments, classically controlled families of quantum channels, classical channels, quantum multimeters, and more. We show that deterministic supermaps between channels of any type can be realised using simple circuits, recovering the previous realisation theorems of [Chiribella et al., EPL 83(3) (2008)] (for deterministic supermaps between quantum channels) and [Bluhm et al. (2024)] (for deterministic supermaps between quantum multimeters) as special cases. To prove this realisation theorem we use the graphical calculus of the 2-category of finite-dimensional 2-Hilbert spaces; the paper includes an accessible and elementary introduction to this graphical calculus, and no prior knowledge of category theory is expected of the reader.
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterization of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and then used to retrieve classic results such as Choi’s characterization of the extreme quantum channels, Parthasarathy’s characterization of the extreme quantum states with given marginals and the quantum version of Birkhoff’s theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalization, and we characterize the extremal POVMs.
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators subject to affine constraints. These examples include sets of quantum states with given expectation values of some observables of interest, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels, and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterisation of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and are used to retrieve classic results such as Choi's characterisation of the extreme quantum channels, Parhasarathy's characterisation of the extreme quantum states with given marginals, and the quantum version of Birkhoff's theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalisation, and we characterise the extremal POVMs.
A pair of quantum channels is said to be incompatible if they cannot be realized as marginals of a single channel. This paper addresses the general structure of the incompatibility of completely positive channels with a fixed quantum input space and with general outcome operator algebras. We define a compatibility relation for such channels by identifying the composite outcome space as the maximal (projective) C*-tensor product of outcome algebras. We show theorems that characterize this compatibility relation in terms of the concatenation and conjugation of channels, generalizing the recent result for channels with quantum outcome spaces. These results are applied to the positive operator valued measures (POVMs) by identifying each of them with the corresponding quantum-classical (QC) channel. We also give a characterization of the maximality of a POVM with respect to the post-processing preorder in terms of the conjugate channel of the QC channel. We consider another definition of compatibility of normal channels by identifying the composite outcome space with the normal tensor product of the outcome von Neumann algebras. We prove that for a given normal channel, the class of normally compatible channels is upper bounded by a special class of channels called tensor conjugate channels. We show the inequivalence of the C*- and normal compatibility relations for QC channels, which originates from the possibility and impossibility of copying operations for commutative von Neumann algebras in C*- and normal compatibility relations, respectively.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.
One of the main recent achievements of quantum information theory [1] is the proof of the equivalence of several (sub-, super-) additivity conjectures for finite-dimensional quantum channels and systems [2] (see the survey in [3]). The main purpose of this paper is to give a generalization of this result to the infinite-dimensional case. Let H and H ′ be separable Hilbert spaces. A quantum channel is a linear completely positive trace-preserving map Φ: T(H ) 7→ T(H ′), where T(H ) is the ideal of all trace class operators on H . In particular, Φ generates an affine map of the convex set S(H ) of states (that is, density operators) on the space H to the set S(H ′) of states on the space H ′ [1]. Important characteristics of a quantum channel are the output entropy HΦ(ρ) = H(Φ(ρ)), a lower-semicontinuous concave function on the input state space with values in [0,+∞], and its convex closure (see [4]), denoted by b HΦ(ρ) and called the b H-function of the channel Φ. It was shown in [5] that the convex closure of the output entropy of an arbitrary quantum channel Φ is given by the expression b HΦ(ρ) = inf μ Z