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.
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.
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.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations and satisfying suitable requirements of normality and complete positivity. Here we present the extension of the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations on generic von Neumann algebras. In this setting, we provide two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. A structure theorem for probability measures with values in the set of quantum supermaps is also illustrated. Finally, some applications are given, and in particular it is shown that all the supermaps defined in this paper can be implemented by connecting quantum devices in quantum circuits.
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.
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.
This work constitutes the systematic and rigorous extension of the entire framework of Meta Operational Mathematics to the domain of quantum operations and their compositional inverses.Quantum mechanics, as the fundamental theory of nature at the microscopic scale, provides a uniquely rich testing ground for the meta-operational philosophy: operations that act upon quantum states—such as completely positive trace-preserving (CPTP) maps, quantum measurements, quantum instruments, and unitary evolutions—can themselves be operated upon, composed, iterated, added, multiplied, differentiated, exponentiated, logarithmized, and subjected to algebraic,analytic, and geometric transformations.We establish a complete four-level hierarchical framework tailored specifically to quantum theory: Level 0 consists of density matrices (quantum states) as elements of the base space F =Tsa(H); Level 1 consists of quantum operations as smooth mappings on F; Level 2 consists of quantum meta-operations as smooth mappings on quantum operations; and Level 3 consists of meta-meta-operations acting upon meta-operations. Within this framework, every fundamental operation of quantum mechanics is canonically lifted to a meta-operation, and the interactions among these meta-operations—through composition, pointwise addition, pointwise multiplication, differentiation, quantum randomization, exponentiation, and logarithm—generate the entire quantum operad QOp in its bornological closure.A fundamental distinction from the hyperbolic, elliptic, Gamma, Beta, Zeta, and probabilistic cases is rigorously established. In quantum theory, the base space F = Tsa(H) is not a subset of the complex plane but a non-commutative Banach space of trace-class self-adjoint operators.The compositional inverse of a quantum channel is not merely a formal inverse but the quantum error correction recovery map, which embodies the fundamental duality between CPTP maps and their Stinespring dilations. The complementary channel operation—corresponding to the environmental output of a Stinespring dilation—satisfies a twisted antipode axiom in the Hopf operad structure, reflecting the irreversible accumulation of information leaked to the environment. The completely positive constraint endows the quantum operad with a non-trivial boundary structure absent in all other cases.The eight fundamental meta-operations generating the entire quantum operad are: composition, pointwise addition, pointwise multiplication, differentiation, quantum randomization, the identity operation, the quantum channel evaluation operation, and the quantum measurement operation. Their irreducibility is proved through a careful analysis of the dependence structure of each generator on the remaining seven, using the Stinespring dilation theorem, the uniqueness of Kraus representations, and the Choi-Jamiołkowski isomorphism for complete positivity. The three essential features of quantum theory—positivity and complete positivity, complementarity and uncertainty relations, and entanglement and inseparability—are systematically elevated to the meta-operational level as algebraic axioms, analytic tools, and geometric objects, constructing a self-contained Quantum Meta-Operational Mathematics. All conjectures and open problems originally stated within this research program have either been resolved as theorems within the body of this work or are precisely formulated as remaining open problems with partial progress rigorously indicated.
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H) of bounded operators on a separable Hilbert space H ) to the space of all operators on H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ can be uniquely extended to a bounded linear operator on the Banach space of all G -bounded operators on H provided that the operation Φ is G -limited: the predual operation Φ∗ maps the set of positive trace class operators ρ with finite value of TrρG into itself. Assuming that G has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the G -limited property. Applications to the theory of Bosonic Gaussian channels are considered.