Quantum operations (QO's) describe any state change allowed in quantum mechanics, such as the evolution of an open system or the state change due to a measurement. We address the problem of which unitary transformations and which observables can be used to achieve a QO with generally different input and output Hilbert spaces. We classify all unitary extensions of a QO and give explicit realizations in terms of free-evolution direct-sum dilations and interacting tensor-product dilations. In terms of Hilbert space dimensionality the free-evolution dilations minimize the physical resources needed to realize the QO, and for this case we provide bounds for the dimension of the ancilla space versus the rank of the QO. The interacting dilations on the other hand, correspond to the customary ancilla-system interaction realization, and for these we derive a majorization relation which selects the allowed unitary interactions between system and ancilla.
Quantum operations (QO's) describe any state change allowed in quantum mechanics, such as the evolution of an open system or the state change due to a measurement. We address the problem of which unitary transformations and which observables can be used to achieve a QO with generally different input and output Hilbert spaces. We classify all unitary extensions of a QO and give explicit realizations in terms of free-evolution direct-sum dilations and interacting tensor-product dilations. In terms of Hilbert space dimensionality the free-evolution dilations minimize the physical resources needed to realize the QO, and for this case we provide bounds for the dimension of the ancilla space versus the rank of the QO. The interacting dilations on the other hand, correspond to the customary ancilla-system interaction realization, and for these we derive a majorization relation which selects the allowed unitary interactions between system and ancilla.
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatization is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "transposed" of a physical transformation. What is new in the present paper with respect to quant-ph/0603011 is the operational deduction of an involution corresponding to the "complex-conjugation" for effects, whose extension to transformations allows to define the "adjoint" of a transformation when the extension is composition-preserving.
We provide an overview of standard "projective" quantum measurements with the goal of elucidating connections between theory and experiment. We make use of a unitary "Stinespring" representation of measurements on a dilated Hilbert space that includes both the physical degrees of freedom and those of the measurement apparatus. We explain how this unitary representation (i) is guaranteed by the axioms of quantum mechanics, (ii) relates to both the Kraus and von Neumann representations, and (iii) corresponds to the physical time evolution of the system and apparatus during the measurement process. The Stinespring representation also offers significant conceptual insight into measurements, helps connects theory and experiment, is particularly useful in describing protocols involving midcircuit measurements and outcome-dependent operations, and establishes that all quantum operations are compatible with relativistic locality, among other insights.
The projection postulate and spectral representation of observables remain central in quantum theory but describe only an idealized measurement limit. Realistic measurements may be inefficient, unsharp, weak, indirect, adaptive, destructive, or informationally incomplete. Positive-operator-valued measures (POVMs) provide the probability rule for processes, whereas quantum instruments specify state transformation. This review traces the hierarchy from von Neumann projective measurement to generalized measurement, emphasizing distinctions among effects, instruments, channels, and implementations. Naimark dilation realizes arbitrary POVMs through unitary coupling to an ancillary meter and projective readout. Stinespring dilation represents completely positive trace-preserving dynamics by unitary evolution on an enlarged system followed by partial trace. A qubit–ancilla interaction yields a partial-measurement instrument, its POVM effects, and its information–disturbance trade-off. It compares projective, generalized, and ancilla-mediated schemes; examines compatibility, instrument incompatibility, and disturbance; and surveys applications to state discrimination, tomography, control, and error correction. Recent developments in dynamic-circuit measurements and noisy-POVM compatibility are integrated with questions concerning measurement back-action and outcome definiteness. The review stresses that a POVM does not uniquely determine state update, dilation theorems establish realizability rather than unique mechanisms, and operational measurement theory does not settle definite outcomes, arguing quantum instruments provide a complete description of measurement.
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of physical experiment and from five simple Postulates concerning experimental accessibility and simplicity. For the infinite dimensional case, on the other hand, a C*‐algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand‐Naimark‐Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in Ref. [1]. The main ingredient of the axiomatization is the postulated existence of faithful states that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the transposed of a physical transformation. What is new in the present paper with respect to Ref. [1], is the operational deduction of an involution corresponding to the complex‐conjugation for effects, whose extension to transformations allows to define the adjoint of a transformation when the extension is composition‐preserving. The existence of such composition‐preserving extension among possible extensions is analyzed.
An operational description of quantum phenomena concerns developing models that describe experimentally observed behaviour. $\textit{Higher-order quantum operations}\unicode{x2014}$quantum operations that transform quantum operations$\unicode{x2014}$are fundamental to modern quantum theory, extending beyond basic state preparations, evolutions, and measurements described by the Born rule. These operations naturally emerge in quantum circuit architectures, correlated open dynamics, and investigations of quantum causality, to name but a few fields of application. This Review Article provides both a pedagogical introduction to the framework of higher-order quantum operations and a comprehensive survey of current literature, illustrated through physical examples. We conclude by identifying open problems and future research directions in this rapidly evolving field.
To implement a positive-operator-valued measurement (POVM), which is defined on the dS-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include dA additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has n elements and ri is the rank of the ith element, then the dimension of the minimal ancilla is ${d}_{A}={\ensuremath{\sum}}_{i=1}^{n}{r}_{i}\ensuremath{-}{d}_{S}$ for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine dA for the unambiguous discrimination of N linearly independent states and provide the full DSE implementation of a state-discriminating POVM for N=2.