The concept of generalized quantum measurement is introduced as a transformation that sets a one-to-one correspondence between the initial states of the measured object system and final states of the object-meter system with the help of a classical informational index, unambiguously linked to a classically compatible set of quantum states. It is shown that the generalized quantum measurement concept covers all key types of quantum measurement---standard projective, entangling, fuzzy, and generalized measurements with a partial or complete destruction of initial information associated with the object. A special class of soft quantum measurements as a basic model for the fuzzy measurements widespread in physics is introduced and its information properties are studied in detail. Also, a special class of partially destructive measurements mapping all states of the Hilbert space of a finite-dimensional quantum system onto the basis states of an infinite-dimensional quantum system is considered.
A concept of the generalized quantum measurement is introduced as the transformation, which establishes a correspondence between the initial states of the object system and final states of the object--measuring device (meter) system with the help of a classical informational index, unambiguously linked to the classically compatible set of states of the object--meter system. It is shown that the generalized measurement covers all the key known quantum measurement concepts--standard projective, entangling, fuzzy and the generalized measurement with the partial or complete destruction of the initial information contained in the object. A special class of partially-destructive measurements that map the continual set of the states in finite-dimensional quantum systems to that one of the infinite-dimensional quantum systems is considered. Their informational essence and some information characteristics are discussed in detail.
The concept of an entangling measurement as the most general form of ideal quantum measurement is introduced. The mathematical definition of an entangling measurement is given in the form of a superoperator transformation, which projects the initial states of an object-apparatus system on the corresponding entangled states appearing after a measurement. The mathematical properties of the superoperator of an entangling measurement are considered. The spectrum of superoperator eigenstates is studied analytically on the example of a two-level system. The algebraic relation ℳ 2 =ℳ AM between the superoperators of entangling and standard measurements is ascertained. The coherent information obtained in the course of measurement is analyzed and the relevant potentialities of entangling measurements are discussed.
Recent theoretical developments are reviewed concentrating on the description of optimal generalised quantum measurement processes in physical systems with finite dimensional Hilbert spaces. Based on a self contained presentation of the required underlying quantum theoretical concepts this review particularly addresses recently explored basic theoretical properties and possible applications of informationally complete symmetric generalised measurements, a concept unifying the description of numerous so far largely separately explored measurement scenarios, such as measurements based on complete systems of mutually unbiased bases or symmetric informationally complete measurements. The recent results concerning the construction and existence of these informationally complete symmetric generalised measurements may also shed new light on still open questions concerning some of their special cases, such as the existence of complete sets of mutually unbiased bases in Hilbert spaces of non-prime power dimensions.
Measurement compatibility asks whether a family of quantum measurements can be simulated by one parent measurement followed by classical post-processing. We extend this measurement compatibility to the notion of \emph{
d-compatibility} for measurements, channels, and instruments whose parent passes a
d-dimensional quantum system and unlimited classical information to the post-processing. We formulate
d-compatibility as a common linear factorization of operator spaces and operator systems and show that
d-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify
d-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a
d-dimensional register. Using factorization theory of operator spaces, we show that measure-and-prepare channels defined in terms of mutually unbiased bases and pinching channels defined in terms of Heisenberg-Weyl operators are
d-incompatible in certain regimes of
d. We also link
d-compatibility of multi-instruments to the preparability of Choi assemblages using channel-state duality, and use this connection to show that for bipartite states the (in)compatibility of instruments can be used to certify Schmidt numbers and the implementability of the instrument using one-way LOCC protocols. Finally, our compatibility functional also bounds the generalized robustness of compatibility, which we interpret through a memory-bounded nontransient preparation game.
In the present paper we consider the problem of the description of an arbitrary generalized quantum measurement with outcomes in a measurable space. Analysing the unitary invariants of a separable statistical realization of a quantum instrument, we present the most general form of a possible integral representation of an instrument. We introduce the notion of a stochastic realization of an instrument and establish a one-to-one correspondence between the class of unitarily and phase equivalent separable statistical realizations and the equivalence class of stochastic realizations of an instrument. We further single out the invariant class of unitarily and phase equivalent separable statistical realizations for which the integral representation of an instrument is the same for all statistical realizations from this class and is wholly determined by the invariants of this class. We call the special form of this integral representation the quantum stochastic representation of an instrument. We show that the description of a generalized direct quantum measurement can be considered in the frame of a new general approach based on the notion of a family of quantum stochastic evolution operators satisfying the orthonormality relation. This approach gives not only the complete statistical description of any generalized direct quantum measurement but the complete description in a Hilbert space of the stochastic behaviour of a quantum system under a generalized direct measurement in the sense of specifying the probabilistic transition law governing the change from the initial state to a final one under a single measurement. Under this approach a unitary evolution of an isolated quantum system is included as a special case. In the frame of the proposed approach, which we call quantum stochastic approach, all possible schemes of measurements upon a quantum system can be considered. In the case of repeated or continuous in time measurements the quantum stochastic approach allows to define, in the most general case, the notion of the family of posterior pure state trajectories (quantum trajectories in discrete or continuous time) in the Hilbert space of a quantum system and to give their probabilistic treatment.