Paolo Perinotti

Informational derivation of quantum theory

We derive quantum theory from purely informational principles. Five elementary axioms - causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning - define a broad class of theories of information processing that can be regarded as standard. One postulate - purification - singles out quantum theory within this class.

Probabilistic theories with purification

We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamiolkowski isomorphism in quantum theory. Such an isomorphism allows one to prove most of the basic features of quantum theory, like, e.g., existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.

Theoretical framework for quantum networks

We present a framework to treat quantum networks and all possible transformations thereof, including as special cases all possible manipulations of quantum states, measurements, and channels, such as, e.g., cloning, discrimination, estimation, and tomography. Our framework is based on the concepts of quantum comb---which describes all transformations achievable by a given quantum network---and link product---the operation of connecting two quantum networks. Quantum networks are treated both from a constructive point of view---based on connections of elementary circuits---and from an axiomatic one---based on a hierarchy of admissible quantum maps. In the axiomatic context a fundamental property is shown, which we call universality of quantum memory channels: any admissible transformation of quantum networks can be realized by a suitable sequence of memory channels. The open problem whether this property fails for some nonquantum theory, e.g., for no-signaling boxes, is posed.

Quantum Computations without Definite Causal Structure

We show that quantum theory allows for transformations of black boxes that cannot be realized by inserting the input black boxes within a circuit in a pre-defined causal order. The simplest example of such a transformation is the classical switch of black boxes, where two input black boxes are arranged in two different orders conditionally on the value of a classical bit. The quantum version of this transformation-the quantum switch-produces an output circuit where the order of the connections is controlled by a quantum bit, which becomes entangled with the circuit structure. Simulating these transformations in a circuit with fixed causal structure requires either postselection, or an extra query to the input black boxes.

Quantum Circuit Architecture

We present a method for optimizing quantum circuits architecture, based on the notion of a quantum comb, which describes a circuit board where one can insert variable subcircuits. Unexplored quantum processing tasks, such as cloning and storing or retrieving of gates, can be optimized, along with setups for tomography and discrimination or estimation of quantum circuits.

Classical randomness in quantum measurements

Similarly to quantum states, also quantum measurements can be ‘mixed’, corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are indecomposable, i.e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each measuring apparatus by a positive operator-valued measure (POVM), which gives the statistics of the outcomes for any input state. The POVMs form a convex set, and in this language the indecomposable apparatuses are represented by extremal points—the analogous of ‘pure states’ in the convex set of states. Differently from the case of states, however, indecomposable POVMs are not necessarily rank-one, e.g. von Neumann measurements. In this paper we give a complete classification of indecomposable apparatuses (for discrete spectrum), by providing different necessary and sufficient conditions for extremality of POVMs, along with a simple general algorithm for the decomposition of a POVM into extremals. As an interesting application, ‘informationally complete’ measurements are analysed in this respect. The convex set of POVMs is fully characterized by determining its border in terms of simple algebraic properties of the corresponding POVMs.

Efficient Use of Quantum Resources for the Transmission of a Reference Frame

We propose a covariant protocol for transmitting reference frames encoded on N spins, achieving sensitivity N-2 without the need of a preestablished reference frame and without using entanglement between sender and receiver. The protocol exploits the use of equivalent representations that were overlooked in the previous literature.

Derivation of the Dirac Equation from Principles of Information Processing

Without using the relativity principle, we show how the Dirac equation in three space dimensions emerges from the large-scale dynamics of the minimal nontrivial quantum cellular automaton satisfying unitarity, locality, homogeneity, and discrete isotropy. The Dirac equation is recovered for small wave vector and inertial mass, whereas Lorentz covariance is distorted in the ultrarelativistic limit. The automaton can thus be regarded as a theory unifying scales from Planck to Fermi. A simple asymptotic approach leads to a dispersive Schrodinger equation describing the evolution of narrowband states at all scales.

Informationally complete measurements and group representation

Informationally complete measurements on a quantum system allow one to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kinds of measurement can be achieved through positive-operator valued measures (POVMs) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help of frame theory we provide a constructive way to evaluate the data-processing function for arbitrary operators.

Clean positive operator valued measures

In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.