Causal structures and the classification of higher order quantum computations

Quantum operations are the most widely used tool in the theory of quantum information processing, representing elementary transformations of quantum states that are composed to form complex quantum circuits. The class of quantum transformations can be extended by including transformations on quantum operations, and transformations thereof, and so on up to the construction of a potentially infinite hierarchy of transformations. In the last decade, a sub-hierarchy, known as quantum combs, was exhaustively studied, and characterised as the most general class of transformations that can be achieved by quantum circuits with open slots hosting variable input elements, to form a complete output quantum circuit. The theory of quantum combs proved to be successful for the optimisation of information processing tasks otherwise untreatable. In more recent years the study of maps from combs to combs has increased, thanks to interesting examples showing how this next order of maps requires entanglement of the causal order of operations with the state of a control quantum system, or, even more radically, superpositions of alternate causal orderings. Some of these non-circuital transformations are known to be achievable and have even been achieved experimentally, and were proved to provide some computational advantage in various information-processing tasks with respect to quantum combs. Here we provide a formal language to form all possible types of transformations, and use it to prove general structure theorems for transformations in the hierarchy. We then provide a mathematical characterisation of the set of maps from combs to combs, hinting at a route for the complete characterisation of maps in the hierarchy. The classification is strictly related to the way in which the maps manipulate the causal structure of input circuits.Comment: 12 pages, revtex styl

Optimal asymptotic cloning machines

We pose the question whether the asymptotic equivalence between quantum cloning and quantum state estimation, valid at the single-clone level, still holds when all clones are examined globally. We conjecture that the answer is affirmative and present a large amount of evidence supporting our conjecture, developing techniques to derive optimal asymptotic cloners and proving their equivalence with estimation in virtually all scenarios considered in the literature. Our analysis covers the case of arbitrary finite sets of states, arbitrary families of coherent states, arbitrary phase- and multiphase-covariant sets of states, and two-qubit maximally entangled states. In all these examples we observe that the optimal asymptotic fidelity enjoys a universality property, as its scaling does not depend on the specific details of the set of input states, but only on the number of parameters needed to specify them.Comment: 27 + 9 pages, corrected one observation about cloning of maximally entangled state

Transforming quantum operations: quantum supermaps

We introduce the concept of quantum supermap, describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and measurements, quantum supermaps describe all possible transformations between elementary quantum objects (quantum systems as well as quantum devices). After giving the axiomatic definition of supermap, we prove a realization theorem, which shows that any supermap can be physically implemented as a simple quantum circuit. Applications to quantum programming, cloning, discrimination, estimation, information-disturbance trade-off, and tomography of channels are outlined.Comment: 6 pages, 1 figure, published versio

Perfect discrimination of no-signalling channels via quantum superposition of causal structures

A no-signalling channel transforming quantum systems in Alice's and Bob's laboratories is compatible with two different causal structures: (A < B) Alice's output causally precedes Bob's input and (B< A) Bob's output causally precedes Alice's input. I show that a quantum superposition of circuits operating within these two causal structures enables the perfect discrimination between no-signalling channels that can not be perfectly distinguished by any ordinary circuit.Comment: 5 + 5 pages, published versio

Joint estimation of real squeezing and displacement

We study the problem of joint estimation of real squeezing and amplitude of the radiation field, deriving the measurement that maximizes the probability density of detecting the true value of the unknown parameters. More generally, we provide a solution for the problem of estimating the unknown unitary action of a nonunimodular group in the maximum likelihood approach. Remarkably, in this case the optimal measurements do not coincide with the so called square-root measurements. In the case of squeezing and displacement we analyze in detail the sensitivity of estimation for coherent states and displaced squeezed states, deriving the asymptotic relation between the uncertainties in the joint estimation and the corresponding uncertainties in the optimal separate measurements of squeezing and displacement. A two-mode setup is also analyzed, showing how entanglement between optical modes can be used to approximate perfect estimation.Comment: 14 pages, 3 eps figures; a section has been added with new results in terms of Heisenberg uncertainty relations for the joint measuremen

Quantum Circuits Architecture

We present a method for optimizing quantum circuits architecture. The method is based on the notion of "quantum comb", which describes a circuit board in which one can insert variable subcircuits. The method allows one to efficiently address novel kinds of quantum information processing tasks, such as storing-retrieving, and cloning of channels.Comment: 10 eps figures + Qcircuit.te

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 pre-established reference frame and without using entanglement between sender and receiver. The protocol exploits the use of equivalent representations, which were overlooked in the previous literature.Comment: 4 pages, no figures; added new references and improved introduction. Accepted for publication on PR

Optimal Probabilistic Simulation of Quantum Channels from the Future to the Past

We introduce the study of quantum protocols that probabilistically simulate quantum channels from a sender in the future to a receiver in the past. The maximum probability of simulation is determined by causality and depends on the amount and type (classical or quantum) of information that the channel can transmit. We illustrate this dependence in several examples, including ideal classical and quantum channels, measure-and-prepare channels, partial trace channels, and universal cloning channels. For the simulation of partial trace channels, we consider generalized teleportation protocols that take N input copies of a pure state in the future and produce M < N output copies of the same state in the past. In this case, we show that the maximum probability of successful teleportation increases with the number of input copies, a feature that was impossible in classical physics. In the limit of asymptotically large N, the probability converges to the probability of simulation for an ideal classical channel. Similar results are found for universal cloning channels from N copies to M > N approximate copies, exploiting a time-reversal duality between universal cloning and partial trace.Comment: 16 pages, 6 figures, published versio