Quantum Computation and Information Seminar 

Our work is on why and how quantum theory should be understood ultimately as a probability theory in a new form, having a prescriptive -- rather than descriptive -- nature. In this talk, the focus will be on two topics contributing to support such an approach, namely, the conditioning of quantum states through the collection of data and the stochasticity of quantum operations, which provides further insight into the latter as uncertainty encoding structures.

I will review some attempts to describe the two- and three-qubit Hilbert space geometries with the help of Hopf fibrations of the $S^7$ and $S^{15}$ high dimensional spheres. In both cases, it is shown that the associated Hopf map is strongly sensitive to states entanglement content. In the two-qubit case, a generalization of the one-qubit celebrated Bloch sphere representation is described.

We consider the problem of detecting ballistic electrons at the edge of a two-dimensional electron gas in the integer quantum Hall regime. This problem is motivated by recent proposals for developing an electronic version of quantum optics with single electrons propagating in nanostructures. A phenomenological model based on a passive RC circuit capacitively coupled to a gate will be considered. Using a quantum description of this circuit, we determine the signal over noise ratio of the detector as well as the backaction on the incident wavepacket in term of the detector characteristics. Using information theory, we define the appropriate notion of quantum limit for such an "on the fly" detector.

We study a specific security property of private quantum channels - ability of the adversary to modify the plaintext in a deterministic way using only operations on ciphertexts (tamper resistance). We also study relation of quality of encryption and quality of tamper resistance on random unitary channels.

This talk will describe an analytical approach for the computation of Long Distance Entanglement (LDE) mediated through one-dimensional quantum spin chains recently found in numerical studies. I review the formalism that allows the computation of LDE for weakly interacting probes with gapped many-body systems and show that, at zero temperature, a DC response function determines the ability of the physical system to develop genuine quantum correlations between the probes. In the second part of the talk, I show that the biquadratic Heisenberg spin-1 chain is able to produce LDE in the thermodynamical limit and that the finite antiferromagnetic Heisenberg chain maximally entangles two spin-1/2 probes very far apart. This is of crucial importance since feasible mechanisms of entanglement extraction from real solid state systems and their ability to transfer entanglement between distant parties are essential ingredients for the implementation of Quantum Information protocols, such as teleportation or superdense coding.

In this work, we generalize the Shannon's zero-error capacity (ZEC) of discrete memoryless channels to quantum channels. The communication protocol restricts codewords to tensor products of input quantum states, whereas entangled measurements can be performed between several channel outputs. Necessary and sufficient conditions for a quantum channel to have positive zero-error capacity is given. From an equivalent graph theoretical definition are given some properties of ensembles of quantum states and measurements attaining the quantum error-free capacity. A channel is said to be non-trivial if two or more channel uses are required in order to reach the capacity, and the ensemble of input states attaining the capacity contains non-orthogonal states. We show an example of quantum channel that gives rise to the pentagon as characteristic graph for a given ensemble of non-orthogonal quantum states and conjecture that its ZEC is that of the classical pentagon.

We present a fair contract signing protocol without communication with a judge during the exchange phase. To this end we use particular Werner states and assume that the agents have no control over the system decoherence. Finally, we show how to implement such Werner states and discuss some limitations and extensions of the proposed protocol (joint work with J. Bouda and N. Paunkovic).

We consider a 1D waveguide where flying qubits can undergo multiple scattering between two stationary, distant (pseudo)spin-s quantum systems. We show that iterated injection of such flying particles in a given internal state, together with appropriate post-selection, is able to conditionally generate the maximally entangled singlet state of the two stationary systems initially prepared in a product state. This task can be attained with a significant probability as well as with a small number of injected particles. Two different physical setups able to implement the protocol are proposed.

The possibility of preparing two-photon hyper-entangled states encoding three or more qubits in each photon leads to the following problem: If we distribute N qubits between two parties, what quantum pure states and distributions of qubits allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell's theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs, and provide all possible proofs up to N=7 qubits. On the other hand, the possibility of preparing N-photon N-qubit graph states leads to the following problems: Which are the best Bell inequalities for graph states? How nonlocality grows with the number of qubits for different graph states? We provide all optimal Mermin-like Bell inequalities for all graph states up to N=5 qubits, and some interesting Bell inequalities for certain relevant classes of graph states of any number of qubits.

Entanglement between light and matter plays an outstanding role in long-distance quantum communication allowing e.g. quantum teleportation from light to matter and a first demonstration of a quantum repeater. However from a more fundamental point of view it opens up also the possibility to perform a first loophole-free test of Bell's inequality with a pair of entangled atoms at remote locations. In this talk I will show first important steps in this direction.

Oracle operators are unitary operators which are used to coherently compute functions. We investigate the problem of unambiguous discrimination among oracle operators for two oracle models: the standard and minimal (or erasing) oracles, where the latter can only be constructed for invertible functions. It is found that the unambiguous distinguishability criterion is the same for both models, and depends only on pairwise relationships between the functions. We show that, except in certain trivial cases, it is impossible to unambiguously discriminate among all standard oracle operators corresponding to integer functions with fixed domain and range. We also find that it is possible to unambiguously discriminate among the Grover oracle operators corresponding to an arbitrarily large unsorted database, which cannot be achieved deterministically. The unambiguous distinguishability of standard oracle operators corresponding to totally indistinguishable functions is analysed in detail. For such functions, there is zero classical discrimination probability even with a memory of the independent variable and the result of the function computation. There must be at least four functions in such a set. Using a graph-theoretic framework, we prove that the standard oracle operators are not unambiguously distinguishable for any finite set of totally indistinguishable functions on a Boolean domain and with fixed range. Finite sets of such functions on a larger domain can, however, have unambiguously distinguishable standard oracle operators. We provide a complete description of sets of four functions with this property. We also examine the possibility of unambiguous oracle operator discrimination with multiple parallel calls, and provide sufficient conditions for this to be possible.

We give a simple, direct proof of the "mother" protocol of quantum information theory. In this new formulation, it is easy to see that the mother, or rather her generalization to the fully quantum Slepian-Wolf protocol, simultaneously accomplishes two goals: quantum communication-assisted entanglement distillation, and state transfer from the sender to the receiver. As a result, in addition to her other "children," the mother protocol generates the state merging primitive of Horodecki, Oppenheim and Winter, a fully quantum reverse Shannon theorem, and a new class of distributed compression protocols for correlated quantum sources which are optimal for sources described by separable density operators. Moreover, the mother protocol described here is easily transformed into the so-called "father" protocol whose children provide the quantum capacity and the entanglement-assisted capacity of a quantum channel, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary: all protocols in the family are children of the mother. Work together with Anura Abeyesinghe, Igor Devetak and Patrick Hayden.

The introduction of the one-way quantum computer established that, in order to realize universal quantum computation, it is sufficient to perform local measurements on a system of qubits which have initially been prepared in a highly entangled cluster state. The computational power of such a measurement-based quantum computer originates in the strong quantum correlations which are initially present in the system. However, it is to date largely unknown which correlations allow such a measurement-based computer to exhibit a computational speed-up with respect to classical devices. Here we provide a new perspective to this question by relating this issue to the field of mathematical logic. We show that the computational power of arbitrary graph state resources for measurement-based quantum computation is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which yield a computational speed-up with respect to classical computation, the underlying graphs — describing the quantum correlations of the states — are associated with undecidable logic theories.

The private quantum channel (PQC) is a framework defining general encryption scheme with quantum plaintext, quantum ciphertext and classical key. Recently Nayak et al derived the most general form of an invertible quantum operation and put it into context of encryption of quantum information using classical key suggesting that encryption with classical key can be more general than the original definition of PQC, that was given without justification of its generality. We use security requirements of encryption of quantum information with classical key and apply the results of Nayak et al to show that the original definition of PQC is the most general one.

Hilbert space operators may be mapped onto a space of ordinary functions (operator symbols) equipped with an associative (but noncommutative) star-product. A unified framework for such maps is reviewed. Because of its clear probabilistic interpretation, a particular class of operator symbols (tomograms) is proposed as a framework for quantum information problems. Qudit states are identified with maps of the unitary group into the simplex. The image of the unitary group on the simplex provides a geometrical characterization of the nature of the quantum states. Generalized measurements, typical quantum channels, entropies, and entropy inequalities are discussed in this setting. Joint work with M. ManŽko and R. Vilela Mendes.

Using quantum non-locality, one can prove the security of quantum cryptography in a "device-independent scenario", that is, without any assumption on the nature of the quantum signal or on the measuring devices. This remarkable generality is a consequence of Bell's theorem applied to cryptography: if the Alice-Bob correlations violate a Bell inequality, then the eavesdropper Eve cannot have full information on Alice's and Bob's symbols, because otherwise Eve's symbol would be a hidden variable that reproduces the correlations. While this idea has been around since Ekert's 1991 paper, quantitative progress has been possible only very recently. In the first works, for mathematical simplicity, Eve was provided with more power than quantum physics, and other strong assumptions were made (zero error rate by Barrett, Hardy, Kent, PRL 2005; individual attacks by Acin et al, PRL 2006, Scarani et al PRA 2006). I shall discuss these pioneering approaches and present the latest achievement: a bound for security in the device-independent scenario against collective attacks of a quantum Eve (Acin et al, work in preparation).

In this talk some recent developments on entanglement theory will be addressed. A simple strategy of entanglement estimation, given by a lower bound on the Generalized Robustness of Entanglement will be explained. A second topic will be entanglement extraction. We will dicuss how to get entanglement from a degenerate Fermi gas, and also describe two distinct experimental proposals, one with ultra-cold neutrons, other with semi-conductors. Finally, the problem of Entanglement Transfer from a pair of qubits to another pair will be described with the exhibition of some recent results.

We will report on work-in-progress on developing a model-checking tool for quantum communication protocols and quantum cryptographic protocols. Our early experiments with probabilistic model checking will be reviewed; then the general model-checking problem for quantum protocols will be formalised and its complexity discussed. This will naturally lead to the question of which protocols can be efficiently simulated on a classical computer. An overview of the stabilizer formalism will be given, followed by a summary of the Aaronson-Gottesman algorithm for simulation of stabilizer circuits. We will present QMC, a tool for automatically verifying properties of such circuits; properties of protocols are expressed using a subset of EQPL, a logic developed by Mateus and Sernadas. This is joint work with Rajagopal Nagarajan (University of Warwick) and Simon Gay (University of Glasgow).

Conclusion of the previous talk.

The first part of the talk will consist of an introduction to some basic aspects of Quantum Field Theory, especially perturbative QFT. Then some combinatorial features of QFT will be addressed, including: (i) symmetries and symmetry factors of Feynman diagrams, and their automatic generation, (ii) combinatorial laws for Green functions.