__COLLOQUIA__

## Umesh Vazirani

Fulkerson Prize 2012

__Testing Quantum Systems__

2-3pm, May 4 2016, Lecture Theater A, Chow Yei Ching Building

The testing of quantum devices is a challenging foundational issue both because of the formidable computational power of quantum systems and the severe limit on the accessible information about the quantum state of a system. Over the past few years there has emerged a new theory of quantum testing that exploits unique features of quantum mechanics to get around these obstacles. This theory has resulted in provably secure quantum cryptography with untrusted quantum devices and certifiable random number generation. It has also resulted in protocols for testing that a claimed quantum computer is truly quantum.

Testing quantum annealers, a type of special purpose quantum computer, has become a pressing practical challenge in view of the announcements by the Canadian company D-Wave of its 108-qubit quantum annealer in 2011, and subsequent scaling and claims of quantum speedups. I will describe recent work on a classical benchmark for quantum annealers, that can be used as part of a quantum Turing Test for such machines.

Testing quantum annealers, a type of special purpose quantum computer, has become a pressing practical challenge in view of the announcements by the Canadian company D-Wave of its 108-qubit quantum annealer in 2011, and subsequent scaling and claims of quantum speedups. I will describe recent work on a classical benchmark for quantum annealers, that can be used as part of a quantum Turing Test for such machines.

## Alexander Holevo

Shannon Award 2016

__Quantum Shannon Theory__

2-3pm, May 5 2016, Lecture Theater A, Chow Yei Ching Building

The notions of channel and capacity are central to the classical Shannon theory. “Quantum Shannon theory” denotes a subfield of quantum information science which uses operator analysis, convexity and matrix inequalities, asymptotic techniques such as large deviations and measure concentration to study mathematical models of communication channels and their information-processing performance. From the mathematical point of view quantum channels are normalized completely positive maps of operator algebras, the analog of Markov maps in the noncommutative probability theory, while the capacities are related to certain norm-like quantities. In applications noisy quantum channels arise from irreversible evolutions of open quantum systems interacting with environment—a physical counterpart of a mathematical dilation theorem.

It turns out that in the quantum case the notion of channel capacity splits into the whole spectrum of numerical information-processing characteristics depending on the kind of data transmitted (classical or quantum) as well as on the additional communication resources. An outstanding role here is played by quantum correlations — entanglement – inherent in tensor-product structure of composite quantum systems. This talk presents a survey of basic coding theorems providing analytical expressions for the capacities of quantum channels in terms of entropic quantities. We also touch upon recent progress in solution of long-standing problems of additivity and Gaussian optimizers, concerning the entropic quantities of theoretically and practically important class of Bosonic Gaussian channels.

It turns out that in the quantum case the notion of channel capacity splits into the whole spectrum of numerical information-processing characteristics depending on the kind of data transmitted (classical or quantum) as well as on the additional communication resources. An outstanding role here is played by quantum correlations — entanglement – inherent in tensor-product structure of composite quantum systems. This talk presents a survey of basic coding theorems providing analytical expressions for the capacities of quantum channels in terms of entropic quantities. We also touch upon recent progress in solution of long-standing problems of additivity and Gaussian optimizers, concerning the entropic quantities of theoretically and practically important class of Bosonic Gaussian channels.

__ABSTRACTS__

## Samson Abramsky

__Quantifying Contextuality__

We present a quantitative measure for degree of contextuality.

This has a number of features:

- Generality: it is applicable to a wide range of contextuality scenarios, including the usual multipartite Bell non-locality scenarios, Kochen-Specker constructions, and more,

- It is efficiently computable using linear programming.

- It is normalized, which enables meaningful comparison between different scenarios.

- It has tight connections with violations of Bell inequalities (themselves generalised to arbitrary contextuality scenarios).

In fact, with a suitable normalization of Bell inequalities, they are bounded within the range of the contextuality measure, and there is a tight Bell inequality corresponding to the contextuality measure for any given empirical model.

We also show how this measure can be naturally extended to cover the case of models with signalling, in such a way as to distinguish the contributions of signalling from genuine contextuality.

The measure is based on the sheaf-theoretic approach to non-locality and contextuality introduced by Abramsky and Brandenburger.

We also make comparisons with the Contextuality-by-Default framework of Dzhafarov and Jujala. We show a precise correspondence with a modified version of the CbD measure, which we argue is more robust and natural.

We also discuss connections with a measure based on "negative probabilities”.

This is a joint work with Rui Soares Barbosa and Shane Mansfield.

This has a number of features:

- Generality: it is applicable to a wide range of contextuality scenarios, including the usual multipartite Bell non-locality scenarios, Kochen-Specker constructions, and more,

- It is efficiently computable using linear programming.

- It is normalized, which enables meaningful comparison between different scenarios.

- It has tight connections with violations of Bell inequalities (themselves generalised to arbitrary contextuality scenarios).

In fact, with a suitable normalization of Bell inequalities, they are bounded within the range of the contextuality measure, and there is a tight Bell inequality corresponding to the contextuality measure for any given empirical model.

We also show how this measure can be naturally extended to cover the case of models with signalling, in such a way as to distinguish the contributions of signalling from genuine contextuality.

The measure is based on the sheaf-theoretic approach to non-locality and contextuality introduced by Abramsky and Brandenburger.

We also make comparisons with the Contextuality-by-Default framework of Dzhafarov and Jujala. We show a precise correspondence with a modified version of the CbD measure, which we argue is more robust and natural.

We also discuss connections with a measure based on "negative probabilities”.

This is a joint work with Rui Soares Barbosa and Shane Mansfield.

## Joonwoo Bae

__Operational Characterization of Divisibility of Dynamical Maps__

Divisibility of dynamical maps turns out to be a fundamental notion in characterizing Markovianity of quantum evolution, although the decision problem for divisibility itself is computationally intractable. In this work, we propose the operational characterization of divisibility of dynamical maps by exploiting distinguishability of quantum channels. We prove that distinguishability for any pair of quantum channels does not increase under divisible maps, and then, in terms of channel distinguishability with entanglement between system and k-dimensional ancillas, provide the operational characterization for the full hierarchy of the so-called k-divisibility. Finally, from the fact that min-entropy corresponds to the information-theoretic measure of distinguishability, the entropic characterization to divisible maps is also provided.

## Caslav Brukner

__Quantum information on indefinite causal structures__

Quantum computation is standardly assumed to happen on a definite causal structure, where the order of the gates in a circuit is fixed in advance and is independent of the states. However, the interplay between general relativity and quantum mechanics might require to consider more general situations in which the metric, and hence the causal structure, is indefinite. Quantum computation on such structures would allow the order in which the gates are applied to be controlled by a quantum state, and analogously quantum communication would allow the direction of communication between parties to be controlled by a quantum state. I will show that these new resources make possible solving specific computational and communication complexity problems more efficiently than any causally ordered quantum circuit.

## Francesco Buscemi

__Reverse Data-Processing Theorems__

Suppose that two random variables X and Y are connected by a process (transition matrix or CPTP map) mapping X in Y. Then, the data-processing inequality states that the information content of Y never exceeds that of its parent X, thus providing a necessary condition for the existence of a process from X to Y. A reverse data-processing theorem aims to the converse, namely, to give a set of sufficient conditions for the existence of a process. In this talk, I review some recent results in this direction and discuss few scenarios in which reverse data-processing theorems play a fundamental role -- in particular, Markov processes, open quantum systems dynamics, and generalized resource theories.

## Bob Coecke

__Causality and connectedness in process theories__

By means of example theorems we investigate the connections between causality and connectedness in general process theories. Causality itself has a purely process-theoretic characterization. These theorems include no-signalling, no-broadcasting, extremity of pure states, and the existence of entanglement.

## Runyao Duan

__Activated zero-error classical capacity of quantum channels in the presence of quantum no-signaling correlations__

Recently the one-shot quantum no-signalling assisted zero-error classical capacity of a quantum channel was formulated as a semidefinite programming (SDP) depending only on the Choi-Kraus operator space of the channel [Duan and Winter, 2014]. In this talk we further study the

This talk is based on a joint work with Xin Wang (http://arxiv.org/abs/1510.05437).

*activated quantum no-signalling assisted zero-error classical capacity*by first allowing the assistance from some noiseless forward communication channel and later paying back the cost of the helper. We show that the one-shot activated capacity can also be formulated as a SDP and derive a number of striking properties of this number. In particular, this number is additive under direct sum, and is always greater than or equal to the super-dense coding bound. As a remarkable consequence, we find that one bit noiseless classical communication is able to fully activate any classical-quantum channel to achieve its asymptotic capacity.This talk is based on a joint work with Xin Wang (http://arxiv.org/abs/1510.05437).

## Masahito Hayashi

__Implementable quadratic enhancement in quantum metrology__

This paper addresses the estimation of the unknown phase parameter.

Our problem is composed of the optimization of the input state and the measurement.

We impose the energy constraint to the input state when the Hamiltonian is given as the number operator in the Boson-Fock space.

Then, we show that quadratic enhancement in the mean squared error of the estimated parameter when the energy is sufficiently large.

We propose an experimental setup to generate an input state achieving for enhanced metrology using squeezing transformations.

This is joint work with Sai Vinjanampathy and L. C. Kwek.

Our problem is composed of the optimization of the input state and the measurement.

We impose the energy constraint to the input state when the Hamiltonian is given as the number operator in the Boson-Fock space.

Then, we show that quadratic enhancement in the mean squared error of the estimated parameter when the energy is sufficiently large.

We propose an experimental setup to generate an input state achieving for enhanced metrology using squeezing transformations.

This is joint work with Sai Vinjanampathy and L. C. Kwek.

## Yeong-Cherng Liang

__Quantum nonlocality from arbitrarily small amount of measurement independence__

The use of Bell’s theorem in any application or experiment relies on the assumption of measurement independence, meaning that the measurement settings can be chosen freely independent of the state of the system to be tested. In this talk, we demonstrate that even in the simplest Bell test—one involving 2 parties each performing 2 binary-outcome measurements—an arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality. In other words, even if we relax the causal structure typically considered in a Bell-type setup by allowing nontrivial dependence of the measurement setting on the state of test system, it is still possible to demonstrate a contradiction with quantum theory. To this end, we will introduce the notion of measurement dependent locality and show that the corresponding correlations form a convex polytope. These correlations can thus be characterized efficiently, for instance, using a finite set of Bell-like inequalities. This observation, in turn, enables the systematic study of quantum nonlocality and related applications under limited measurement independence.

## Renbao Liu

__Using qubit decoherence to study thermodynamics in the Complex Plane__

In two foundational papers published in 1952, Lee and Yang established the connection between the analytic properties of thermodynamic functions and phase transitions. In particular, they proved that the zero points of the partition functions of lattice gases (called Lee-Yang zeros) are all located along the unit circle in the complex plane of fugacity (Lee-Yang theorem), or equivalently, along the imaginary axis of magnetic field for ferromagnetic Ising models. The Lee-Yang zeros, however, had not been observed in experiments until recently. The difficulty is that complex parameters are generally regarded as unphysical. Recently, we discovered that the quantum coherence of a probe spin coupled to a bath is equivalent to the partition function of the bath, with the evolution time corresponding to an imaginary physical parameter [1]. This makes it possible to experimentally study Lee-Yang zeros in particular and thermodynamics for complex parameters in general. Based on this idea, Lee-Yang zeros have been experimentally observed for the first time [2]. We further find that the Yang-Lee edges, i.e., the starting and ending points of the Lee-Yang zeros, lead to a new type of phase transitions, namely, time-domain phase transitions, which manifest themselves in the probe coherence as sudden changes when the bath approaches to the thermodynamic limit [1]. Starting from that, we developed a systematic theory on phase transitions in the complex plane of physical parameters, which can be measured as abrupt changes of the probe coherence evolutions even at temperatures higher than the critical points for conventional phase transitions [3]. We also established the thermodynamic holography in which the thermodynamic properties of a system in an area of physical parameters are fully determined by the properties along the boundary [4]. We expect a wealth of new, experimentally verifiable physics to be explored in the complex plane of physical parameters.

This work was supported by Hong Kong RGC. I acknowledge collaborations with B. B. Wei, X. H. Peng, S. W. Chen, H. C. Po, H. Zhou, J. Cui, and J. Du.

This work was supported by Hong Kong RGC. I acknowledge collaborations with B. B. Wei, X. H. Peng, S. W. Chen, H. C. Po, H. Zhou, J. Cui, and J. Du.

## Tomoyuki Morimae

__Verification of quantum computing__

Alice, who does not have enough quantum technology, asks Bob, who has a full-fledged universal quantum computer, to perform her quantum computing. Alice, however, does not trust Bob. Can she verify the correctness of his quantum computing in spite that her quantum technology is severely limited? In this talk, I show it is possible by introducing several protocols using blind quantum computing and measurement-based quantum computing. I will also explain some applications of these results to Quantum Merlin-Arthur (QMA) which a quantum analog of NP and an important complexity class in quantum computing complexity.

## Mio Murao

__Entanglement assisted classical communication simulates "classical communication'' without causal order__

Phenomena induced by the existence of entanglement, such as nonlocal correlations between spatially separated parties sharing entanglement, exhibit characteristic properties of quantum mechanics distinguishing from classical theories. When entanglement is accompanied by classical communication, it enhances the power of quantum operations jointly performed by two spatially separated parties. Such a power has been analyzed by the gap between the performances of joint quantum operations implementable by local operations at each party connected by classical communication with and without the assistance of entanglement. In this work, we present a new formulation for joint quantum operations connected by classical communication beyond special relativistic causal order but without entanglement and still within quantum mechanics. Using the formulation, we show that entanglement assisting classical communication necessary for implementing a class of joint quantum operations called separable maps can be interpreted to simulate "classical communication'' not respecting causal order. Our results reveal a new counter-intuitive aspect of entanglement related to spacetime.

## Mikio Nakahara

__Decoherence Free Subspace, Noiseless Subsystem and Group Representation__

Quantum information is fragile against external noise. It is possible, however, to protect qubits/qudits from external noise if all the qubits/qudits suffer from the same noise operator. This applies to the case when (1) a group of photons are transmitted through an optical fibre with a fixed imperfection, (2) the system size is much smaller than the wave length of the external noise and (3) qubits/qudits are exchanged between two parties without a shared reference frame. Then the error operators are invariant under permutations of qubits/qudits. By making use of the group representation theory of the symmetric group, we can identify subspace/subsystem for which such error operators act as a unit matrix, and hence quantum information encoded in the subspace/subsystem is protected. This scheme is known as the decoherence free subspace and the noiseless subsystem when the relevant irreducible representation is 1-dimensional and k-dimensional (k >1), respectively. For a given d for qudits and the number n of physical qudits, we have successfully identified such subspace/subsystem with the maximal encoding efficiency, which approaches to 1 as n grows larger. Note, however, that encoding in such subsystem/subspace is challenging for large n in general. We also worked out a scheme, with which encoding/decoding quantum circuit can be implemented recursively by reducing the asymptotic encoding efficiency to 1/d.

This talk is based on works done in collaboration with

This talk is based on works done in collaboration with

__Utkan__Güngördü, Chi-__Kwong__Li,__Yiu__-__Tung____Poon__and__Nung__-Sing__Sze__.## Ognyan Oreshkov

__Causal and causally separable processes__

The possibility for indefinite causal structures in quantum theory and more general probabilistic theories has recently attracted a lot of interest, both from a foundational perspective and in the context of information processing. The process framework for local experiments without predefined causal order provides a concrete paradigm within which this phenomenon can be formally studied. Although significant progress has been made in understanding violations of causality in this framework, a general theory of causality in it has so far been lacking. In this talk, I will present such a theory which includes the multipartite case and captures the possibility for dynamical causal relations, where the order of a set of local experiments can depend on other experiments performed in the past. I will develop both the purely operational, theory-independent concept of causality, as well as different device-dependent notions, referred to as causal separability, which are of particular interest in the context of quantum theory.

## Masanao Ozawa

__Quantum theoretical generalization of the root-mean-square__

error for measurements

error for measurements

The notion of root-mean-square (rms) error for measurements, originally introduced by Gauss, is a well-established notion for the mean error of a measurement in classical physics. A generalization to quantum measurements was introduced as the rms of the noise operator in the recent study of the universal uncertainty principle. This talk will review its basic properties with a historical survey and respond to a recent criticism that this notion is useful only for a limited class of measurements based on comparison with distances between probability distributions. In particular, we introduce the following requirements for any quantum mechanical generalizations of the classical root-mean square error (quantum rms errors):

(i) Device-independent definability. The error measure should be definable by the POVM of the measuring apparatus, the observable to be measured, and the state of the object.

(ii) Correspondence principle. The error measure should be identical with the classical rms error in the classical case where the POVM and the observable commute.

(iii) Soundness. The error measure should take the value zero for any error-free measurements.

It is shown that the rms of the noise operator satisfies all the above requirements, whereas any error measures based on the distance of probability distributions, such as Wasserstein metrics, satisfy (i) and (iii) but do not satisfy (ii). This suggests that the criticism stems from an improper comparison between those error notions.

It is also shown that there are modifications of the rms of the noise operator that in addition to (i) - (iii) satisfy the following condition:

(iv) Completeness. The error measure should take the value zero if and only if the measurement is error-free.

We also show that universal uncertainty relations shown for the rms of the noise operator are also satisfied by a complete quantum root-mean-square error. Thus, we conclude that the recent criticism against state-dependent formulations of measurement uncertainty relations is not tenable.

(i) Device-independent definability. The error measure should be definable by the POVM of the measuring apparatus, the observable to be measured, and the state of the object.

(ii) Correspondence principle. The error measure should be identical with the classical rms error in the classical case where the POVM and the observable commute.

(iii) Soundness. The error measure should take the value zero for any error-free measurements.

It is shown that the rms of the noise operator satisfies all the above requirements, whereas any error measures based on the distance of probability distributions, such as Wasserstein metrics, satisfy (i) and (iii) but do not satisfy (ii). This suggests that the criticism stems from an improper comparison between those error notions.

It is also shown that there are modifications of the rms of the noise operator that in addition to (i) - (iii) satisfy the following condition:

(iv) Completeness. The error measure should take the value zero if and only if the measurement is error-free.

We also show that universal uncertainty relations shown for the rms of the noise operator are also satisfied by a complete quantum root-mean-square error. Thus, we conclude that the recent criticism against state-dependent formulations of measurement uncertainty relations is not tenable.

## Prakash Panangaden

__Topology, Order and Causal Structure__

The causal structure of

In a remarkable example of serendipity, order theory has been developed by computer scientists and mathematicians in order to capture

We applied domain theory to the problem of reconstructing the

We prove that a globally hyperbolic

In this talk I will not assume any prior familiarity with domain theory and will not say anything about computation. This was joint work with

__spacetime__defines a (pair of) natural order structures on the underlying set of events. Much of the analysis of__causal__structure involves a delicate interplay between order, topology and geometry. In view of the fundamental role of the causal order in certain approaches to quantum gravity one can ask whether the topology can be derived from pure order theoretic considerations. Usually one thinks of topology as being about continuity and of order as being quintessentially discrete.In a remarkable example of serendipity, order theory has been developed by computer scientists and mathematicians in order to capture

__computability__concepts. Dana Scott developed a notion of a continuous lattice or continuous__poset__with a view to capturing__computability__as continuity with a suitable topology that has come to be known as the Scott topology. This subject has acquired the somewhat odd name of ``domain theory.''We applied domain theory to the problem of reconstructing the

__spacetime__topology from the order and came up with a number of results about reconstruction of__spacetime__structure from just a countable dense set.We prove that a globally hyperbolic

__spacetime__with its causality relation is a__bicontinuous____poset__whose interval topology is the manifold topology. From this one can show that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic__spacetime__in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic__spacetimes__belong to a category that is equivalent to a special category of domains called interval domains.In this talk I will not assume any prior familiarity with domain theory and will not say anything about computation. This was joint work with

__Keye__Martin of Naval Research Laboratories.## Xinhua Peng

__Controlling spin systems in complex, correlated environment: reachable sets, polarization enhancement, and quantum state engineering__

Precisely characterizing and controlling realistic open quantum systems is one of the most exciting frontiers in quantum sciences and technologies. Here, we present methods of approximately computing reachable sets for coherently controlled dissipative systems, which is very useful for assessing control performances. Using the theoretical results, we implement some tasks of quantum control in open systems: increasing polarization and preparing pseudo-pure states. Our work shows interesting and promising applications of environment-assisted quantum dynamics.

## Miklos Santha

__Linear time algorithm for quantum 2SAT__

A canonical result in complexity theory is that the 2SAT problem, the satisfiability of boolean formulae where every close has two literals, can be solved in linear time. In the quantum 2SAT problem, we are given a family of 2qubit projectors on a system of n qubits, and the task is to decide whether the Hamiltonian, which is the sum of the projectors, has a 0eigenvalue, or it is larger than inverse polynomial in the number of qubits. The problem is not only a natural extension of classical 2SAT to the quantum case, but is also equivalent to the problem of finding the ground state of 2local frustrationfree Hamiltonians of spin 1/2, a wellstudied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown a few years ago that quantum 2SAT has a classical polynomial time algorithm, the running time of his algorithm is O(n^4). We present a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.

Joint work with Itai Arad, Aarthi Sundaram and Shengyu Zhang.

Joint work with Itai Arad, Aarthi Sundaram and Shengyu Zhang.

## Valerio Scarani

__Device-independent self-testing__

Device-independent self-testing refers to the following remarkable fact: some statistics of outcomes of quantum measurements determine fully the state and the measurements being performed (up to local isometries, that is, ancillary systems and local unitaries). The most famous examples are the two self-testing criteria for the singlet, namely CHSH=2√2 and the Mayers-Yao correlations.

The original works dealt with self-testing of the singlet state and the corresponding measurements. Recent developments have focused on robustness (i.e. certification when the observed statistics deviate from the ideal case) and versatility (i.e. the application of self-testing to other states). Besides, self-testing is used as a theoretical primitive to prove the possibility of more complex tasks like randomness amplification and blind computing.

There is a lot of room for further research. We conjecture that all extremal points of the quantum set of correlations self-test some state, and that every pure state can be self-tested. Besides, self-testing certifies also the measurements, but comparatively little emphasis has been put on analysing this information. Finally, for really practical uses it's important to improve the robustness bounds, especially for the many-copy case.

The original works dealt with self-testing of the singlet state and the corresponding measurements. Recent developments have focused on robustness (i.e. certification when the observed statistics deviate from the ideal case) and versatility (i.e. the application of self-testing to other states). Besides, self-testing is used as a theoretical primitive to prove the possibility of more complex tasks like randomness amplification and blind computing.

There is a lot of room for further research. We conjecture that all extremal points of the quantum set of correlations self-test some state, and that every pure state can be self-tested. Besides, self-testing certifies also the measurements, but comparatively little emphasis has been put on analysing this information. Finally, for really practical uses it's important to improve the robustness bounds, especially for the many-copy case.

## Zidan Wang

__Realizing Universal Quantum Gates with Topological Bases in Quantum-simulated Superconducting Chains__

Exploration of exotic one-dimensional time-reversal-invariant topological superconductors has recently become a hot spot of condensed matter physics, due to its fundamental importance in physics and promising applications in fault-tolerant topological quantum computing. In this talk, I will first introduce a symmetry-protected hard-core boson quantum-simulator of highly nontrivial time-reversal-invariant spinful topological superconductors, noting that the most existing studies have only focused on the simulation of spinless fermionic models. By using the dispersive dynamic modulation approach developed by us, not only a faithful quantum-simulation of the above systems is achieved, but also a set of universal quantum gates are topologically constructed, while only one kind of specific topological quantum gate has been addressed for the well-know Kitaev-type models before. Moreover, physical implementation of our scheme with superconducting quantum circuits is also elaborated, showing its feasibility with the current technology.

This work was supported by the GRF (HKU7045/13P&HKU173051/14P) and the CRF (HKU8/11G) of Hong Kong.

This work was supported by the GRF (HKU7045/13P&HKU173051/14P) and the CRF (HKU8/11G) of Hong Kong.

## Howard Wiseman

__Experimentally manifesting quantum nonlocality.__

Bell’s 1964 theorem is (as I will discuss) that there are quantum phenomena that violate the joint assumptions of determinism and locality. Thus the data from an experimental Bell-inequality-violation does not in itself show nonlocality, in the sense Bell used the term in 1964. To conclude that there is nonlocality we would have to assume determinism (or something similar). To see nonlocality from experimental data we would have to somehow enforce a deterministic interpretation on the experiment. The best known deterministic interpretation of quantum mechanics is Bohm’s theory, involving particle trajectories. It was shown by me in 2007 that an experimenter can ‘discover’ such trajectories, with minimal assumptions about quantum physics, by making use of weak measurements. Now, we [1] have mapped out such trajectories, for the first time with an entangled state of two particles. These trajectories make the nonlocality manifest: the choice of action by the experimenter on one particle changes the trajectories of the other, distant, particle, and hence the distribution of results obtained by the other party. This experiment certainly does not prove that Bohmian mechanics is correct, or that nonlocality (in this sense) is at work in the world, but it does offer a compelling visualisation of that nonlocality when a deterministic interpretation is enforced on the experiment.

[1] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories”

Science Adv. 2, e1501466 (2016). DOI: 10.1126/science.1501466

[1] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories”

Science Adv. 2, e1501466 (2016). DOI: 10.1126/science.1501466

## Haidong Yuan

__Geometrical distance on quantum channels__

Quantifying the distance between quantum channels is an essential component for many quantum information processing tasks. In this talk I will present a geometrical distance on quantum channels which can be efficiently computed using semi-definite programming, and show how this distance determines the ultimate precision limit in quantum metrology and the minimum number of uses needed to perfectly discriminating two arbitrary quantum channels. This geometrical distance thus provides a framework for these two related fields. In particular I will show that the precision limit in quantum metrology can be seen as a manifestation of the distance between quantum channels, and the minimum number of uses needed for perfect discrimination between two quantum channels is exactly the counterpart of the precision limit in quantum metrology. With the connection provided by this geometrical distance we also show how useful lower bounds for minimum number of uses needed for perfect channel discrimination can be obtained via a bridge to the precision limit in quantum metrology.

## Man-Hong Yung

__Network Bifurcation Triggered by Quantum Entanglement__

Quantum entanglement is ubiquitous in many physical, chemical, and even biological systems where the role of entanglement in biochemical processes has become a vivid research topic. However, a quantitative understanding on how entanglement can impact biological or sociological processes remains a major challenge. As a step towards this goal, here we consider a physical model of evolutionary network where the interactions among the nodes are locally supplied with quantum entanglement. We found that the equilibrium state can ben drastically changed from one type of global demographic condensation to another, through a small change in the amount of entanglement around a critical value. To further investigate, we formulated a mean-field theory that describes the process as a dynamical phase transition with a critical value highly consistent with the results obtained from numerical simulation, indicating the universality of the bifurcation phenomenon in networks with different topologies. Our model can be extended to describe a broad range of applications, such as Bose-Einstein condensation, plasma oscillations, and kinetics of chemical reactions, suggesting the potential existence of the universal bifurcation mechanism beyond evolutionary networks.

## Shengyu Zhang

__Quantum nonlocality meets complexity theory__

Traditional quantum nonlocality focuses on what quantum entanglement can do beyond the classical shared randomness in generating random variables correlated with inputs. In a recent line of work, we studied a finer-grained version of this by taking the complexity into the consideration. This talk will define the complexity measures, survey some known results and raise some questions for future studies.