Chaos i Informacja Kwantowa - Archiwum

Extensive work is being done both to construct a full-scale quantum computer, and to design quantum algorithms to run on such a device. One of the major applications will be in the simulation of the dynamics of quantum systems, where exponential advantage over classical methods is expected. However, the Schrödinger equation is ultimately a particular linear, first-order differential equation with Hamiltonian structure, and so one could ask: to what degree does quantum advantage extend to more general differential equations? In this talk I will describe our current understanding and expectations for such quantum advantage. I will also present recent PsiQuantum results that provide the first explicit costing of solving a general linear ordinary differential equation on a quantum computer, and theoretical advances that show that stable classical systems can always be "fast-forwarded" with sub-linear simulation time. I will conclude by briefly discussing the prospects for simulating high-impact classical systems, such as in fluid dynamics and plasma physics

Approximate t-design are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of unitaries up to the order t. As such, they find numerous applications throughout quantum information, including randomized benchmarking, efficient estimation of properties of quantum states, decoupling, information transmission and quantum state discrimination. In this talk I will characterize how finite random gate-sets mimic the Haar measure. The talk will be based on: P. Dulian, A. Sawicki, A random matrix model for random approximate t-designs, IEEE Transactions on Information Theory, vol. 70, no. 4, pp. 2637-2654, (2024)

We introduce a method for characterising a universal qudit gate set, including controlled T gates. Our scheme, along with the associated gate set, is designed to estimate the average gate fidelity of a T gate within a randomised benchmarking framework. For the qubit case, our approach reduces to dihedral benchmarking. The feasibility of our scheme is substantiated by a celebrated mathematical identity; the identity links the k-th Bell number with a sum over integer partitions of an integer n greater than k. Our scheme completes the characterisation of a universal gate set, a result particularly relevant for gates acting on encoded qudits. Additionally, our gate set supports the extension of interleaved benchmarking and shadow estimation, techniques essential for assessing other performance metrics.

What are the fundamental limits and advantages of using a catalyst to aid thermodynamic transformations between quantum systems? In this seminar we will consider the role of catalysts in quantum thermodynamics with particular focus on thermal operations, focusing on their limits and advantages in facilitating transformations between quantum systems. We investigate transformations among energy-incoherent states under various energy-conserving interactions, with the catalyst returning unperturbed and uncorrelated. Using the framework of thermomajorization, we identify the set of states accessible through catalytic processes, establish lower bounds on catalyst dimensionality under thermal processes, and quantify the catalytic advantage in terms of volume of the future cone. Finally, we apply our findings to generation of entanglement under thermal operations, demonstrating the advantage of a catalyst in such a setting. The talk will be based on a work under the same title, available on ArXiv: https://arxiv.org/abs/2403.04845

Models of quantum many-body dynamics for which quantities such as correlation functions and spectral statistics can be exactly computed are rare. In the recent past there has been growing interest in building models that retain essential features of generic systems, and are solvable in a sense that several physically relevant quantities can be computed exactly. In this talk, I will discuss solvable models that are discrete both in space and time using generalized dual unitary circuits. These are locally interacting quantum circuits built-out of special two-qubit/qudit unitary gates that treat space and time on an equal footing. I will illustrate how this spacetime duality leads to solvability in generalized dual unitary circuits, and discuss some constructions of two-qubit/qudit unitary gates that are used to build these circuits.

In 2008 Hastings provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy. The example will be based on a random unitary channel which will be introduced and discussed during the talk. We will focus on the details of the Hastings’ construction, the connection with other additivity conjectures and optimality of the parameters. Talk will be based on several references and the central place is given to “Comments on Hastings’ Additivity Counterexamples” by D.Moser, C.King, and M.Fukuda.

Quantum computational complexity measures the difficulty of performing a quantum mechanical task in terms of some elementary steps. For example, in a quantum circuit picture, complexity can be a measure of the optimal number of elementary quantum gates needed to prepare a final state from a simple initial state. I will discuss a new notion known as the Krylov complexity, a natural measure for the growth of an operator (Heisenberg picture) or a state (Schrodinger picture) along quantum mechanical time evolution. It measures how quickly an initial operator or state scrambles in the Hilbert space. By looking at the behaviour of the Krylov complexity with respect to time, we can learn about the underlying physics of a system. The system information input is the Hamiltonian. As I will discuss, the Krylov complexity can distinguish between an integrable and a chaotic evolution. It can also probe various phase transitions within a system. I will discuss how it compares with respect to other well-known probes of chaos, for example, the out-of-time-ordered correlated variables and level statistics. Finally, I will briefly mention some of our works in this direction, generalising the ideas to dissipative open and measurement-induced quantum systems.

A rank of a tensor is analyzed in context of quantum entanglement. We define and discuss various notions of tensor ranks: generic, maximal and border ones, and review selected results for the low dimensions. A relation between different ranks and norms of a tensor and the entanglement of the corresponding quantum state is presented.

I will discuss quantum entanglement in ground states of inhomogeneous quantum chains, and the physics that results from the interplay of the inhomogeneity with defects, phase transitions, and topological phases of matter by proposing different inhomogeneous models. Their characterization must be done in terms of the strength of the inhomogeneity, so I will present and compare the strong and weak inhomogeneity regimes, which respectively are obtained with real-space renormalisation group schemes or with an appropriate continuum limit of the lattice model.

I am going to discuss the results of the entanglement entropy of proton's constituents as have been obtained recently within Kharzeev-Levin approach. The resulting entropy formula can be shown to describe measured hadronic entropy data as collected in inclusive and diffractive Deep Inelastic Scattering. The talk is based on the following papers: Eur. Phys. J. C 82, 111 (2022) Eur. Phys. J. C 82, 1147 (2022) Phys. Rev. Lett. 131, 241901

Bipartite bound entanglement represents a peculiar and fascinating quantum phenomenon that sits at the intersection of quantum information theory and quantum entanglement. Bound entangled states are characterized by the fact that they require entanglement to be created, yet paradoxically, no entanglement can be distilled from them using only local operations and classical communication (LOCC). This means that while they possess correlations that are non-classical, extracting and using those correlations for quantum informational tasks is highly non-trivial. Its discovery has profound implications for our understanding of quantum entanglement's structure and the operational boundaries of quantum information processing, challenging researchers to find novel ways to harness such entangled states for practical applications. In this talk we will summarize recent findings, such as the frequency of bound entanglement, but also dive into the substructure of entanglement and discuss its role in quantum teleportation.

This work examines the problem of learning an inversion of unknown unitary transformation of dimension d from a finite number of copies of a given unitary channel. We focus of the probabilistic approach, where we postprocess the output of this procedure and require that it is exactly equal to the inverse operation of the input. The goal is to calculate the maximal probability of successful inversion calculated over all possible parallel learning schemes.

One-way quantum computers (1WQC) use reversible, unitary evolution and treat boundary conditions in asymmetric way: allowing to fix only the initial states by state preparation. We discuss their hypothetical two-way enhancement (2WQC) adding CPT symmetry analog of such state preparation to fix some final states with physical constraints – analogs like pull/push, negative/positive pressure (e.g. radiation), stimulated emission/absorption enforcing excitation/ deexcitation. For hydrodynamics realizations it could be done by connecting such chip into a circuit with pump: both pushing into with positive pressure, and pulling from with negative pressure. Mathematically hydrodynamics is governed by similar wave-like equation as electromagnetism. Hence I focus on photonic quantum computers, with unidirectional ring laser acting as pump for photons. If successful, thanks to a better control of information flow, analogously to Shor algorithm, such 2WQC could attack NP-complete problems.

In the present work we find tests of real separability in contrast to complex separability. This implies an improved tool for ascertaining, based on the experimental outcome of Bell inequalities, whether quantum physics based solely on the field of real numbers is possible or not.

The transfer of quantum information between different locations is key to many quantum information processing tasks. Whereas, the transfer of a single qubit state has been extensively investigated, the transfer of a many-body system configuration has insofar remained elusive. We address the problem of transferring the state of n interacting qubits [1]. Both the exponentially increasing Hilbert space dimension, and the presence of interactions significantly scale-up the complexity of achieving high-fidelity transfer. By employing tools from random matrix theory and using the formalism of quantum dynamical maps, we derive a general expression for the average and the variance of the fidelity of an arbitrary quantum state transfer protocol for n interacting qubits. We find that the average fidelity decreases with the amount and the type of entanglement in the sender state [2]. Finally, by adopting a weak-coupling scheme in a spin chain, we obtain the explicit conditions for high-fidelity transfer of 3 and 4 interacting qubits. [1] Tony J G Apollaro et al, Quantum transfer of interacting qubits, 2022 New J. Phys. 24 083025 https://iopscience.iop.org/article/10.1088/1367-2630/ac86e7 [2] Tony J G Apollaro et al, Entangled States Are Harder to Transfer than Product States, Entropy 2023, 25(1), 46; https://doi.org/10.3390/e25010046

Catalysis plays a key role in many scientific areas, most notably in chemistry and biology. In this talk, I will present a catalytic process in a paradigmatic quantum optics setup, namely the Jaynes-Cummings model, where an atom interacts with an optical cavity. The atom plays the role of the catalyst, and allows for the deterministic generation of non-classical light in the cavity. Considering a cavity prepared in a "classical'' coherent state, and choosing appropriately the atomic state and the interaction time, we obtain an evolution with the following properties. First, the state of the cavity has been modified, and now features non-classicality, as witnessed by sub-Poissonian statistics or Wigner negativity. Second, the process is catalytic, in the sense that the atom is deterministically returned to its initial state exactly, and could then in principle be re-used multiple times. We investigate the mechanism of this catalytic process, in particular highlighting the key role of correlations and quantum coherence.

A method for the detection of the energy levels of arbitrary spin systems on a quantum device will be presented. The method is based on studies of the evolution of a probe spin. The results of quantum calculations of the energy levels of spin chain in a magnetic field, triangle spin cluster will be discussed. The results of observation of the spin-1 tunneling and splitting of energy levels as a result of tunneling on IBM's quantum computer will be analyzed. The energy level splitting is observed on the basis of the proposed quantum algorithm. Geometric properties of the evolutionary quantum graph states generated by the operator of evolution with Ising Hamiltonian will be discussed. Namely, the curvature and the torsion will be considered. We obtain that the geometric properties of the graph states are related to the total number of edges, triangles, and squares in the corresponding graphs. On the basis of this result algorithm for the detection of edges, triangles, and squares in a graph on a quantum device is proposed. It is also found that another property of graphs, the vertex degree is related to the geometric measure of entanglement of a spin with other spins in the graph state. We quantify the geometric measure of entanglement of quantum graph states on IBM's quantum computer.

Graph states are a cornerstone of quantum information theory. Existing invariants characterizing the local Clifford (LC) equivalence classes of graph states are however computationally inefficient and call for a more tractable approach. This paper introduces the foliage partition, an easy-to-compute LC-invariant of computational complexity O(n^3) in the number of qubits. Inspired by the foliage of a graph, our invariant has a natural graphical representation in terms of leaves, axils, and twins. It captures both, the connection structure of a graph and the $2$-body marginal properties of the associated graph state. We relate the foliage partition to the size of LC-orbits and use it to bound the number of LC-automorphisms of graphs. We also show the invariance of the foliage partition when generalized to weighted graphs and qudit graph states. Joint work with Frederik Hahn.

A multipartite quantum system consisting of two or more components/subsystems that interact with each other results in entanglement among the subsystems. In this talk, I focus on a special class of multipartite pure quantum states describing a multipartite system in which all subsystems are maximally mixed and are called absolutely maximally entangled (AME) states. The existence of AME states is not known in general and a surprising result concerning two-level quantum systems or qubits is that there is no AME state of four qubits. AME states of four qudits (d-level quantum systems) can be constructed from combinatorial designs called orthogonal Latin squares that exist for all sizes except two and six. The non-existence of orthogonal Latin squares of size six is the reason that the 36 officers problem of Euler has no solution. Despite this negative result, an AME state of four quhexes (six-level systems) can be constructed and I discuss the constructions of AME states beyond (classical) orthogonal Latin squares. Based on a notion of equivalence, it is shown that for four qutrits (three-level systems) there is only one AME state. However for ququads (four-level system) and beyond, including quhexes, uncountably infinite AME states exist that are not equivalent.

I will review different kind of uncertainty relations and their significance in Quantum Information, e.g. for entanglement and non-locality characterization. Then I will explain how to obtain (theoretically tight) bounds for these relations using the framework of state polynomial optimization.

In Quantum Computing one of the big problems it is to correct errors. We will see why the techniques used in classical computation does not work in quantum and more precisely, how can we correct errors that appear in bursts.

Universal quantum gates play a central role in quantum computing. It is well known that in order to construct a universal set of gates for many qudits it is enough to take a universal set for one qudit and extend it by a two-qudit entangling gate. On the other hand, it is a great challenge to find a time efficient procedure that enables deciding if a given set of one-qudit gates is universal. In this talk I will connect the universality problem with the theory of t-designs and provide a universality checking procedure whose complexity scales polynomially with the dimension of the qudit. The talk will be based on: A. Sawicki, L. Mattioli, Z. Zimboras Phys. Rev. A 105, 052602, 2022

Quantifying quantum states' complexity is a key problem in various subfields of science, from quantum computing to black-hole physics. I will present two approaches to understand the behavior and the operational significance of quantum complexity in a many-body quantum system. First, I'll show that the quantum complexity of a random circuit grows linearly in time until saturating at a value that is exponential in the number of qubits, proving a version of a conjecture by Brown and Susskind in the context of quantum gravity. Second, I'll show that the necessary resources to reset an n-qubit memory register to the pure all-zero computational basis state exhibits a trade-off between complexity cost and thermodynamic work cost. This trade-off is governed by a new entropy measure that quantifies the amount of randomness that a system appears to have to an observer whose observations are computationally limited. Beyond applications in many-body physics, our methods help us understand the set of quantum states that can be reached with a fixed number of gates, with anticipated applications to quantum pseudorandomness, variational quantum machine learning, quantum error correction, and tasks in quantum information and quantum thermodynamics subject to computational limitations.

Hybrid codes simultaneously encode both quantum and classical information together, giving an advantage over coding schemes where the quantum and classical information are transmitted separately. We construct the first known families of hybrid codes that are guaranteed to provide an advantage over quantum codes, as well as also giving a construction of hybrid codes from subsystem codes that allow for different minimum distances for the encoded quantum and classical information . We also show how hybrid codes can be applied to the problem of faulty syndrome measurements and lead to the construction of new quantum data-syndrome codes.

We introduce a family of multipartite entanglement measures called "concentratable entanglements". These are connected to previously defined entanglement measures and have an operational interpretation in terms of probabilistic concentration of entanglement into Bell pairs. Furthermore, we show that these quantities can be estimated on a quantum computer by implementing a parallelized SWAP test, opening a research direction for measuring multipartite entanglement measures on quantum devices.

Device-independent quantum key distribution (DIQKD) constitutes the most pragmatic approach to quantum cryptography that does not put any trust in the inner workings of the devices. This is possible by constructing security proofs at the level of correlations being shared by the end-users, leveraging from the phenomenon of Bell nonlocality. In particular, quantum nonlocality allows one then to lower-bound the asymptotically achievable key rates, even in the presence of the most general eavesdropping attacks. However, only recently first proof-of-principle implementations of DIQKD have been demonstrated, as the device-independent framework imposes very stringent requirements on the noise tolerance, even in the absence of any eavesdroppers. In our works, we follow a complementary approach in which we propose an easy-to-optimise attack on any DIQKD protocol, with help of which we construct upper bounds on the asymptotic key. On one hand, it allows us to disprove a long-standing conjecture that any form of Bell nonlocality is sufficient for distributing secret keys in a device-independent manner. On the other, it allows us to verify that current state-of-the-art implementations already operate very close to the ultimate noise thresholds, and cannot be thus improved by resorting to more profound security-proof techniques.

The quantum marginal problem consists in deciding whether a given set of marginal reductions is compatible with the existence of a global quantum state or not. In this work, we formulate the problem from the perspective of dynamical systems theory and study its advantages with respect to the standard approach. The introduced formalism allows us to analytically determine global quantum states from a wide class of self-consistent marginal reductions in any multipartite scenario. In particular, we show that any self-consistent set of multipartite marginal reductions is compatible with the existence of a global quantum state, after passing through a depolarizing channel. This result reveals that the complexity associated with the marginal problem can be drastically reduced when restricting the attention to sufficiently mixed marginals. We also formulate the marginal problem in a compressed way, in the sense that the total number of scalar constraints is smaller than the one required by the standard approach. This fact suggests an exponential speedup in runtime when considering semi-definite programming techniques to solve it, in both classical and quantum algorithms. Finally, we reconstruct n-qubit quantum states from all the $\binom{n}{k}$ marginal reductions to k parties, generated from randomly chosen mixed states. Numerical simulations reveal that the fraction of cases where we can find a global state equals 1 when $5\leq n\leq12$ and $\lfloor(n-1)/\sqrt{2}\rfloor\leq k\leq n-1$, where $\lfloor\cdot\rfloor$ denotes the floor function.

The ability to generate, modify, and retrieve a quantum state is of paramount importance for quantum information. Conventional physical implementations of the schemes, enabling realization of the tasks, employ single microscopic objects (atoms, photons, superconducting circuits, etc.). However, operation with such objects presents many experimental challenges. In the seminar, an alternative approach, enabling realization of quantum-state engineering and tomography using the collective state of many atoms (10^9), will be presented.

The discovery of holographic codes established a surprising connection between quantum error correction and the AdS/CFT correspondence. Recent technological progress in artificial quantum systems renders the experimental realization of such holographic codes now within reach. Formulating the hyperbolic pentagon code in terms of a stabilizer graph code, we propose an experimental implementation that is tailored to systems with long-range interactions. We show how to obtain encoding and decoding circuits for the hyperbolic pentagon code [Pastawski et al., JHEP 2015:149 (2015)], before focusing on a small instance of the holographic code on twelve qubits. Our approach allows to verify holographic properties by partial decoding operations, recovering bulk degrees of freedom from their nearby boundary.

The standard dynamical approach to quantum thermodynamics is based on Markovian master equations describing the thermalization of a system weakly coupled to a large environment. Here, based on the newly introduced notion of continuous thermomajorization, we obtain necessary and sufficient conditions for the existence of such a thermalization process transforming between given initial and final energy populations of the system. These include standard entropy production relations as a special case, but also yield a complete, continuous family of entropic functions that need to monotonically increase during the dynamics. Importantly, we significantly simplify these conditions by identifying a finitely verifiable set of constraints governing non-equilibrium transformations under master equations. What is more, the framework is also constructive, i.e., it returns an explicit set of elementary controls realizing any allowed transformation. Finally, we also present an algorithm that in a finite number of steps allows one to construct all energy distributions achievable from a given initial state via Markovian thermalization processes, and illustrate its usefulness by employing it to investigate the role of non-Markovianity in important thermodynamic protocols.

Probabilistic quantum error correction (pQEC) is an error-correcting procedure which uses postselection to determine if the encoded information was successfully restored. In this talk, I will show an application of pQEC for various types of noise channels, e.g. Schur channels, channels with bounded Choi rank and random channels. I will present how to correct errors caused by an arbitrary unitary interaction with an auxiliary qubit system. Finally, I will discuss the optimization problem related with pQEC procedure.

The quantum degree of freedom of the rotor is the only one that pairs a continuous, periodic position with a discrete momentum (azimuth and angular momentum). For all other quantum degrees of freedom, the two basic complementary observables are either both discrete or both continuous. Standard procedures, such as those for constructing exhaustive sets of pairwise complementary observables or for introducing an analog of Wigner's phase space function, do not work for the quantum rotor; modified procedures are required and available.

Jensen–Shannon divergence is an important distinguishability measure between probability distributions that finds interesting applications within the context of Information Theory. In particular, this classical divergence belongs to a remarkable class of divergences known as Csiszár or f -divergences. In this talk I analyze the problem of obtaining a distance measure between two quantum states starting from the classical Jensen–Shannon divergence between two probability distributions. Considering the Jensen–Shannon divergence as a Csiszár divergence, I first focus on the problem of distinguishability between two pure quantum states. It is found a quantum version of the classical Jensen–Shannon divergence that differs from the previously introduced Quantum Jensen–Shannon Divergence. The two quantum versions of Jensen–Shannon divergence have different interpretations within the framework of Quantum Information Theory. Whereas the former quantum version of Jensen–Shannon divergence can be interpreted as the Holevo bound, the alternative quantum version obtained in this work equals the accessible information. Furthermore, it is obtained a monoparametric family of metrics between two quantum pure states. Finally, it is presented an extension of this family of metrics to the case of mixed quantum states by means of the concept of purification

In this talk I will motivate Local Operations and Shared Randomness (LOSR) as a paradigm to quantify non-classical resources, in contrast to Local Operations and Classical Communication (LOCC). I will provide examples of the resource-theoretic study of entanglement, Bell non-classicality, and Einstein-Podolsky-Rosen steering, that stems from this LOSR approach. In particular, I'll discuss how this triggers a whole distinct new branch of entanglement theory.

We provide a proof that entanglement of any density matrix which block diagonal in subspaces which are disjoint in terms of the Hilbert space of one of the two potentially entangled subsystems can simply be calculated as the weighted average of entanglement present within each block. This is especially useful for thermal-equilibrium states which always inherit the symmetries present in the Hamiltonian, since block-diagonal Hamiltonians are common as are interactions which involve only a single degree of freedom of a greater system. We exemplify our method on a simple Hamiltonian, showing the diversity in possible temperature-dependencies of Gibbs state entanglement which can emerge in different parameter ranges.

Recently there appeared many works on modified Wigner's Friend paradoxes, which suggest that quantum theory cannot consistently describe the scenario with many observers. In this presentation I will show an alternative approach to this problem, which indicates that the paradoxes are in fact apparent, and the source of confusion is the undefined status of the measurement process. The talk will be based on recently published work "Physics and Metaphysics of Wigner’s Friends: Even Performed Premeasurements Have No Results" by Marek Żukowski and Marcin Markiewicz, Phys. Rev. Lett. 126, 130402 (2021).

We introduce operational distance measures between quantum states, measurements, and channels based on their average-case distinguishability. To this end, we analyze the average Total Variation Distance (TVD) between statistics of quantum protocols in which quantum objects are intertwined with random circuits and subsequently measured in a computational basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects, which we call average-case distances. The so-defined distances capture average-case distinguishability via moderate-depth random quantum circuits and satisfy many natural properties. We apply them to analyze the effects of noise in quantum advantage experiments and in the context of efficient discrimination of high-dimensional quantum states and channels without quantum memory. Furthermore, based on analytical and numerical examples, we argue that average-case distances are better suited for assessing the quality of NISQ devices than conventional distance measures such as trace distance and the diamond norm. The talk is based on recent preprints: arXiv:2112.14283 and arXiv:2112.14284.

Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example is the Bloch ball, the geometrical representation for the state space of a two-level system (or qubit).However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this talk we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter.

Magic states are key ingredients in schemes to realize universal fault-tolerant quantum computation. Theories of magic states attempt to quantify this computational element via monotones and determine how these states may be efficiently transformed into useful forms. Here, we develop a statistical mechanical framework based on majorization to describe Wigner negative magic states for qudits of odd prime dimension processed under Clifford circuits. We show that majorization allows us to both quantify disorder in the Wigner representation and derive upper bounds for magic distillation. These bounds are shown to be tighter than other bounds, such as from mana and thauma, and can be used to incorporate hardware physics, such as temperature dependence and system Hamiltonians. We also show that a subset of single-shot Rényi entropies remain well-defined on quasi-distributions, are fully meaningful in terms of data processing and can acquire negative values that signal magic.

I will explain how to compute the average with respect to the Haar measure of polynomial functions on compact matrix groups. I do not assume any knowledge beyond the existence of the Haar measure, the definition of tensor products, and elementary linear algebra. Time allowing, I will discuss the large n dimension behavior and applications to random matrix theory or quantum information theory.

I'm going to discuss some recent work on tensor network decoders for arbitrary 2D Pauli codes. Tensor network (TN) decoders are a powerful framework that allows one to systematically construct decoders that well-approximate the optimal decoder for large families of codes. I am going to talk about some recent work demonstrating that TN decoding for several families of 2D Pauli codes can not only be performed efficiently, but can yield state-of-the-art accuracies, in some cases out- performing bespoke code-specific decoders. Core to this is a novel tensor network contraction algorithm inspired by computational geometry methods, which I will also talk about.
See more: http://arxiv.org/abs/2101.04125 and https://github.com/chubbc/SweepContractor.jl

The aim of a quantum memory is to protect an encoded quantum state against errors for long periods of time. In this talk, I will consider the effect that temperature has on states encoded in the degenerate ground states of two dimensional models, such as the toric code. I will explain how to estimate the lifetime of such memories, and present some negative results: for models belonging to a very large class, the resulting memory is not very good, in the sense that their lifetime at finite temperature is bounded and cannot be extended arbitrarily.

Generalized probabilistic theories (GPT) give us a framework to describe the operational features of classical and quantum physics, in particular, their state spaces and measurement processes. Within this approach we define the morphophoric measurements with the property that the measurement map transforming states into distributions of the measurement results is a similarity. In quantum case, SIC-POVMs or, more generally, 2-design POVMs belong to this class. We give some examples of morphophoric measurements for other GPTs and try to answer the question to what extent the theory of these measurements retains the chief features of the QBism approach to the basics of quantum mechanics.

The concept of deformation of Riemannian geometry is reviewed, with applications to quantum mechanics. The deformation of a metric is ruled by the Ricci flow equation. Associated quantities are the Perelman’s Functional and the Perelman’s Entropy, which also are discussed in the context quantum systems. The Ricci flow equation has an interesting meaning for certain kind of metrics. One of them is the Fubini-Study metric, which is a relevant Riemannian metric in the realm of quantum mechanics. We analyze the meaning of deformation for a particular example in which the Fubini-Study metric can be explicitly evaluated. Finally we present a general approach to the study of quantum dynamics from the Ricci flow equation.

Though the celebrated Gorini-Kossakowski-Lindblad-Sudarshan equation can describe a wide range of evolutions, and not all of them, one may almost always find a quantum channel that maps the initial state of an open system to its state at time t. In this sense, quantum channels cover a broader class of evolutions rather than Markovian ones. The problem of Markovianity then asks for a given channel whether there exists a Markovian dynamical semigroup passing through the assumed channel at some time t. We will review the problem of Markovianity and its slightly different version; the Accessibility problem.

Methods for dissipatively engineering open quantum dynamics have both a fundamental significance from a general control-theoretic standpoint and are finding practical application across a growing range of quantum information processing tasks. In this talk, I will showcase some of the challenges and opportunities that arise in this broad context, by focusing on Markovian quantum dynamics. Specifically, I will first consider the task of quantum state stabilization and show that, surprisingly, the set of pure states that are stabilizable by quasi-local Markovian dynamics does not coincide with the set of pure states that are unique ground states of a Hamiltonian, subject to the same locality constraints. Next, I will formalize the task of dissipative quantum encoding and point to possible advantages that dissipative schemes may afford in terms of extra robustness against initialization errors. Time permitting, I will outline constructions of dissipative quantum encoders for stabilizer quantum error-correcting codes.

A study of the measure of not completely positive maps brings surprises, even in the qubit case. Non-Markovianity, defined by the completely positive-indivisibility of the intermediate map of a given noisy channel, accentuates this surprise in terms of the non-convexity of the set of (non-)Markovian channels, and also in the counterintuitive fact that mixing channels that deviate to a larger degree from quantum dynamical semigroup structure can result in a larger measure of Markovian channels. In particular, this behavior is seen in the case of mixing the three Pauli qubit channels. The implication of such a result for the physical significance of quantum memory and a resource theory of quantum non-Markovianity is discussed.

Quantum Bayesian Networks (QBNs) provide a method for obtaining the statistics of multipoint quantities in quantum coherent systems, while avoiding the measurement backaction. These quantities play a particularly important role in quantum thermodynamics, since concepts such as heat and work do not depend on the state of a system, but rather on the process it undergoes. Although QBNs cannot be directly measured from a single experiment, in this talk we show how they can be recovered from experiments on identical copies of a system, via measurement post-selection. The potential applications of this tool to quantum thermodynamics are also discussed. In particular, we report on a recent experiment which uses QBNs for determining the full statistics of the heat exchanged between two quantum correlated bodies.

Quantum catalysis is a fascinating concept that demonstrates how certain transformations can only become possible when given access to a specific resource that has to be returned unaffected. It was first discovered in the context of entanglement theory, and since then, it has been applied in a number of resource-theoretic frameworks, including quantum thermodynamics. Although, in that case, the necessary (and sometimes also sufficient) conditions on the existence of a catalyst are known, almost nothing is known about the precise form of the catalyst state required by the transformation. In particular, it is not clear whether it has to have some special properties or be finely tuned to the desired transformation. In this work, we describe a surprising property of multicopy states: We show that in resource theories governed by majorization, all resourceful states are catalysts for all allowed transformations. In quantum thermodynamics, this means that the so-called “second laws of thermodynamics” do not require a fine-tuned catalyst; rather, any state, given sufficiently many copies, can serve as a useful catalyst. These analytic results are accompanied by several numerical investigations that indicate that neither a multicopy form nor a very-large-dimension catalyst is required to activate most allowed transformations catalytically.

A quantum analogon of the famous problem of 36 officers of Euler is presented. Although it is well known that the classical problem has no solutions, we found an explicit analytic solution of the quantum version of this problem, in which officers are allowed to be entangled. This unexpected result provides constructive solutions to the related problem of the existence of absolutely maximally entangled state AME(4,6) of four subsystems with six levels each. The existence of such a state is equivalent to the existence of a 2-unitary matrix of size 36, or a perfect tensor with four indices, each running from one to six, or a pure quantum error correction code $((4,1,3))_6$. Our result sheds some light on how to construct non-additive quantum error correction codes when the stabilizer approach fails.

SIC-POVMs are generalised quantum measurements which are of particular interest in the context of quantum state tomography and quantum key distribution. Alternatively, they can be described by d^2 normalised vectors in the d-dimensional complex vector space such that the inner product between any pair of vectors has constant modulus. It has been conjectured that SIC-POVMs exist for all dimensions and that they can be constructed as orbits of a so-called fiducial vector under the Weyl-Heisenberg group. Despite a lot of effort, numerical or exact fiducial vectors are only known for a finite, though growing list of dimensions. Currently, numerical ones have been found for all dimensions up to 193. Solutions in larger dimensions have been found as part of conjectured families obeying additional symmetries. After an introduction to SIC-POVMs, the talk will highlight the methods used to find numerical and exact SIC-POVMs. Links to number-theoretic conjectures in this context allow to convert numerical solutions of moderate precision into exact solutions, including dimensions 844 and 1299. We will also briefly address recent results obtained in collaboration with Marcus Appleby, Ingemar Bengtsson, Michael Harrison, and Gary McConnell.

We identify a fundamental difference between classical and quantum dynamics in the linear response regime by showing that the latter is in general contextual. This allows us to provide an example of a quantum engine whose favorable power output scaling unavoidably requires nonclassical effects in the form of contextuality. Furthermore, we describe contextual advantages for local metrology. Given the ubiquity of linear response theory, we anticipate that these tools will allow one to certify the nonclassicality of a wide array of quantum phenomena.

We use binary linear codes to store and transmit classical information. These are length one chain complexes over a finite field. Quantum (CSS) stabilizer codes can be seen as length two (or more) chain complexes over a finite field. There are no complex numbers to be seen here, so what is quantum about quantum codes? One possible answer is that these codes, as chain complexes, live in a symmetric closed monoidal category enriched over itself. This also applies to the category of finite dimensional complex vector spaces, the usual context in which we do quantum information theory. In this talk I will discuss these parallels using string diagram notation. As might be suspected, they key is the behaviour of the tensor product in both contexts.

The optimal transport problem, established by Monge and refined by Kantorovich and Wasserstein, has ubiquitous applications in statistics, machine learning, computer vision and early Universe reconstruction. Recently, several approaches towards its quantum version have been developed. In the talk I will provide an overview of the quantum optimal transport problem basing on the recent joint work (arXiv:2102.07787). The general results will be illustrated with a more detailed study of the single-qubit transport problem. In particular, I will show that the quantum optimal transport induces a new metric on the Bloch ball with intriguing properties.

Coherent errors and cross-talk are two fundamental obstacles in the development of Quantum Technologies, and especially Quantum Computing. Full channel tomography is exponentially hard, and moreover suffers from state-preparation and measurement errors. To circumvent such obstacles powerful techniques have been developed in the field of Randomized Benchmarking in which one can estimate key features of a given quantum channel, such as its average gate fidelity. To supplement average gate fidelity quantities such as the unitarity of a quantum channel have been proposed to quantify the coherent structure of a channel, however we still lack such a handle on the correlative aspects of quantum channels. In this talk I will present a unitarity-based measure of coherent correlations in quantum channels. This has the key advantage that it can be efficiently estimated experimentally and is tailored to NISQ-era devices where it could be used to determine the structure of 2-qubit gates, or to measure correlation errors. This correlated unitarity is also of interest from an abstract structural perspective, as it satisfies a range of desirable properties. For example it is invariant under local changes of basis, monotonic under local operations, provides a witness of non-separability in quantum channels, and provides a 'monogamy of unitarity' relation similar to the Information-Disturbance Theorem that is amenable to experimental verification.

In this talk, we investigate the detection of entanglement via inequivalent sets of MUBs (Mutually Unbiased Bases). These MUB-witnesses have turned out to be already successful in experiments, e.g. in detecting for the first time bipartite bound (=PPT) entanglement in a system of two photons entangled in orbital angular momentum. As we discovered recently, these MUB-witnesses have also lower bounds on the full set of separable states. In contrast to the upper bounds they depend strongly on the dimension d and on the set of MUBs. Surprisingly, unextendible sets of MUBs are more powerful in detecting entanglement than extendible sets of MUBs though they can obviously never be used for quantum state verification. Furthermore, these lower bounds allow for a systematic method to search for inequivalent MUBs and show that such sets occur regularly within the Heisenberg-Weyl MUBs, as the dimension increases.

In the last two decades, significant interest has been focused on the so-called AdS/CFT correspondence, which connects gravitational theories with conformal field theories. Even more recently, quantum information scientists have designed toy models of the AdS/CFT correspondence, in the form of tensor networks that give rise to quantum error correction codes. In particular, it was shown that an important feature of the AdS/CFT correspondence---the Ryu--Takayanagi formula that relates the entanglement entropy of the boundary of the network to the network geometry---holds in appropriately chosen tensor network codes. These toy models heavily rely on an important class of multipartite quantum states, absolutely maximally entangled (AME) states, that are useful not only for quantum codes, but also in secret sharing schemes, and are of generic interest in entanglement theory. It was already known that AME states give rise to quantum error correction codes, and the AdS/CFT toy models are based on the concatenation of such codes. In this talk, we will analyse the concatenation of quantum error correction codes based on AME states, and describe an algorithm for obtaining the explicit form of stabiliser generators and logical operators of the resulting codes. Using this algorithm, we will analyse the spread of quantum information in a specific AME-based tensor network code. We will calculate corrections to the Ryu--Takayanagi formula in the case of initial entanglement in the input quantum state, and find that in our specific case, AME states saturate a bound on these corrections. We conjecture that this saturation holds for generic tensor network codes, that is, AME states (when they exist) give the maximal value of the entanglement entropy of the boundary state.

Holevo's theorem constitutes a remarkable result within quantum information science: it establishes an upper bound for the accessible information, i.e. the amount of classical information encoded into a collection of quantum states. Additionally, in quantum theory states are not directly observable. Thus nowadays there exist a variety of different measures of distinguishability between quantum states. In this talk, we present a Holevo's theorem generalization through distance measures between quantum states, showing that each of these leads to an alternative Holevo inequality. Besides, we consider three distinguishability notions relevant to quantum cryptography, applying the resulting new inequalities to qubits ensembles. For some well-known distinguishability notion, we show that for any ensemble of two qubits the generalized quantities $I_d$ and $X_d$ are equal. Additionally, by using a paradigmatic example, we show that the corresponding quantities to the Bures distance are sensible to the non-commutativity of the ensemble but also its purity.

We present an efficient algorithm that solves the quantum state tomography problem from an arbitrary taken set of rank-one projective measurements in any finite dimension d. The algorithm is flexible enough to allow us to impose any desired rank r to the state to be reconstructed, ranging from pure (r=1) to full rank (r=d) quantum states. The method exhibits successful and fast convergence under the presence of realistic errors in both state preparation and measurement stages, and also when considering overcomplete sets of observables. We demonstrate that our method outperforms maximum likelihood quantum state estimation when reconstructing N qubit states from tensor product of Pauli bases, for N ranging from 2 to 7. Furthermore, our algorithm exhibits ultrafast convergence when considering measurements based on mutually unbiased bases