arXiv subjects

Michał Horodecki

Publications and source records attributed to Michał Horodecki.

At least 37 records · Page 2Linked to original sources

arXiv2023

The Josephson junction as a quantum engine

We treat the Cooper pairs in the superconducting electrodes of a Josephson junction (JJ) as an open system, coupled via Andreev scattering to external baths of electrons. The disequilibrium between the baths generates the direct-current bias applied to the JJ. In the weak-coupling limit we obtain a Markovian master equation that provides a simple dynamical description consistent with the main features of the JJ, including the form of the current-voltage characteristic, its hysteresis, and the appearance under periodic voltage driving of discrete Shapiro steps. For small dissipation, our model also exhibits a self-oscillation of the JJ's electrical dipole with frequency $Ω= 2 e V / \hbar$ around mean voltage $V$. This self-oscillation, associated with "hidden attractors" of the nonlinear equations of motion, explains the observed production of monochromatic radiation with frequency $Ω$ and its harmonics. We argue that this picture of the JJ as a quantum engine resolves open questions about the Josephson effect as an irreversible process and could open new perspectives in quantum thermodynamics and in the theory of dynamical systems.

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arXiv2023

An approximation scheme and non-Hermitian re-normalization for description of atom-field system evolution

Interactions between a source of light and atoms are ubiquitous in nature. The study of them is interesting on the fundamental level as well as for applications. They are in the core of Quantum Information Processing tasks and in Quantum Thermodynamics protocols. However, even for two-level atom interacting with field in rotating wave approximation there exists no exact solution. This touches as basic problem in quantum field theory, where we can only calculate the transitions in the time asymptotic limits (i.e. minus and plus infinity), while we are not able to trace the evolution. In this paper we want to get more insight into the time evolution of a total system of a two-level atom and a continuous-mode quantum field. We propose an approximation, which we are able to apply systematically to each order of Dyson expansion, resulting in greatly simplified formula for the evolution of the combined system at any time. Our tools include a proposed novel, {\it non-Hermitian} re-normalization method. As a sanity check, by applying our framework, we derive the known optical Bloch equations.

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arXiv2023

Correlations constrained by composite measurements

How to understand the set of correlations admissible in nature is one outstanding open problem in the core of the foundations of quantum theory. Here we take a complementary viewpoint to the device-independent approach, and explore the correlations that physical theories may feature when restricted by some particular constraints on their measurements. We show that demanding that a theory exhibits {a composite} measurement imposes a hierarchy of constraints on the structure of its sets of states and effects, which translate to a hierarchy of constraints on the allowed correlations themselves. We moreover focus on the particular case where one demands the existence of a correlated measurement that reads out the parity of local fiducial measurements. By formulating a non-linear Optimisation Problem, and semidefinite relaxations of it, we explore the consequences of the existence of such a parity reading measurement for violations of Bell inequalities. In particular, we show that in certain situations this assumption has surprisingly strong consequences, namely, that Tsirelson's bound can be recovered.

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arXiv2023

Autonomous Quantum Devices: When Are They Realizable without Additional Thermodynamic Costs?

The resource theory of quantum thermodynamics has been a very successful theory and has generated much follow-up work in the community. It requires energy-preserving unitary operations to be implemented over a system, bath, and catalyst as part of its paradigm. So far, such unitary operations have been considered a "free" resource in the theory. However, this is only an idealization of a necessarily inexact process. Here, we include an additional auxiliary control system which can autonomously implement the unitary by turning an interaction "on or off". However, the control system will inevitably be degraded by the backaction caused by the implementation of the unitary. We derive conditions on the quality of the control device so that the laws of thermodynamics do not change and prove - by utilizing a good quantum clock - that the laws of quantum mechanics allow the backreaction to be small enough so that these conditions are satisfiable. Our inclusion of nonidealized control into the resource framework also raises interesting prospects, which were absent when considering idealized control. Among other things, the emergence of a third law without the need for the assumption of a light cone. Our results and framework unify the field of autonomous thermal machines with the thermodynamic quantum resource-theoretic one, and lay the groundwork for all quantum processing devices to be unified with fully autonomous machines.

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arXiv2022

Extraction of ergotropy: free energy bound and application to open cycle engines

The second law of thermodynamics uses change in free energy of macroscopic systems to set a bound on performed work. Ergotropy plays a similar role in microscopic scenarios, and is defined as the maximum amount of energy that can be extracted from a system by a unitary operation. In this analysis, we quantify how much ergotropy can be induced on a system as a result of system's interaction with a thermal bath, with a perspective of using it as a source of work performed by microscopic machines. We provide the fundamental bound on the amount of ergotropy which can be extracted from environment in this way. The bound is expressed in terms of the non-equilibrium free energy difference and can be saturated in the limit of infinite dimension of the system's Hamiltonian. The ergotropy extraction process leading to this saturation is numerically analyzed for finite dimensional systems. Furthermore, we apply the idea of extraction of ergotropy from environment in a design of a new class of stroke heat engines, which we label open-cycle engines. Efficiency and work production of these machines can be completely optimized for systems of dimensions 2 and 3, and numerical analysis is provided for higher dimensions.

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arXiv2022

The asymptotic emergence of the Second Law for a repeated charging process

In one of its versions, the Second Law states: "It is impossible to construct an engine which will work in a complete cycle, and produces no effect except the raising of a weight and cooling of a heat reservoir." While the Second Law is considered as one of the most robust laws of Nature, it is still challenging how to interpret it in a fully quantum domain. Here we unpack the true meaning of the "cyclicity" and formulate the Second Law for a generic quantum battery via its asymptotic properties of a charging process rather than in terms of a single cycle. As a paradigm, we propose a machine consisting of a battery that repeatedly interacts with identically prepared systems. We then propose the Second Law in the form: The ergotropy of the battery may increase indefinitely if and only if systems are in a non-passive state. One of the most interesting features of this new formulation is the appearance of the passive states that naturally generalize the notion of the heat bath. In this paper, we provide a handful of results that supports this formulation for diagonal systems. Interestingly, our methodology meets a well-known theory of Markov chains, according to which we classify the general charging processes based on the passivity/non-passivity of charging systems. In particular, the adopted mathematics allows us to distinguish a subtle asymptotic difference between the indefinite increase of the battery's energy (induced by the maximally mixed states) and of ergotropy (induced by the non-passive states) in terms of the so-called null-recurrent versus transient Markov chains.

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arXiv2022

Efficient multi port-based teleportation schemes

In this manuscript we analyse generalised port-based teleportation (PBT) schemes, allowing for transmitting more than one unknown quantum state (or a composite quantum state) in one go, where the state ends up in several ports at Bob's side. We investigate the efficiency of our scheme discussing both deterministic and probabilistic case, where parties share maximally entangled states. It turns out that the new scheme gives better performance than various variants of the optimal PBT protocol used for the same task. All the results are presented in group-theoretic manner depending on such quantities like dimensions and multiplicities of irreducible representations in the Schur-Weyl duality. The presented analysis was possible by considering the algebra of permutation operators acting on n systems distorted by the action of partial transposition acting on more than one subsystem. Considering its action on the n-fold tensor product of the Hilbert space with finite dimension, we present construction of the respective irreducible matrix representations, which are in fact matrix irreducible representations of the Walled Brauer Algebra. I turns out that the introduced formalism, and symmetries beneath it, appears in many aspects of theoretical physics and mathematics - theory of anti ferromagnetism, aspects of gravity theory or in the problem of designing quantum circuits for special task like for example inverting an unknown unitary.

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arXiv2022

Fluctuation-dissipation relations for thermodynamic distillation processes

The fluctuation-dissipation theorem is a fundamental result in statistical physics that establishes a connection between the response of a system subject to a perturbation and the fluctuations associated with observables in equilibrium. Here we derive its version within a resource-theoretic framework, where one investigates optimal quantum state transitions under thermodynamic constraints. More precisely, we first characterise optimal thermodynamic distillation processes, and then prove a relation between the amount of free energy dissipated in such processes and the free energy fluctuations of the initial state of the system. Our results apply to initial states given by either asymptotically many identical pure systems or arbitrary number of independent energy-incoherent systems, and allow not only for a state transformation, but also for the change of Hamiltonian. The fluctuation-dissipation relations we derive enable us to find the optimal performance of thermodynamic protocols such as work extraction, information erasure and thermodynamically-free communication, up to second-order asymptotics in the number $N$ of processed systems. We thus provide a first rigorous analysis of these thermodynamic protocols for quantum states with coherence between different energy eigenstates in the intermediate regime of large but finite $N$.

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arXiv2021

Multiport based teleportation -- transmission of a large amount of quantum information

We analyse the problem of transmitting a number of unknown quantum states or one composite system in one go. We derive a lower bound on the performance of such process, measured in the entanglement fidelity. The obtained bound is effectively computable and outperforms the explicit values of the entanglement fidelity calculated for the pre-existing variants of the port-based protocols, allowing for teleportation of a much larger amount of quantum information. The comparison with the exact formulas and similar analysis for the probabilistic scheme is also discussed. In particular, we present the closed-form expressions for the entanglement fidelity and for the probability of success in the probabilistic scheme in the qubit case in the picture of the spin angular momentum.

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arXiv2021

Epsilon-nets, unitary designs and random quantum circuits

Epsilon-nets and approximate unitary $t$-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. The former constitute subsets of unitary channels that are epsilon-close to any unitary channel in the diamond norm. The latter are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of unitary channels up to order $t$. In this work we establish quantitative connections between these two notions. Specifically, we prove that, for a fixed dimension $d$ of the Hilbert space, unitaries constituting $δ$-approximate $t$-expanders form $ε$-nets for $t\simeq\frac{d^{5/2}}ε$ and $δ=\left(\frac{ε^{3/2}}{d}\right)^{d^2}$. We also show that $ε$-nets can be used to construct $δ$-approximate unitary $t$-designs for $δ= εt$. Finally, we prove that the degree of an exact unitary $t$-design necessary to obtain an $ε$-net must grow at least fast as $\frac1ε$ (for fixed $d$) and not slower than $d^2$ (for fixed $ε$). This shows near optimality of our result connecting $t$-designs and $ε$-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures. Our gate sets need not to be symmetric (i.e. contain gates together with their inverses) or consist of gates with algebraic entries. We also show a non-constructive version of the Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.

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arXiv2021

Second law of thermodynamics for batteries with vacuum state

In stochastic thermodynamics work is a random variable whose average is bounded by the change in the free energy of the system. In most treatments, however, the work reservoir that absorbs this change is either tacitly assumed or modelled using unphysical systems with unbounded Hamiltonians (i.e. the ideal weight). In this work we describe the consequences of introducing the ground state of the battery and hence -- of breaking its translational symmetry. The most striking consequence of this shift is the fact that the Jarzynski identity is replaced by a family of inequalities. Using these inequalities we obtain corrections to the second law of thermodynamics which vanish exponentially with the distance of the initial state of the battery to the bottom of its spectrum. Finally, we study an exemplary thermal operation which realizes the approximate Landauer erasure and demonstrate the consequences which arise when the ground state of the battery is explicitly introduced. In particular, we show that occupation of the vacuum state of any physical battery sets a lower bound on fluctuations of work, while batteries without vacuum state allow for fluctuation-free erasure.

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arXiv2020

Thermodynamics of Minimal Coupling Quantum Heat Engines

The minimal-coupling quantum heat engine is a thermal machine consisting of an explicit energy storage system, heat baths, and a working body, which alternatively couples to subsystems through discrete strokes -- energy-conserving two-body quantum operations. Within this paradigm, we present a general framework of quantum thermodynamics, where a work extraction process is fundamentally limited by a flow of non-passive energy (ergotropy), while energy dissipation is expressed through a flow of passive energy. It turns out that small dimensionality of the working body and a restriction only to two-body operations make the engine fundamentally irreversible. Our main result is finding the optimal efficiency and work production per cycle within the whole class of irreversible minimal-coupling engines composed of three strokes and with the two-level working body, where we take into account all possible quantum correlations between the working body and the battery. One of the key new tools is the introduced "control-marginal state" -- one which acts only on a working body Hilbert space, but encapsulates all features regarding work extraction of the total working body-battery system. In addition, we propose a generalization of the many-stroke engine, and we analyze efficiency vs extracted work trade-offs, as well as work fluctuations after many cycles of the running of the engine.

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arXiv2020

Practical No-Signalling proof Randomness Amplification using Hardy paradoxes and its experimental implementation

Device-Independent (DI) security is the best form of quantum cryptography, providing information-theoretic security based on the very laws of nature. In its highest form, security is guaranteed against adversaries limited only by the no-superluminal signalling rule of relativity. The task of randomness amplification, to generate secure fully uniform bits starting from weakly random seeds, is of both cryptographic and foundational interest, being important for the generation of cryptographically secure random numbers as well as bringing deep connections to the existence of free-will. DI no-signalling proof protocols for this fundamental task have thus far relied on esoteric proofs of non-locality termed pseudo-telepathy games, complicated multi-party setups or high-dimensional quantum systems, and have remained out of reach of experimental implementation. In this paper, we construct the first practically relevant no-signalling proof DI protocols for randomness amplification based on the simplest proofs of Bell non-locality and illustrate them with an experimental implementation in a quantum optical setup using polarised photons. Technically, we relate the problem to the vast field of Hardy paradoxes, without which it would be impossible to achieve amplification of arbitrarily weak sources in the simplest Bell non-locality scenario consisting of two parties choosing between two binary inputs. Furthermore, we identify a deep connection between proofs of the celebrated Kochen-Specker theorem and Hardy paradoxes that enables us to construct Hardy paradoxes with the non-zero probability taking any value in $(0,1]$. Our methods enable us, under the fair-sampling assumption of the experiment, to realize up to $25$ bits of randomness in $20$ hours of experimental data collection from an initial private source of randomness $0.1$ away from uniform.

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arXiv2020

Gadget structures in proofs of the Kochen-Specker theorem

The Kochen-Specker theorem is a fundamental result in quantum foundations that has spawned massive interest since its inception. We show that within every Kochen-Specker graph, there exist interesting subgraphs which we term $01$-gadgets, that capture the essential contradiction necessary to prove the Kochen-Specker theorem, i.e,. every Kochen-Specker graph contains a $01$-gadget and from every $01$-gadget one can construct a proof of the Kochen-Specker theorem. Moreover, we show that the $01$-gadgets form a fundamental primitive that can be used to formulate state-independent and state-dependent statistical Kochen-Specker arguments as well as to give simple constructive proofs of an "extended" Kochen-Specker theorem first considered by Pitowsky.

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arXiv2019

Quantum error correction codes and absolutely maximally entangled states

For every stabiliser $N$-qudit absolutely maximally entangled state, we present a method for determining the stabiliser generators and logical operators of a corresponding quantum error correction code. These codes encode $k$ qudits into $N-k$ qudits, with $k\leq \left \lfloor{N/2} \right \rfloor$, where the local dimension $d$ is prime. We use these methods to analyse the concatenation of such quantum codes and link this procedure to entanglement swapping. Using our techniques, we investigate the spread of quantum information on a tensor network code formerly used as a toy model for the AdS/CFT correspondence. In this network, we show how corrections arise to the Ryu-Takayanagi formula in the case of entangled input state, and that the bound on the entanglement entropy of the boundary state is saturated for absolutely maximally entangled input states.

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arXiv2019

Operational foundations of complementarity and uncertainty relations

The so-called preparation uncertainty can be understood in purely operational terms. Namely, it occurs when for some pair of observables, there is no preparation, for which they both exhibit deterministic statistics. However, the right-hand side of uncertainty relation is generally not operational as it depends on the quantum formalism. Also, while joint non-measurability of observables is an operational notion, the complementarity in Bohr sense (i.e. excess of information needed to describe the system) has not yet been expressed in purely operational terms. In this paper we propose a solution to these problems, by introducing an operational definition for complementarity, and further postulating uncertainty as a necessary price for complementarity in physical theories. In other words, we propose to put the (operational) complementarity as the right-hand side of uncertainty relation. Concretely, we first identify two different notions of uncertainty and complementarity for which the above principle holds in quantum mechanics. We also introduce postulates for the general measures of uncertainty and complementarity. In order to define quantifiers of complementarity we first turn to the simpler notion of independence that is defined solely in terms of statistics two observables. We also use our framework to define new complementarity indicators based on (i) performance of random access codes, (ii) geometrical properties of the body of observed statistics, and (iii) variation of information. We then show that they can be used to state uncertainty relations. Moreover, we show that non-signaling and uncertainty relation expressed by complementarity of type (ii) leads to the Tsirelson bound for CHSH inequality. Lastly, we show that a variant of Information Causality called Information Content Principle, can be interpreted as uncertainty relation in the above sense.

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arXiv2019

Generalized XOR non-locality games with graph description on a square lattice

We propose a family of non-locality unique games for 2 parties based on a square lattice on an arbitrary surface. We show that, due to structural similarities with error correction codes of Kitaev for fault tolerant quantum computation, the games have classical values computable in polynomial time for $d=2$ measurement outcomes. By representing games in their graph form, for arbitrary $d$ and underlying surface we provide their classification into equivalence classes with respect to relabeling of measurement outcomes, for a selected set of permutations which define the winning conditions. A case study of games with periodic boundary conditions is presented in order to verify their impact on classical and quantum values of the family of games. It suggests that quantum values suffer independently from presence of different winning conditions that can be imposed due to periodicity, as long as no local restrictions are in place.

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arXiv2018

Decomposability and Convex Structure of Thermal Processes

We present an example of a Thermal Process for a system of $d$ energy levels, which cannot be performed without an instant access to the whole energy space. This Thermal Process is uniquely connected with a transition between some states of the system, that cannot be performed without access to the whole energy space even when approximate transitions are allowed. Pursuing the question about the decomposability of Thermal Processes into convex combinations of compositions of processes acting non-trivially on smaller subspaces, we investigate transitions within the subspace of states diagonal in the energy basis. For three level systems, we determine the set of extremal points of these operations, as well as the minimal set of operations needed to perform an arbitrary Thermal Process, and connect the set of Thermal Processes with thermomajorization criterion. We show that the structure of the set depends on temperature, which is associated with the fact that Thermal Processes cannot increase deterministically extractable work from a state -- the conclusion that holds for arbitrary $d$ level system. We also connect the decomposability problem with detailed balance symmetry of an extremal Thermal Processes.

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