My research

Abstract: We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel \(\cal{E}\) mapping \(\rho_1^{\otimes n}\) into \(\rho_2^{\otimes R_nn}\) with an error \(\epsilon_n\) (measured by trace distance) and \(\sigma_1^{\otimes n}\) into \(\sigma_2^{\otimes R_n n}\) exactly, for a large number \(n\). We derive second-order asymptotic expressions for the optimal transformation rate \(R_n\) in the small, moderate, and large deviation error regimes, as well as the zero-error regime, for an arbitrary pair \((\rho_1,\sigma_1)\) of initial states and a commuting pair \((\rho_2,\sigma_2)\) of final states. We also prove that for \(\sigma_1\) and \(\sigma_2\) given by thermal Gibbs states, the derived optimal transformation rates in the first three regimes can be attained by thermal operations. This allows us, for the first time, to study the second-order asymptotics of thermodynamic state interconversion with fully general initial states that may have coherence between different energy eigenspaces. Thus, we discuss the optimal performance of thermodynamic protocols with coherent inputs and describe three novel resonance phenomena allowing one to significantly reduce transformation errors induced by finite-size effects. What is more, our result on quantum dichotomies can also be used to obtain, up to second-order asymptotic terms, optimal conversion rates between pure bipartite entangled states under local operations and classical communication.

Abstract: We develop a resource-theoretic framework that allows one to bridge the gap between two approaches to quantum thermodynamics based on Markovian thermal processes (which model memoryless dynamics) and thermal operations (which model arbitrarily non-Markovian dynamics). Our approach is built on the notion of memory-assisted Markovian thermal processes, where memoryless thermodynamic processes are promoted to non-Markovianity by explicitly modelling ancillary memory systems initialised in thermal equilibrium states. Within this setting, we propose a family of protocols composed of sequences of elementary two-level thermalisations that approximate all transitions between energy-incoherent states accessible via thermal operations. We prove that, as the size of the memory increases, these approximations become arbitrarily good for all transitions in the infinite temperature limit, and for a subset of transitions in the finite temperature regime. Furthermore, we present solid numerical evidence for the convergence of our protocol to any transition at finite temperatures. We also explain how our framework can be used to quantify the role played by memory effects in thermodynamic protocols such as work extraction. Finally, our results show that elementary control over two energy levels at a given time is sufficient to generate all energy-incoherent transitions accessible via thermal operations if one allows for ancillary thermal systems.

Abstract: The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future and incomparable regions. In this work, we analyse the structure of the resulting thermal cones, i.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a \(d\)-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the incomparable region. Moreover, we provide a detailed analysis of their behaviour based on newly introduced thermodynamic monotones given by the volumes of thermal cones. Finally, we point out that our results also apply to other majorisation-based resource theories (such as that of entanglement and coherence), since the partial ordering describing allowed state transformations is then the opposite of the thermodynamic order in the infinite temperature limit.

Abstract: We present a rigorous approach, based on the concept of continuous thermomajorisation, to algorithmically characterise the full set of energy occupations of a quantum system accessible from a given initial state through weak interactions with a heat bath. The algorithm can be deployed to solve complex optimization problems in out-of-equilibrium setups and it returns explicit elementary control sequences realizing optimal transformations. We illustrate this by finding optimal protocols in the context of cooling, work extraction and catalysis. The same tools also allow one to quantitatively assess the role played by memory effects in the performance of thermodynamic protocols. We obtained exhaustive solutions on a laptop machine for systems with dimension \(d\leq 7\), but with heuristic methods one could access much higher \(d\).

Abstract: 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, and on tools such as entropy production relations. Here we introduce a new framework overcoming the limitations that the current dynamical and information theory approaches encounter when applied to this setting. More precisely, based on a newly introduced notion of continuous thermomajorization, we obtain necessary and sufficient conditions for the existence of a Markovian thermal process transforming between given initial and final energy distributions of the system. These lead to a complete set of generalized entropy production inequalities including the standard one as a special case. Importantly, these conditions can be reduced to 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 explicit protocols realizing any allowed transformation. These protocols use as building blocks elementary thermalizations, which we prove to be universal controls. Finally, we also present an algorithm constructing the full set of energy distributions achievable from a given initial state via Markovian thermal processes and provide a \(\texttt{Mathematica}\) implementation solving \(d=6\) on a laptop computer in minutes.

Abstract: 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 \(N\).

Abstract: We study work extraction processes mediated by finite-time interactions with an ambient bath — partial thermalizations — as continuous time Markov processes for two-level systems. Such a stochastic process results in fluctuations in the amount of work that can be extracted and is characterized by the rate at which the system parameters are driven in addition to the rate of thermalization with the bath. We analyze the distribution of work for the case where the energy gap of a two-level system is driven at a constant rate. We derive analytic expressions for average work and lower bound for the variance of work showing that such processes cannot be fluctuation-free in general. We also observe that an upper bound for the Monte Carlo estimate of the variance of work can be obtained using Jarzynski's fluctuation-dissipation relation for systems initially in equilibrium. Finally, we analyse work extraction cycles by modifying the Carnot cycle, incorporating processes involving partial thermalizations and obtain efficiency at maximum power for such finite-time work extraction cycles under different sets of constraints.

Abstract: We identify and explore the intriguing property of resource resonance arising within resource theories of entanglement, coherence and thermodynamics. While the theories considered are reversible asymptotically, the same is generally not true in realistic scenarios where the available resources are bounded. The finite-size effects responsible for this irreversibility could potentially prohibit small quantum information processors or thermal machines from achieving their full potential. Nevertheless, we show here that by carefully engineering the resource interconversion process any such losses can be greatly suppressed. Our results are predicted by higher order expansions of the trade-off between the rate of resource interconversion and the achieved fidelity, and are verified by exact numerical optimizations of appropriate approximate majorization conditions.

Abstract: We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorisation relation, e.g. the resource theories of entanglement and coherence (for pure state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a non-trivial trade-off between the rate \(r_n\) at which \(n\) copies of a resource state \(\rho\) can be transformed into \(nr_n\) copies of another resource state \(\sigma\), and the error level \(\epsilon_n\) of the interconversion process, as a function of \(n\). In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion \(r_n\) approaches its optimum in the asymptotic limit of unbounded resources (\(n\to\infty\)), while the error \(\epsilon_n\) vanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behaviour which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite \(n\) regime.

Abstract: Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.

Abstract: In this work we analyse the structure of the thermodynamic arrow of time, defined by transformations that leave the thermal equilibrium state unchanged, in classical (incoherent) and quantum (coherent) regimes. We note that in the infinite-temperature limit the thermodynamic ordering of states in both regimes exhibits a lattice structure. This means that when energy does not matter and the only thermodynamic resource is given by information, the thermodynamic arrow of time has a very specific structure. Namely, for any two states at present there exists a unique state in the past consistent with them and with all possible joint pasts. Similarly, there also exists a unique state in the future consistent with those states and with all possible joint futures. We also show that the lattice structure in the classical regime is broken at finite temperatures, i.e., when energy is a relevant thermodynamic resource. Surprisingly, however, we prove that in the simplest quantum scenario of a two-dimensional system, this structure is preserved at finite temperatures. We provide the physical interpretation of these results by introducing and analysing the history erasure process, and point out that quantum coherence may be a necessary resource for the existence of an optimal erasure process.

Abstract: The interplay between quantum-mechanical properties, such as coherence, and classical notions, such as energy, is a subtle topic at the forefront of quantum thermodynamics. The traditional Carnot argument limits the conversion of heat to work; here we critically assess the problem of converting coherence to work. Through a careful account of all resources involved in the thermodynamic transformations within a fully quantum-mechanical treatment, we show that there exist thermal machines extracting work from coherence arbitrarily well. Such machines only need to act on individual copies of a state and can be reused. On the other hand, we show that for any thermal machine with finite resources not all the coherence of a state can be extracted as work. However, even bounded thermal machines can be reused infinitely many times in the process of work extraction from coherence.

Abstract: The first law of thermodynamics imposes not just a constraint on the energy-content of systems in extreme quantum regimes, but also symmetry-constraints related to the thermodynamic processing of quantum coherence. We show that this thermodynamic symmetry decomposes any quantum state into mode operators that quantify the coherence present in the state. We then establish general upper and lower bounds for the evolution of quantum coherence under arbitrary thermal operations, valid for any temperature. We identify primitive coherence manipulations and show that the transfer of coherence between energy levels manifests irreversibility not captured by free energy. Moreover, the recently developed thermo-majorization relations on block-diagonal quantum states are observed to be special cases of this symmetry analysis.

Abstract: In this paper we introduce and investigate the concept of a perfect quantum protractor, a pure quantum state \(|\psi\rangle\in\mathcal{H}\) that generates three different orthogonal bases of \(\mathcal{H}\) under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum \(j\), and we prove that they do not exist for \(j\in\{1/2,2,5/2\}\), but they do exist for \(j\in\{1,3/2,3\}\) (with numerical evidence for their existence when \(j=7/2\). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not \(\textit{a priori}\) known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around \(x\), \(y\) or \(z\) axis, and then employ it to optimally estimate the rotation angle.

Abstract: The classical embeddability problem asks whether a given stochastic matrix \(T\), describing transition probabilities of a \(d\)-level system, can arise from the underlying homogeneous continuous-time Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given \(T\). More precisely, we aim at characterising the set of quantum-embeddable stochastic matrices that arise from memoryless continuous-time quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantum-embeddable but not classically-embeddable, as well as families of stochastic matrices that are not quantum-embeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a non-zero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantum-embeddable.

Abstract: In this paper we aim to push the analogy between thermodynamics and quantum resource theories one step further. Previous inspirations were based on thermodynamic considerations concerning scenarios with a single heat bath, neglecting an important part of thermodynamics that studies heat engines operating between two baths at different temperatures. Here, we investigate the performance of resource engines, which replace the access to two heat baths at different temperatures with two arbitrary constraints on state transformations. The idea is to imitate the action of a two-stroke heat engine, where the system is sent to two agents (Alice and Bob) in turns, and they can transform it using their constrained sets of free operations. We raise and address several questions, including whether or not a resource engine can generate a full set of quantum operations or all possible state transformations, and how many strokes are needed for that. We also explain how the resource engine picture provides a natural way to fuse two or more resource theories, and we discuss in detail the fusion of two resource theories of thermodynamics with two different temperatures, and two resource theories of coherence with respect to two different bases.

Abstract: We introduce and analyse the problem of encoding classical information into different resources of a quantum state. More precisely, we consider a general class of communication scenarios characterised by encoding operations that commute with a unique resource destroying map and leave free states invariant. Our motivating example is given by encoding information into coherences of a quantum system with respect to a fixed basis (with unitaries diagonal in that basis as encodings and the decoherence channel as a resource destroying map), but the generality of the framework allows us to explore applications ranging from super-dense coding to thermodynamics. For any state, we find that the number of messages that can be encoded into it using such operations in a one-shot scenario is upper-bounded in terms of the information spectrum relative entropy between the given state and its version with erased resources. Furthermore, if the resource destroying map is a twirling channel over some unitary group, we find matching one-shot lower-bounds as well. In the asymptotic setting where we encode into many copies of the resource state, our bounds yield an operational interpretation of resource monotones such as the relative entropy of coherence and its corresponding relative entropy variance.

Abstract: We characterise a class of environmental noises that decrease coherent properties of quantum channels by introducing and analysing the properties of dephasing superchannels. These are defined as superchannels that affect only non-classical properties of a quantum channel \(\mathcal{E}\), i.e., they leave invariant the transition probabilities induced by \(\mathcal{E}\) in the distinguished basis. We prove that such superchannels \(\Xi_C\) form a particular subclass of Schur-product supermaps that act on the Jamiołkowski state \(J(\mathcal{E})\) of a channel \(\mathcal{E}\) via a Schur product, \(J'=J\circ C\). We also find physical realizations of general \(\Xi_C\) through a pre- and post-processing employing dephasing channels with memory, and show that memory plays a non-trivial role for quantum systems of dimension \(d>2\). Moreover, we prove that coherence generating power of a general quantum channel is a monotone under dephasing superchannels. Finally, we analyse the effect dephasing noise can have on a quantum channel \(\mathcal{E}\) by investigating the number of distinguishable channels that \(\mathcal{E}\) can be mapped to by a family of dephasing superchannels. More precisely, we upper bound this number in terms of hypothesis testing channel divergence between \(\mathcal{E}\) and its fully dephased version, and also relate it to the robustness of coherence of \(\mathcal{E}\).

Abstract: We investigate the problem of simulating classical stochastic processes through quantum dynamics, and present three scenarios where memory or time quantum advantages arise. First, by introducing and analysing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory. Second, by extending the notion of space-time cost of a stochastic process \(P\) to the quantum domain, we prove an advantage of the quantum cost of simulating \(P\) over the classical cost. Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to a potential advantage in cooling protocols.

Abstract: To what extent does Noether's principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law, and show that this relates to physically impossible transformations in quantum theory, such as time-reversal and spin-inversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group \(G\) becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel. In particular, we investigate in detail the U(1) and SU(2) symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin-\(j_A\) system can be used to polarise a larger spin-\(j_B\) system, and on how much one can invert spin polarisation using a rotationally-symmetric operation. Finally, we also establish novel links between unitarity, complementary channels and purity that are of independent interest.

Abstract: We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between \(M\) quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of \(M\) quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to time-energy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of \(M\) perfectly distinguishable states (channels) that are classically indistinguishable.

Abstract: Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel \(\Phi\), i.e., we look for channels \(\Phi^{\mathcal{C}}\) that induce the same classical transitions \(T\), but are "more coherent". To quantify the coherence of a channel \(\Phi\) we measure the coherence of the corresponding Jamiołkowski state \(J_{\Phi}\). We show that the classical transition matrix \(T\) can be coherified to reversible unitary dynamics if and only if \(T\) is unistochastic. Otherwise the Jamiołkowski state \(J_\Phi^{\mathcal{C}}\) of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To asses the extent to which an optimal process \(\Phi^{\mathcal{C}}\) is indeterministic we find explicit bounds on the entropy and purity of \(J_\Phi^{\mathcal{C}}\), and relate the latter to the unitarity of \(\Phi^{\mathcal{C}}\). We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel \(\Phi\) and reduces its rank (the minimal number of required Kraus operators) from \(d^2\) to \(d\).

Abstract: Both conservation laws and practical restrictions impose symmetry constraints on the dynamics of open quantum systems. In the case of time-translation symmetry, which arises naturally in many physically relevant scenarios, the quantum coherence between energy eigenstates becomes a valuable resource for quantum information processing. In this work we identify the minimum amount of decoherence compatible with this symmetry for a given population dynamics. This yields a generalisation to higher-dimensional systems of the relation \(T_2\leq 2T_1\) for qubit decoherence and relaxation times. It also enables us to witness and assess the role of non-Markovianity as a resource for coherence preservation and transfer. Moreover, we discuss the relationship between ergodicity and the ability of Markovian dynamics to indefinitely sustain a superposition of different energy states. Finally, we establish a formal connection between the resource-theoretic and the master equation approaches to thermodynamics, with the former being a non-Markovian generalisation of the latter. Our work thus brings the abstract study of quantum coherence as a resource towards the realm of actual physical applications.

Abstract: Quantum catalysis, the ability to enable previously impossible transformations by using auxiliary systems without degrading them, has emerged as a powerful tool in various resource theories. Although catalytically enabled state transformations have been formally characterized by the monotonic behaviour of entropic quantifiers (e.g., the von Neumann entropy or non-equilibrium free energy), such characterizations often rely on unphysical assumptions, namely the ability of using catalysts of infinitely large dimension. This approach offers very limited insights into the practical significance of using catalysis for quantum information processing. Here, we address this problem across a broad class of quantum resource theories. Leveraging quantum information tools beyond the asymptotic regime, we establish sufficient conditions for the existence of catalytic transformations with finite-size catalysts. We further unveil connections between finite-size catalysis and multi-copy transformations. Notably, we discover a fascinating phenomenon of catalytic resonance: by carefully tailoring the catalysts's state, one can drastically reduce the required dimension of the catalyst, thus enabling efficient catalytic transformations with minimal resources. Finally, we illustrate our findings with examples from the resource theories of entanglement and thermodynamics, as well in the context of catalytic unitary transformations.

Abstract: We present a classical algorithm for simulating universal quantum circuits composed of "free" nearest-neighbour matchgates or equivalently fermionic-linear-optical (FLO) gates, and "resourceful" non-Gaussian gates. We achieve the promotion of the efficiently simulable FLO subtheory to universal quantum computation by gadgetizing controlled phase gates with arbitrary phases employing non-Gaussian resource states. Our key contribution is the development of a novel phase-sensitive algorithm for simulating FLO circuits. This allows us to decompose the resource states arising from gadgetization into free states at the level of statevectors rather than density matrices. The runtime cost of our algorithm for estimating the Born-rule probability of a given quantum circuit scales polynomially in all circuit parameters, except for a linear dependence on the newly introduced FLO extent, which scales exponentially with the number of controlled-phase gates. More precisely, as a result of finding optimal decompositions of relevant resource states, the runtime doubles for every maximally resourceful (e.g., swap or CZ) gate added. Crucially, this cost compares very favourably with the best known prior algorithm, where each swap gate increases the simulation cost by a factor of approximately 9. For a quantum circuit containing arbitrary FLO unitaries and \(k\) controlled-Z gates, we obtain an exponential improvement \(O(4.5k)\) over the prior state-of-the-art.

Abstract: We present two classical algorithms for the simulation of universal quantum circuits on \(n\) qubits constructed from \(c\) instances of Clifford gates and \(t\) arbitrary-angle \(Z\)-rotation gates such as \(T\) gates. Our algorithms complement each other by performing best in different parameter regimes. The \(\tt{Estimate}\) algorithm produces an additive precision estimate of the Born rule probability of a chosen measurement outcome with the only source of run-time inefficiency being a linear dependence on the stabilizer extent (which scales like \(\approx 1.17t\) for \(T\) gates). Our algorithm is state-of-the-art for this task: as an example, in approximately 25 hours (on a standard desktop computer), we estimated the Born rule probability to within an additive error of 0.03, for a 50 qubit, 60 non-Clifford gate quantum circuit with more than 2000 Clifford gates. The \(\tt{Compute}\) algorithm calculates the probability of a chosen measurement outcome to machine precision with run-time \(O(2t−r(t−r)t)\) where \(r\) is an efficiently computable, circuit-specific quantity. With high probability, \(r\) is very close to \(\min\{t,n−w\}\) for random circuits with many Clifford gates, where \(w\) is the number of measured qubits. \(\tt{Compute}\) can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+\(T\) circuits with \(n=55\), \(w=5\), \(c=105\) and \(t=80\) \(T\) gates, and then compute the Born rule probability with a run-time consistently less than 104 seconds using a single core of a standard desktop computer. We provide a C+Python implementation of our algorithms.

Abstract: The Birkhoff polytope \(\mathcal{B}_d\) consisting of all bistochastic matrices of order \(d\) assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset \(\mathcal{U}_d\) of unistochastic matrices, determined by squared moduli of unitary matrices, is of a particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantised. In order to investigate the problem of unistochasticity we introduce the set \(\mathcal{L}_d\) of bracelet matrices that forms a subset of \(\mathcal{B}_d\), but a superset of \(\mathcal{U}_d\). We prove that for every dimension \(d\) this set contains the set of factorisable bistochastic matrices \(\mathcal{F}_d\) and is closed under matrix multiplication by elements of \(\mathcal{F}_d\). Moreover, we prove that both \(\mathcal{L}_d\) and \(\mathcal{F}_d\) are star-shaped with respect to the flat matrix. We also analyse the set of \(d\times d\) unistochastic matrices arising from circulant unitary matrices, and show that their spectra lie inside \(d\)-hypocycloids on the complex plane. Finally, applying our results to small dimensions, we fully characterise the set of circulant unistochastic matrices of order \(d\leq 4\), and prove that such matrices form a monoid for \(d=3\).

Abstract: Which quantum states minimise the unavoidable uncertainty arising from the non-commutativity of two observables? The immediate answer to such a question is: it depends. Due to the plethora of uncertainty measures there are many answers. Here, instead of restricting our study to a particular measure, we present plausible axioms for the set \(\mathcal{F}\) of bona-fide information-theoretic uncertainty functions. Then, we discuss the existence of states minimising uncertainty with respect to all members of \(\mathcal{F}\), i.e., universal minimum uncertainty states (MUS). We prove that such states do not exist within the full state space and study the effect of classical noise on the structure of minimum uncertainty states. We present an explicit example of a qubit universal MUS that arises when purity is constrained by introducing a threshold amount of noise. For higher dimensional systems we derive several no-go results limiting the existence of noisy universal MUS. However, we conjecture that universality may emerge in an approximate sense. We conclude by discussing connections with thermodynamics, and highlight the privileged role that non-equilibrium free energy \(F_2\) plays close to equilibrium.

Abstract: How much of the uncertainty in predicting measurement outcomes for non-commuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two non-commuting observables into a classical component, and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits "purity-based" lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a non-trivial interplay between quantum and classical randomness in any finite-dimensional state.

Abstract: We argue for an operational requirement that all state-dependent measures of disturbance should satisfy. Motivated by this natural criterion, we prove that in any \(d\)-dimensional Hilbert space and for any pair of non-commuting operators, \(A\) and \(B\), there exists a set of at least \(2^{d−1}\) zero-noise, zero-disturbance (ZNZD) states, for which the first observable can be measured without noise and the second will not be disturbed. Moreover, we show that it is possible to construct such ZNZD states for which the expectation value of the commutator \([A,B]\) does not vanish. Therefore any state-dependent error-disturbance relation, based on the expectation value of the commutator as a lower bound, must violate the operational requirement. We also discuss Ozawa's state-dependent error-disturbance relation in light of our results and show that the disturbance measure used in this relation exhibits unphysical properties. We conclude that the trade-off is inevitable only between state-independent measures of error and disturbance.

Abstract: We propose a quantum-state transfer protocol in a spin chain that requires only the control of the spins at the ends of the quantum wire. The protocol is to a large extent insensitive to inhomogeneity caused by local magnetic fields and perturbation of exchange couplings. Moreover, apart from the free evolution regime, it allows one to induce an adiabatic spin transfer, which provides the possibility of performing the transfer on demand. We also show that the amount of information leaking into the central part of the chain is small throughout the whole transfer process (which protects the information sent from being eavesdropped) and can be controlled by the magnitude of the external magnetic field.

Abstract: We develop a theoretical description of the spin dynamics of resident holes in a p-doped semiconductor quantum well (QW) subject to a magnetic field tilted from the Voigt geometry. We find the expressions for the signals measured in time-resolved Faraday rotation (TRFR) and resonant spin amplification (RSA) experiments and study their behavior for a range of system parameters. We find that an inversion of the RSA peaks can occur for long hole spin dephasing times and tilted magnetic fields. We verify the validity of our theoretical findings by performing a series of TRFR and RSA experiments on a \(p\)-modulation doped GaAs/Al\(_{0.3}\)Ga\(_{0.7}\)As single QW and showing that our model can reproduce experimentally observed signals.

Abstract: Understanding and controlling the spin dynamics in semiconductor heterostructures is a key requirement for the design of future spintronics devices. In GaAs-based heterostructures, electrons and holes have very different spin dynamics. Some control over the spin-orbit fields, which drive the electron spin dynamics, is possible by choosing the crystallographic growth axis. Here, (110)-grown structures are interesting, as the Dresselhaus spinorbit fields are oriented along the growth axis and therefore, the typically dominant Dyakonov-Perel mechanism is suppressed for spins oriented along this axis, leading to long spin depasing times. By contrast, hole spin dephasing is typically very rapid due to the strong spin-orbit interaction of the p-like valence band states. For localized holes, however, most spin dephasing mechanisms are suppressed, and long spin dephasing times may be observed. Here, we present a study of electron and hole spin dynamics in GaAs-AlGaAs-based quantum wells. We apply the resonant spin amplification (RSA) technique, which allows us to extract all relevant spin dynamics parameters, such as \(g\) factors and dephasing times with high accuracy. A comparison of the measured RSA traces with the developed theory reveals the anisotropy of the spin dephasing in the (110)-grown two-dimensional electron systems, as well as the complex interplay between electron and hole spin and carrier dynamics in the two-dimensional hole systems.

Abstract: We propose a quantum dot (QD) implementation of a quantum state transfer channel. The proposed channel consists of \(N\) vertically stacked QDs with the nearest neighbor tunnel coupling, placed in an axial electric field. We show that the system supports high-fidelity transfer of the state of a terminal dot both by free evolution and by adiabatic transfer. The protocol is to a large extent insensitive to inhomogeneity of the energy parameters of the dots and requires only a global electric field.

Abstract: We investigate spin dynamics of resident holes in a \(p\)-modulation-doped GaAs/Al\(_{0.3}\)Ga\(_{0.7}\)As single quantum well. Time-resolved Faraday and Kerr rotation, as well as resonant spin amplification, are utilized in our study. We observe that nonresonant or high-power optical pumping leads to a resident hole spin polarization with opposite sign with respect to the optically oriented carriers, while low-power resonant optical pumping only leads to a resident hole spin polarization if a sufficient in-plane magnetic field is applied. The competition between two different processes of spin orientation strongly modifies the shape of resonant spin amplification traces. Calculations of the spin dynamics in the electron-hole system are in good agreement with the experimental Kerr rotation and resonant spin amplification traces and allow us to determine the hole spin polarization within the sample after optical orientation, as well as to extract quantitative information about spin dephasing processes at various stages of the evolution.