Currently, there is no systematic way to describe a quantum process with memory solely in terms of experimentally accessible quantities. However, recent technological advances mean we have control over systems at scales where memory effects are non-negligible. The lack of such an operational description has hindered advances in understanding physical, chemical and biological processes, where often unjustified theoretical assumptions are made to render a dynamical description tractable. This has led to theories plagued with unphysical results and no consensus on what a quantum Markov (memoryless) process is. Here, we develop a universal framework to characterise arbitrary non-Markovian quantum processes. We show how a multi-time non-Markovian process can be reconstructed experimentally, and that it has a natural representation as a many body quantum state, where temporal correlations are mapped to spatial ones. Moreover, this state is expected to have an efficient matrix product operator form in many cases. Our framework constitutes a systematic tool for the effective description of memory-bearing open-system evolutions.

Can collective quantum effects make a difference in a meaningful thermodynamic operation? Focusing on energy storage and batteries, we demonstrate that quantum mechanics can lead to an enhancement in the amount of work deposited per unit time, i.e., the charging power, when N batteries are charged collectively. We first derive analytic upper bounds for the collective quantum advantage in charging power for two choices of constraints on the charging Hamiltonian. We then demonstrate that even in the absence of quantum entanglement this advantage can be extensive. For our main result, we provide an upper bound to the achievable quantum advantage when the interaction order is restricted; i.e., at most k batteries are interacting. This constitutes a fundamental limit on the advantage offered by quantum technologies over their classical counterparts.

We derive a necessary and sufficient condition for a quantum process to be Markovian which coincides with the classical one in the relevant limit. Our condition unifies all previously known definitions for quantum Markov processes by accounting for all potentially detectable memory effects. We then derive a family of measures of non-Markovianity with clear operational interpretations, such as the size of the memory required to simulate a process, or the experimental falsifiability of a Markovian hypothesis.In classical probability theory, a stochastic process is the collection of joint probability distributions of a system's state (described by random variable X) at different times, {P (X k , t k ; X k−1 , t k−1 ; . . . ; X 1 , t 1 ; X 0 , t 0 ) ∀k ∈ N}; to be a valid process, these distributions must additionally satisfy the Kolmogorov consistency conditions [1]. A Markov process is one where the state X k of the system at any time t k only depends conditionally on the state of the system at the previous time step, and not on the remaining history. That is, the conditional probability distributions satisfy P (X k , t k |X k−1 , t k−1 ;. . .; X 0 , t 0 ) = P (X k , t k |X k−1 , t k−1 ) (1) for all k. This simple looking condition has profound implications, leading to a massively simplified description of the stochastic process. The study of such processes forms an entire branch of mathematics, and the evolution of physical systems is frequently approximated to be Markov (when it is not exactly so). This is in part due to the fact that the properties of Markov processes make them easier to manipulate analytically and computationally [2].Implicit in this description of a classical process is the assumption that the value of X j at a given time can be observed without affecting the subsequent evolution. This assumption cannot be valid for quantum processes. In quantum theory, a measurement must be performed to infer the state of system. And the measurement process, in general, must disturb that state. Therefore, unlike its classical counterpart, a generic quantum stochastic process cannot be described without interfering with it [3]. These complications make it challenging to define the process independently of the control operations of the experimenter. From a technical perspective, a serious consequence of this is that joint probability distributions of quantum observables at different times do not satisfy the Kolmogorov conditions [1], and do not constitute stochastic processes in the classical sense.Nevertheless, temporal correlations between observables do play an important role in the dynamics of many open quantum systems, e.g. in the emission spectra of quantum dots [4] and in the vibrational motion of interacting molecular fluids [5]. Quantifying memory effects, and clearly defining * felix.pollock@monash.edu † kavan.modi@monash.edu the boundary between Markovian and non-Markovian quantum processes, represents an important challenge in describing such systems. Attempts at solving this problem tend to take a...

Recently, it has been shown that energy can be deposited on a collection of quantum systems at a rate that scales super-extensively. Some of these schemes for 'quantum batteries' rely on the use of global many-body interactions that take the batteries through a correlated short cut in state space. Here, we extend the notion of a quantum battery from a collection of a priori isolated systems to a many-body quantum system with intrinsic interactions. Specifically, we consider a one-dimensional spin chain with physically realistic two-body interactions. We find that the spin-spin interactions can yield an advantage in charging power over the non-interacting case, and we demonstrate that this advantage can grow super-extensively when the interactions are long ranged. However, we show that, unlike in previous work, this advantage is a mean-field interaction effect that does not involve correlations and that relies on the interactions being intrinsic to the battery. arXiv:1712.03559v1 [quant-ph]

Abstract. This special volume celebrates the 40th anniversary of the discovery of the Gorini-Kossakowski-Sudarshan-Lindblad master equation, which is widely used in quantum physics and quantum chemistry. The present contribution aims to celebrate a related discovery -also by Sudarshan -that of Quantum Maps (which had their 55th anniversary in the same year). Nowadays, much like the master equation, quantum maps are ubiquitous in physics and chemistry. Their importance in quantum information and related fields cannot be overstated. Here, we motivate quantum maps from a tomographic perspective, and derive their well-known representations. We then dive into the murky world beyond these maps, where recent research has yielded their generalisation to non-Markovian quantum processes.

Conventional quantum speed limits perform poorly for mixed quantum states: They are generally not tight and often significantly underestimate the fastest possible evolution speed. To remedy this, for unitary driving, we derive two quantum speed limits that outperform the traditional bounds for almost all quantum states. Moreover, our bounds are significantly simpler to compute as well as experimentally more accessible. Our bounds have a clear geometric interpretation; they arise from the evaluation of the angle between generalized Bloch vectors.Quantum speed limits (QSLs) set fundamental bounds on the shortest time required to evolve between two quantum states [1][2][3]. The earliest derivation of minimal time of evolution was in 1945 by Mandelstam and Tamm [4] with the aim of operationalising the famous (but oft misunderstood) timeenergy uncertainty relations [5-9] ∆t ≥ /∆E, relating the standard deviation of energy with the time it takes to go from one state to another. QSLs were originally derived for the unitary evolution of pure states [10][11][12]; since then they have been generalized to the case of mixed states [13][14][15][16], nonunitary evolution [17][18][19], and multi-partite systems [20][21][22][23][24].Extending their original scope, their significance has evolved from fundamental physics to practical relevance, defining the limits of the rate of information transfer [25] and processing [26], entropy production [27], precision in quantum metrology [28] and time-scale of quantum optimal control [29]. For example, in [30], the authors use QSLs to calculate the maximal rate of information transfer along a spin chain; similarly, Reich et al. show that optimization algorithms and QSLs can be used together to achieve quantum control over a large class of physical systems [31]. In Refs. [32][33][34] QSLs are used to bound the charging power of non-degenerate multi-partite systems, which are treated as batteries. The latter results imply a significant speed advantage for entangling over local unitary driving of quantum systems, given the same external constraints.Combining the Mandelstam-Tamm result with the results by Margolus and Levitin, along with elements of quantum state space geometry [35], leads to a unified QSL [36]. It bounds the shortest time required to evolve a (mixed) state ρ to another state σ by means of a unitary operator U t generated by some time-dependent Hamiltonian H twhere L(ρ, σ) = arccos(F(ρ, σ)) is the Bures angle, a measure of the distance between states ρ and σ, F(ρ, σ) = tr[ √ ρσ √ ρ] is the Uhlmann root fidelity [37,38]; For pure states ρ = |ψ ψ| and σ = |φ φ|, the Bures angle reduces to the Fubini-Study distance d(|ψ , |φ ) = arccos | ψ|φ | [35,39,40]. Under this condition Eq. (1) is provably tight [36]. An insightful geometric interpretation of QSLs for pure states (in a Hilbert space of any dimension) is that the geodesic connecting initial and final states lives on a complex projective line CP 1 (isomorphic to a 2-sphere S 2 ), defined by the linear combinations of |ψ an...

In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, noninteracting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we relate the influence functional to the process tensor -a more general representation of a quantum stochastic process -describing the evolution. We then use this connection to motivate a tensor network algorithm for the simulation of multi-time correlations in open systems, building on recent work where the influence functional is represented in terms of time evolving matrix product operators. By exploiting the symmetries of the influence functional, we are able to use our algorithm to achieve orders-of-magnitude improvement in the efficiency of the resulting numerical simulation. Our improved algorithm is then applied to compute exact phonon emission spectra for the spin-boson model with strong coupling, demonstrating a significant divergence from spectra derived under commonly used assumptions of memorylessness.

Unitary transformations can allow one to study open quantum systems in situations for which standard, weak-coupling type approximations are not valid. We develop here an extension of the variational (polaron) transformation approach to open system dynamics, which applies to arbitrarily large exciton transport networks with local environments. After deriving a time-local master equation in the transformed frame, we go on to compare the population dynamics predicted using our technique with other established master equations. The variational frame dynamics are found to agree with both weak coupling and full polaron master equations in their respective regions of validity. In parameter regimes considered difficult for these methods, the 7