An approach to the description of subdynamics inside non-relativistic quantum field theory is presented, in which the notions of relevant observable, time scale and complete positivity of the time evolution are stressed. A scattering theory derivation of the subdynamics of a microsystem interacting through collisions with a macrosystem is given, leading to a master-equation expressed in terms of the operator-valued dynamic structure factor, a two-point correlation function which compactly takes the statistical mechanics properties of the macrosystem into account. For the case of a free quantum gas the dynamic structure factor can be exactly calculated and in the long wavelength limit a Fokker-Planck equation for the description of quantum dissipation and in particular quantum Brownian motion is obtained, where peculiar corrections due to quantum statistics can be put into evidence.
The formalism of subdynamics is extended to the functional approach of quantum systems, and used for the Friedrichs model, in which diagonal singularities in states and observables are included. We compute in this approach the generalized eigenvectors and eigenvalues of the Liouvulle-Von Newmann operator, using an iterative scheme. As complex generalized eigenvalues are obtained, the decay rates of unstable modes are included in the spectral decomposition.
Bell's theorem is a statement by which averages obtained from specific types of statistical distributions must conform to a family of inequalities. These models, in accordance with the EPR argument, provide for the simultaneous existence of quantum mechanically incompatible quantities. We first recall several contradictions arising between the assumption of a joint distribution for incompatible observables and the probability structure of quantum-mechanics, and conclude that Bell's theorem is not expected to be relevant to quantum phenomena described by non-commuting observables, irrespective …
We address the problem of understanding from first principles the conditions under which a quantum system equilibrates rapidly with respect to a concrete observable. On the one hand previously known general upper bounds on the time scales of equilibration were unrealistically long, with times scaling linearly with the dimension of the Hilbert space. These bounds proved to be tight, since particular constructions of observables scaling in this way were found. On the other hand, the computed equilibration time scales for certain classes of typical measurements, or under the evolution of typical …
Understanding how correlations can be used for quantum communication protocols is a central goal of quantum information science. While many authors have linked global measures of correlations such as entanglement or discord to the performance of specific protocols, in general the latter may require only correlations between specific observables. In this work, we first introduce a general measure of correlations for two-qubit states based on the classical mutual information between local observables. We then discuss the role of the symmetry in the state's correlations distribution and according…
The concept of weak invariants has recently been introduced in the context of conserved quantities in finite-time processes in nonequilibrium quantum thermodynamics. A weak invariant itself has a time-dependent spectrum, but its expectation value remains constant under time evolution defined by a relevant master equation. Although its expectation value is thus conserved by definition, its fluctuation is not. Here, time evolution of such a fluctuation is studied. It is shown that if the subdynamics is given by a completely positive map, then the fluctuation of the associated weak invariant does…
We ask which is the best strategy to reveal uncertainty relations between comple- mentary observables of a continuous variable system for coarse-grained measurements. This leads to the derivation of new uncertainty relations for coarse-grained measurements that are always valid, even for detectors with low precision. These relations should be particularly relevant in experimental demonstrations of squeezing in quantum optics, quantum state reconstruction, and the development of trustworthy entanglement criteria.
It is argued that the partition of a quantum system into subsystems is dictated by the set of operationally accessible interactions and measurements. The emergence of a multi-partite tensor product structure of the state-space and the associated notion of quantum entanglement are then relative and observable-induced. We develop a general algebraic framework aimed to formalize this concept. We discuss several cases relevant to quantum information processing and decoherence control.
Homodyne tomography provides a way for measuring generic field-operators. Here we analyze the determination of the most relevant quantities: intensity, field, amplitude and phase. We show that tomographic measurements are affected by additional noise in comparison with the direct detection of each observable by itself. The case of of coherent states has been analyzed in details and earlier estimations of tomographic precision are critically discussed.
The concept of physical twin observables (PTO) for bipartite quantum states,introduced and proved relevant for quantum information theory in recent work, is substantially simplified. The relation of observable and state is studied in detail from the point of view of coherence entropy. Properties of this quantity are further explored. It is shown that, besides for pure states, quantum discord (measure of entanglement) can be expressed through the coherence entropy of a PTO complete in relation to the state.
Under the principle that quantum mechanical observables are invariant under relevant symmetry transformations, we explore how the usual, non-invariant quantities may capture measurement statistics. Using a relativisation mapping, viewed as the incorporation of a quantum reference frame, we show that the usual quantum description approximates the relative one precisely when the reference system admits an appropriate localisable quantity and a localised state. From this follows a new perspective on the nature and reality of quantum superpositions and optical coherence.
Recently, a striking experimental demonstration [G. Kirchmair \emph{et al.}, Nature, \textbf{460}, 494(2009)] of the state-independent quantum mechanical violation of non-contextual realist models has been reported for any two-qubit state using suitable choices of \emph{nine} product observables and \emph{six} different measurement setups. In this report, a considerable simplification of such a demonstration is achieved by formulating a scheme that requires only \emph{five} product observables and \emph{two} different measurement setups. It is also pointed out that the relevant empirical data …
The quantum-mechanical state vector is not directly observable even though it is the fundamental variable that appears in Schrodinger's equation. In conventional time-dependent perturbation theory, the state vector must be calculated before the experimentally-observable expectation values of relevant operators can be computed. We discuss an alternative form of time-dependent perturbation theory in which the observable expectation values are calculated directly and expressed in the form of nested commutators. This result is consistent with the fact that the commutation relations determine the p…
A spin system on a lattice can usually be modelled at large scales by an effective quantum field theory. A key mathematical result relating the two descriptions is the quantum central limit theorem, which shows that certain spin observables satisfy an algebra of bosonic fields under certain conditions. Here, we show that these particular observables and conditions are the relevant ones for an observer with certain limited abilities to resolve spatial locations as well as spin values. This is shown by computing the asymptotic behaviour of a quantum Fisher information metric as function of the r…
Joint or simultaneous measurements of non-commuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for defining what an "optimal" joint measurement is, with corresponding different tradeoff relations for the measurements. Understanding the limitations of such measurements is of fundamental interest and relevant for quantum technology. Here, we experimentally test a tradeoff relation for the sharpness of qubit measurements, a relation which refers directly to the form of the measurement operators, rather than to er…
The determination of a quantum observable from the first and second moments of its measurement outcome statistics is investigated. Operational conditions for the moments of a probability measure are given which suffice to determine the probability measure. Differential operators are shown to lead to physically relevant cases where the expectation values of large classes of noncommuting observables do not distinguish superpositions of states and, in particular, where the full moment information does not determine the probability measure.
We introduce a technique for recovering noise-free observables in noisy quantum systems by combining the results of many slightly different experiments. Our approach is applicable to a variety of quantum systems but we illustrate it with applications to quantum computing and quantum sensing. The approach corresponds to repeating the same quantum evolution many times with known variations on the underlying systems' error properties, e.g. the spontaneous emission and dephasing times, $T_1$ and $T_2$. As opposed to standard quantum error correction methods, which have an overhead in the number of…
The so-called classical limit of quantum mechanics is generally studied in terms of the decoherence of the state operator that characterizes a system. This is not the only possible approach to decoherence. In previous works we have presented the possibility of studying the classical limit in terms of the decoherence of relevant observables of the system. On the basis of this approach, in this paper we introduce the classical limit from a logical perspective, by studying the way in which the logical structure of quantum properties corresponding to relevant observables acquires Boolean character…
Shock wave gives back reaction to spacetime and the information is stored in the memory of background. Quantum memory effect of localised shock wave on Minkowski metric is investigated here. We find that the Wightman function for massless scalar field, on both sides of the wave location, turns out to be the same for usual Minkowski spacetime. Therefore the observables obtained from this, for any kind of observer, do not show the signature of the shock. Moreover, the vacuum states of field on both sides are equivalent. On the contrary, the correlator for the non-vacuum state does memorise the c…
We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is non-unitary. This allows to describe observables that are initially non-commutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally pres…
In an ensemble of two-level atoms that can be described in terms of a collective spin, entangled states can be used to enhance the sensitivity of interferometric precision measurements. While non-Gaussian spin states can produce larger quantum enhancements than spin-squeezed Gaussian states, their use requires the measurement of observables that are nonlinear functions of the three components of the collective spin. In this paper we develop strategies that achieve the optimal quantum enhancements using non-Gaussian states produced by a nonlinear one-axis-twisting Hamiltonian, and show that mea…
We investigate the dynamics of the rate function and of local observables after a quench in models which exhibit phase transitions between a superfluid and an insulator in their ground states. Zeros of the return probability, corresponding to singularities of the rate functions, have been suggested to indicate the emergence of dynamical criticality and we address the question of whether such zeros can be tied to the dynamics of physically relevant observables and hence order parameters in the systems. For this we first numerically analyze the dynamics of a hard-core boson gas in a one-dimensio…
We consider a quantum system with a time-independent Hamiltonian parametrized by a set of unknown parameters $α$. The system is prepared in a general quantum state by an evolution operator that depends on a set of unknown parameters $P$. After the preparation, the system evolves in time, and it is characterized by a time-dependent observable ${\cal O}(t)$. We show that it is possible to obtain closed-form expressions for the gradients of the distance between ${\cal O}(t)$ and a calculated observable with respect to $α$, $P$ and all elements of the system density matrix, whether for pure or mix…
A sudden change of the Hamiltonian parameter drives a quantum system out of equilibrium. For a finite-size system, expectations of observables start fluctuating in time without converging to a precise limit. A new equilibrium state emerges only in probabilistic sense, when the probability distribution for the observables expectations over long times concentrate around their mean value. In this paper we study the full statistic of generic observables after a small quench. When the quench is performed around a regular (i.e. non-critical) point of the phase diagram, generic observables are expect…
We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity - some of which new - in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We cl…
When deriving exact generalized master equations for the evolution of a reduced set of degrees of freedom, one is free to choose what quantities are relevant by specifying projection operators. However, obtaining a reduced description does not always need to be achieved through projections--one can also use conservation laws for this purpose. Such an operation should be considered as distinct from any kind of projection; that is, projection onto a single observable yields a different form of master equation compared to that resulting from a projection followed by the application of a constrain…
We numerically examine slow and hierarchical relaxation dynamics of interacting bosons described by a tilted two-band Bose-Hubbard model. The system is found to exhibit signatures of quantum chaos within the spectrum and the validity of the eigenstate thermalization hypothesis for relevant physical observables is demonstrated for certain parameter regimes. Using the truncated Wigner representation in the semiclassical limit of the system, dynamics of relevant observables reveal hierarchical relaxation and the appearance of prethermalized states is studied from the perspective of statistics of …
The notion that any physical quantity is defined and measured relative to a reference frame is traditionally not explicitly reflected in the theoretical description of physical experiments where, instead, the relevant observables are typically represented as "absolute" quantities. However, the emergence of the resource theory of quantum reference frames as a new branch of quantum information science in recent years has highlighted the need to identify the physical conditions under which a quantum system can serve as a good reference. Here we investigate the conditions under which, in quantum t…
We introduce a measure Q of bipartite quantum correlations for arbitrary two-qubit states, expressed as a state-independent function of the density matrix elements. The amount of quantum correlations can be quantified experimentally by measuring the expectation value of a small set of observables on up to four copies of the state, without the need for a full tomography. We extend the measure to 2 x d systems, providing its explicit form in terms of observables and applying it to the relevant class of multiqubit states employed in the deterministic quantum computation with one quantum bit model…
In physics we attempt to infer the rules governing a system given only the results of imprecise measurements. This is an ill-posed problem because certain features of the system's state cannot be resolved by the measurements. However, by ignoring the irrelevant features, an effective theory can be made for the remaining observable relevant features. We explain how these relevant and irrelevant degrees of freedom can be concretely characterised using quantum distinguishability metrics, thus solving the ill-posed inference problem. This framework then allows us to provide an information-theoreti…
A quantum control landscape is defined as the observable as a function(al) of the system control variables. Such landscapes were introduced to provide a basis to understand the increasing number of successful experiments controlling quantum dynamics phenomena. This paper extends the concept to encompass the broader context of the environment having an influence. For the case that the open system dynamics are fully controllable, it is shown that the control landscape for open systems can be lifted to the analysis of an equivalent auxiliary landscape of a closed composite system that contains th…
We show that the critical point of the two-dimensional Bose-Hubbard model can be easily found through studies of either on-site atom number fluctuations or the nearest-neighbor two-point correlation function (the expectation value of the tunnelling operator). Our strategy to locate the critical point is based on the observation that the derivatives of these observables with respect to the parameter that drives the superfluid-Mott insulator transition are singular at the critical point in the thermodynamic limit. Performing the quantum Monte Carlo simulations of the two-dimensional Bose-Hubbard…
Quantifying coherence is an essential endeavour for both quantum foundations and quantum technologies. Here the robustness of coherence is defined and proven a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is fina…
