Johan Åberg

The thermodynamic meaning of negative entropy

The heat generated by computations is not only an obstacle to circuit miniaturization but also a fundamental aspect of the relationship between information theory and thermodynamics. In principle, reversible operations may be performed at no energy cost; given that irreversible computations can always be decomposed into reversible operations followed by the erasure of data, the problem of calculating their energy cost is reduced to the study of erasure. Landauer’s principle states that the erasure of data stored in a system has an inherent work cost and therefore dissipates heat. However, this consideration assumes that the information about the system to be erased is classical, and does not extend to the general case where an observer may have quantum information about the system to be erased, for instance by means of a quantum memory entangled with the system. Here we show that the standard formulation and implications of Landauer’s principle are no longer valid in the presence of quantum information. Our main result is that the work cost of erasure is determined by the entropy of the system, conditioned on the quantum information an observer has about it. In other words, the more an observer knows about the system, the less it costs to erase it. This result gives a direct thermodynamic significance to conditional entropies, originally introduced in information theory. Furthermore, it provides new bounds on the heat generation of computations: because conditional entropies can become negative in the quantum case, an observer who is strongly correlated with a system may gain work while erasing it, thereby cooling the environment.

Min- and Max-Entropy in Infinite Dimensions

We consider an extension of the conditional min- and max-entropies to infinite-dimensional separable Hilbert spaces. We show that these satisfy characterizing properties known from the finite-dimensional case, and retain information-theoretic operational interpretations, e.g., the min-entropy as maximum achievable quantum correlation, and the max-entropy as decoupling accuracy. We furthermore generalize the smoothed versions of these entropies and prove an infinite-dimensional quantum asymptotic equipartition property. To facilitate these generalizations we show that the min- and max-entropy can be expressed in terms of convergent sequences of finite-dimensional min- and max-entropies, which provides a convenient technique to extend proofs from the finite to the infinite-dimensional setting.

Robustness of the adiabatic quantum search

The effects of open system evolution on single particle interferometry, quantum computation, and the adiabatic approximation are investigated.Single particle interferometry: Three concepts concerning completely positive maps (CPMs) and trace preserving CPMs (channels), named subspace preserving (SP) CPMs, subspace local channels, and gluing of CPMs, are introduced. SP channels preserve probability weights on given orthogonal sum decompositions of the Hilbert space of a quantum system. Subspace locality determines what channels act locally with respect to such decompositions. Gluings are the possible total channels obtainable if two evolution devices, characterized by channels, act jointly on a superposition of a particle in their inputs. It is shown that gluings are not uniquely determined by the two channels. We determine all possible interference patterns in single particle interferometry for given channels acting in the interferometer paths. It is shown that the standard interferometric setup cannot distinguish all gluings, but a generalized setup can.Quantum computing: The robustness of local and global adiabatic quantum search subject to decoherence in the instantaneous eigenbasis of the search Hamiltonian, is examined. In both the global and local search case the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search with scaling N, and that the global search scales like N3/2 , where N is the list length. We consider success probabilities p<1 and prove bounds on the run-time with the same scaling as in the conditions for the p → 1 limit.Adiabatic evolution: We generalize the adiabatic approximation to the case of open quantum systems in the joint limit of slow change and weak open system disturbances.

Noncyclic geometric changes of quantum states

Non-Abelian quantum holonomies, i.e., unitary state changes solely induced by geometric properties of a quantum system, have been much under focus in the physics community as generalizations of the Abelian Berry phase. Apart from being a general phenomenon displayed in various subfields of quantum physics, the use of holonomies has lately been suggested as a robust technique to obtain quantum gates; the building blocks of quantum computers. Non-Abelian holonomies are usually associated with cyclic changes of quantum systems, but here we consider a generalization to noncyclic evolutions. We argue that this open-path holonomy can be used to construct quantum gates. We also show that a structure of partially defined holonomies emerges from the open-path holonomy. This structure has no counterpart in the Abelian setting. We illustrate the general ideas using an example that may be accessible to tests in various physical systems.

Quantum adiabatic search with decoherence in the instantaneous energy eigenbasis

In Phys. Rev. A 71, 060312(R) (2005), the robustness of the local adiabatic quantum search to decoherence in the instantaneous eigenbasis of the search Hamiltonian was examined. We expand this analysis to include the case of the global adiabatic quantum search. As in the case of the local search the asymptotic time complexity for the global search is the same as for the ideal closed case, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, we find that the time complexity of the global quantum adiabatic search scales like N{sup 3/2}, where N is the list length. We moreover extend the analysis to include success probabilities p<1 and prove bounds on the run time with the same scaling as in the conditions for the p{yields}1 limit. We supplement the analytical results by numerical simulations of the global and local search.

Operational approach to the Uhlmann holonomy

We suggest a physical interpretation of the Uhlmann amplitude of a density operator. Given this interpretation we propose an operational approach to obtain the Uhlmann condition for parallelity. This allows us to realize parallel transport along a sequence of density operators by an iterative preparation procedure. At the final step the resulting Uhlmann holonomy can be determined via interferometric measurements.

Non-Abelian generalization of off-diagonal geometric phases

If a quantum system evolves in a noncyclic fashion the corresponding geometric phase or holonomy may not be fully defined. Off-diagonal geometric phases have been developed to deal with such cases. Here, we generalize these phases to the non-Abelian case, by introducing off-diagonal holonomies that involve evolution of more than one subspace of the underlying Hilbert space. Physical realizations of the off-diagonal holonomies in adiabatic evolution and interferometry are put forward.

Adiabatic geometric phases in hydrogenlike atoms

We examine the effect of spin-orbit coupling on geometric phases in hydrogenlike atoms exposed to a slowly varying magnetic field. The marginal geometric phases associated with the orbital angular ...

Fidelity and Coherence Measures from Interference

By utilizing single particle interferometry, the fidelity or coherence of a pair of quantum states is identified with their capacity for interference. We consider processes acting on the internal degree of freedom (e.g., spin or polarization) of the interfering particle, preparing it in states rho_{A} or rho_{B} in the respective path of the interferometer. The maximal visibility depends on the choice of interferometer, as well as the locality or nonlocality of the preparations, but otherwise depends only on the states rho_{A} and rho_{B} and not the individual preparation processes themselves. This allows us to define interferometric measures which probe locality and correlation properties of spatially or temporally separated processes, and can be used to differentiate between processes that cannot be distinguished by direct process tomography using only the internal state of the particle.

Subspace preserving completely positive maps

A class of quantum channels and completely positive maps (CPMs) are introduced and investigated. These, which we call subspace preserving (SP) CPMs has, in the case of trace preserving CPMs, a simple interpretation as those which preserve probability weights on a given orthogonal sum decomposition of the Hilbert space of a quantum system. Several equivalent characterizations of SP CPMs are proved and an explicit construction of all SP CPMs, is provided. For a subclass of the SP channels a construction in terms of joint unitary evolution with an ancilla system, is presented.Gluings of completely positive maps (CPMs) are defined and investigated. As a brief description of this concept consider a pair of ‘evolution machines’, each with the ability to evolve the internal state of a ‘particle’ inserted into its input. Each of these machines is characterized by a channel describing the operation the internal state has experienced when the particle is returned at the output. Suppose a particle is put in a superposition between the input of the first and the second machine. Here it is shown that the total evolution caused by a pair of such devices is not uniquely determined by the channels of the two machines. Such ‘global’ channels describing the machine pair are examples of gluings of the two single machine channels. Under the limiting assumption that all involved Hilbert spaces are finite-dimensional, an expression which generates all subspace preserving gluings of a given pair of CPMs, is derived. The nature of the non-uniqueness of gluings and its relation to a proposed definition of subspace locality, is discussed. PACS numbers: 03.65.-w, 03.67.-a