The Power of Quantum Systems on a Line

We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses. Supported by Israel Science Foundation grant number 039-7549, Binational Science Foundation grant number 037-8404, and US Army Research Office grant number 030-7790. Supported by CIFAR, by the Government of Canada through NSERC, and by the Province of Ontario through MRI. Partially supported by NSF Grant CCR-0514082. This work was mainly done while the author was at CNRS and LRI, University of Paris-Sud, Orsay, France. Partially supported by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848, by an ANR AlgoQP grant of the French Research Ministry, by an Alon Fellowship of the Israeli Higher Council of Academic Research, by an Individual Research grant of the ISF, and by a European Research Council (ERC) Starting Grant.     

On the Orthocomplementation of State–Property–Systems of Contextual Systems

We adopt an operational approach to quantum mechanics in which a physical system is defined by the mathematical structure of its set of states and properties. We present a model in which the maximal change of state of the system due to interaction with the measurement context is controlled by a parameter which corresponds with the number N of possible outcomes in an experiment. In the case N=2 the system reduces to a model for the spin measurements on a quantum spin-1/2 particle. In the limit N→∞ the system is classical, i.e. the experiments are deterministic and its set of properties is a Boolean lattice. For intermediate situations the change of state due to measurement is neither ‘maximal’ (i.e. quantum) nor ‘zero’ (i.e. classical). We show that two of the axioms used in Piron’s representation theorem for quantum mechanics are violated, namely the covering law and weak modularity. Next, we discuss a modified version of the model for which it is even impossible to define an orthocomplementation on the set of properties. Another interesting feature for the intermediate situations of this model is that the probability of a state transition in general not only depends on the two states involved, but also on the measurement context which induces the state transition.     

Scaled-energy spectroscopy of a ｜M｜= 1 Rydberg barium atom in an electric field

We observe strong energy-dependent quantum defects in the scaled-energy Stark spectra for |M| = 1 Rydberg states of barium atoms at three scaled energies: ε = 2.000,ε = 2.500 and ε = 3.000.In an attempt to explain the observations,theoretical calculations of closed orbit theory based on a model potential including core effect are performed for non-hydrogenic atoms.While such a potential has been uniformly successful for alkali atoms with a single valence electron,it fails to match experimental results for barium atoms in the 6snp Rydberg states with two valence electrons.Our study points out that this discrepancy is due to the strong perturbation from the 5d8p state,which voids the simple approximation for constant quantum defects of principle quantum number n.     

Teleportation of a continuous variable

Measurements used in quantum teleportation are examined from the standpoint of the general theory of quantum-mechanical measurements. It is shown that in order to find a teleported state, it is sufficient to know only the resolution of the identity operator (positive operator-valued measure) generated by the respective instrument (the quantum operation determining the change in the state of the system as a result of the measurement) in the state space of the system, rather than the instrument itself. A protocol for quantum teleportation of the state of a system with a nondegenerate continuous spectrum based on a measurement which corresponds to a certain nonorthogonal resolution of the identity operator is proposed. Zh. éksp. Teor. Fiz. 116, 777–792 (September 1999)     

Interface <Emphasis Type="Italic">D</Emphasis><Superscript>−</Superscript> complexes in a two-dimensional electron system

The excitation spectrum of a two-dimensional electron system in high-quality AlGaAs/GaAs quantum wells has been studied by Raman scattering. New Raman lines due to the excitation of interface D ⁻ complexes in which two electrons localized in a quantum well are coupled to a charged impurity at the quantum well interface have been identified. The ground state of the interface D ⁻ complexes has been found to change in the transverse magnetic field from spin-singlet to spin-triplet, similar to a change in the ground state of the system of two electrons localized in a harmonic potential.     

Interface D − complexes in a two-dimensional electron system

The excitation spectrum of a two-dimensional electron system in high-quality AlGaAs/GaAs quantum wells has been studied by Raman scattering. New Raman lines due to the excitation of interface D ^? complexes in which two electrons localized in a quantum well are coupled to a charged impurity at the quantum well interface have been identified. The ground state of the interface D ^? complexes has been found to change in the transverse magnetic field from spin-singlet to spin-triplet, similar to a change in the ground state of the system of two electrons localized in a harmonic potential.

     

Unlearning quantum information

Measurement can drive quantum dynamics, for example in ancilla driven quantum computation where unitary evolution is generated by measurements that extract no information. Where a measurement does reveal some information about the system, it may sometimes be possible to “unlearn” this information and restore unitary evolution through subsequent measurements. Here we analyse two methods of quantum “unlearning” and present a simplified proof of the bound on the probability of successfully applying the required correction operators. The probability of successful recovery is inversely related to the ability of the initial measurement to exclude the possibility of a state. As a consequence there exist unrecoverable measurements that provide little information gain.     

Spontaneous recoherence of quantum states after decoherence

In this work, we identify the set of time-dependent pure states building the statistical mixture to which a system, initially in a pure state, is driven by the reservoir. This set of time-dependent pure states, composing what we term a pure basis, are those that diagonalize the reduced density operator of the system. Next, we show that the evolution of the pure-basis states reveals an interesting phenomenon as the system, after decoherence, evolves toward the equilibrium: the spontaneous recoherence of quantum states. Around our defined recoherence time, the statistical mixture associated with a special kind of initial states termed even-symmetric, spontaneously undergoes a recoherence process, by which the initial state of the system emerges from the mixture except for its reduced excitation drained into the reservoir. This phenomenon reveals that the reservoir only shuffle the original information carried out by the initial state of the system instead of erasing it. Moreover, as the spontaneously recohered state occurs only for asymptotic time, we also present a protocol to extract it from the mixture through specific projective measurements. The password to retrieve the original information stems is the knowledge of both the initial state itself and the associated pure basis. A definition of the decoherence time of an N-state superposition is also presented.     

Undecidable Classical Properties of Observers

A property of a system is called actual, if the observation of the outcome of the test that pertains to that property yields an affirmation with certainty. We formalize the act of observation by assuming the outcome itself is an actual property of the state of the observer after the act of observation and correlates with the state of the system. For an actual property this correlation needs to be perfect. A property is called classical if either the property or its negation is actual. We show by a diagonal argument that there exist classical properties of an observer that he cannot observe perfectly. Because states are identified with the collection of properties that are actual for that state, it follows no observer can perfectly observe his own state. Implications for the quantum measurement problem are briefly discussed. PACS: 02.10-v, 03.65.Ta     

Noise sensing via stochastic quantum Zeno

The dynamics of any quantum system is unavoidably influenced by the external environment. Thus, the observation of a quantum system (probe) can allow the measure of the environmental features. Here, to spectrally resolve a noise field coupled to the quantum probe, we employ dissipative manipulations of the probe, leading to so-called Stochastic Quantum Zeno (SQZ) phenomena. A quantum system coupled to a stochastic noise field and subject to a sequence of protective Zeno measurements slowly decays from its initial state with a survival probability that depends both on the measurement frequency and the noise. We present a robust sensing method to reconstruct the unknown noise power spectral density by evaluating the survival probability that we obtain when we additionally apply a set of coherent control pulses to the probe. The joint effect of coherent control, protective measurements and noise field on the decay provides us the desired information on the noise field.     

Spectroscopy of molecular states in a few-electron double quantum dot

Semiconductor quantum dots, so-called artificial atoms, have attracted considerable interest as mesoscopic model systems and prospective building blocks of the “quantum computer”. Electrons are trapped locally in quantum dots, forming controllable and coherent mesoscopic atom- and moleculelike systems. Electrostatic definition of quantum dots by use of top gates on a GaAs/AlGaAs heterostructure allows wide variation of the potential in the underlying two-dimensional electron gas. By distorting the trapping potential of a single quantum dot, a strongly tunnel-coupled double quantum dot can be defined. Transport spectroscopy measurements on such a system charged with N=0,1,2,… electrons are presented. In particular, the tunnel splitting of the double well potential for up to one trapped electron is unambiguously identified. It becomes visible as a pronounced level anticrossing at finite source drain voltage. A magnetic field perpendicular to the two-dimensional electron gas also modulates the orbital excitation energies in each individual dot. By tuning the asymmetry of the double well potential at finite magnetic field the chemical potentials of an excited state of one of the quantum dots and the ground state of the other quantum dot can be aligned, resulting in a second level anticrossing with a larger tunnel splitting. In addition, data on the two-electron transport spectrum are presented.     

Limited Holism and Real-Vector-Space Quantum Theory

Quantum theory has the property of “local tomography”: the state of any composite system can be reconstructed from the statistics of measurements on the individual components. In this respect the holism of quantum theory is limited. We consider in this paper a class of theories more holistic than quantum theory in that they are constrained only by “bilocal tomography”: the state of any composite system is determined by the statistics of measurements on pairs of components. Under a few auxiliary assumptions, we derive certain general features of such theories. In particular, we show how the number of state parameters can depend on the number of perfectly distinguishable states. We also show that real-vector-space quantum theory, while not locally tomographic, is bilocally tomographic.     

Unitary state preparation,local position measurements,and spin in quantum mechanics

The orthodox presentation of quantum theory often includes statements on state preparation and measurements without mentioning how these processes can be achieved. The often quoted projection postulate is regarded by many as problematical. This paper presents a systematic framework for state preparation and measurement. Within the existing Hilbert space formulation of quantum mechanics for spinless particles we show that it is possible (1)to prepare an arbitrary state and (2)to reduce all quantum measurements to local position measurements in an asymptotic way by unitary evolution processes without recourse to the projection postulate. A generalization to spin-1/2particles is also given. The theory presented provides a general mathematical and theoretical foundation for many practical schemes for state preparation and measurement.     

Stochastic differential equations anda posteriori states in quantum mechanics

In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is atrajectoryfor one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to timet has been observed. The response to this question is the notion ofa posteriori states and afilteringequation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).     

Preservation of quantum states via a super-Zeno effect on ensemble quantum computers

Following a recent proposal by Dhar et al (2006 Phys. Rev. Lett. 96 100405), we demonstrate experimentally the preservation of quantum states in a two-qubit system based on a super-Zeno effect using liquid-state nuclear magnetic resonance techniques. Using inverting radiofrequency pulses and delicately selecting time intervals between two pulses, we suppress the effect of decoherence of quantum states. We observe that preservation of the quantum state |11\rg with the super-Zeno effect is three times more efficient than the ordinary one with the standard Zeno effect.     

Quantum decoherence from adiabatic entanglement with external one or a few degrees of freedom

Based on the Born-Oppenhemer approximation, the concept of adiabatic quantum entanglement is introduced to account for quantum decoherence of a quantum system due to its interaction with a large system of one or a few degrees of freedom. In the adiabatic limit, it is shown that the wave function of the total system formed by the quantum system plus the large system can be factorized as an entangled state with correlation between adiabatic quantum states and quasi-classical motion configurations of the large system. In association with a novel viewpoint about quantum measurement, which has been directly verified by most recent experiments [e.g., S. Durr et al., Nature 33, 359 (1998)], it is shown that the adiabatic entanglement is indeed responsible for the quantum decoherence and thus can be regarded as a “clean” quantum measurement when the large system behaves as a classical object. By taking the large system respectively to be a macroscopically distinguishable spatial variable, a high spin system and a harmonic oscillator with a coherent initial state, three illustrations are presented with their explicit solutions in this paper. Received 26 February 2000 and Received in final form 14 July 2000     

Quantum beats in fluorescence for multi-level atomic system

For Λ-type three-level atomic systems we have clarified using diagram that (1) it is impossible to observe quantum beats due to the ground state sublevels by measuring the time dependence of the fluorescence intensity, and (2) why it is physically possible to observe and how we can observe quantum beats in the ground state sublevels by using fluorescence. Generalization of the results shows that we can determine from which state (the ground state or the excited state) the quantum beats are originated. Analytical result is shown for four-level atomic systems.     

Teleportation on a quantum dot array

We present a model of quantum teleportation protocol based on a double quantum dot array. The unknown qubit is encoded using a pair of quantum dots, with one excess electron, coupled by tunneling. It is shown how to create a maximally entangled state using an adiabatically increasing Coulomb repulsion between different dot pairs. This entangled state is exploited to perform teleportation again using an adiabatic coupling between itself and the incoming unknown state. Finally, a sudden separation of Bob's qubit allows a time evolution of Alice's, which amounts to a modified version of standard Bell measurement. A transmission over a long distance could be obtained by considering the entangled state of a chain of N coupled double quantum dots. The system is shown to be increasingly robust with N against decoherence due to phonons.     

Quantum theory of state reduction and measurement

The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrödinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrödinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.     

Non-Markovian dynamics and quantum jumps

Experiments with trapped particles have demonstrated the existence of quantum jumps and the discrete nature of single-system dynamics in quantum mechanics. The concept of jumps is also a powerful tool for simulating and understanding open quantum systems. In non-Markovian systems jump probabilities can become negative due to memory effects between the system and its environment. We discuss a recently presented method that can handle both positive and negative probabilities and provides powerful insight into the dynamics of open systems with memory. The key element is a reversed quantum jump to a system state that was, in principle, already destroyed by an earlier normal jump. Instead of using artificial extensions of the system or exploiting hidden variables we take advantage of the information stored in the quantum ensemble itself.