The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg's `quantum-theoretical Umdeutung (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schroedinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h -> 0 of small Planck's constant (in a finite system), in the limit N goes to infinity of a large system with $N$ degrees of freedom (at fixed h), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C*-algebras and the concepts of superselection sectors and macroscopic observables. In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain ``classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are ``einselected", i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories. This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it.

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Between classical and quantum

Quantum metrology provides a route to overcome practical limits in sensing devices. It holds particular relevance to biology, where sensitivity and resolution constraints restrict applications both in fundamental biophysics and in medicine. Here, we review quantum metrology from this biological context, focusing on optical techniques due to their particular relevance for biological imaging, sensing, and stimulation. Our understanding of quantum mechanics has already enabled important applications in biology, including positron emission tomography (PET) with entangled photons, magnetic resonance imaging (MRI) using nuclear magnetic resonance, and bio-magnetic imaging with superconducting quantum interference devices (SQUIDs). In quantum metrology an even greater range of applications arise from the ability to not just understand, but to engineer, coherence and correlations at the quantum level. In the past few years, quite dramatic progress has been seen in applying these ideas into biological systems. Capabilities that have been demonstrated include enhanced sensitivity and resolution, immunity to imaging artifacts and technical noise, and characterization of the biological response to light at the single-photon level. New quantum measurement techniques offer even greater promise, raising the prospect for improved multi-photon microscopy and magnetic imaging, among many other possible applications. Realization of this potential will require cross-disciplinary input from researchers in both biology and quantum physics. In this review we seek to communicate the developments of quantum metrology in a way that is accessible to biologists and biophysicists, while providing sufficient detail to allow the interested reader to obtain a solid understanding of the field. We further seek to introduce quantum physicists to some of the central challenges of optical measurements in biological science.

Quantum metrology and its application in biology

We propose a general method to embed target states into the middle of the energy spectrum of a many-body Hamiltonian as its energy eigenstates. Employing this method, we construct a translationally invariant local Hamiltonian with no local conserved quantities, which does not satisfy the eigenstate thermalization hypothesis. The absence of eigenstate thermalization for target states is analytically proved and numerically demonstrated. In addition, numerical calculations of two concrete models also show that all the energy eigenstates except for the target states have the property of eigenstate thermalization, from which we argue that our models thermalize after a quench even though they do not satisfy the eigenstate thermalization hypothesis.

Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis.

The purpose of a phase-preserving linear amplifier is to make a small signal larger, regardless of its phase, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise ratio. Quantum mechanics limits how well this can be done: A high-gain linear amplifier must degrade the signal-to-noise ratio; the noise added by the amplifier, when referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we derive the quantum constraints on the entire distribution of added noise: We show that any phase-preserving linear amplifier is equivalent to a parametric amplifier with a physical state for the ancillary mode; the noise added to the amplified field mode is distributed according to the Wigner function of the ancilla state.

康治伏見

Papers

Some formal properties of the density matrix

康治 伏見

Papers

Some formal properties of the density matrix

康治伏見