Session P09: Quantum Foundations II

Objectivity In Different Quantum Reference Frames

Quantum Darwinism and its frameworks describe objectivity of quantum systems via their information in the environment. However, the observation of these environments must be done with respect to a frame of reference. Given so, is objectivity consistent across different reference frames? Here, we take the quantum reference frame structure of [1] and consider the scenario where system and environment are static particles distributed across space. In this prescription, entanglement and coherence are interchangeable frame-dependent properties, and statistical mixedness and classical correlations are equivalently interchangeable. We find that the consistency of objectivity depends on the distinguishability of states in all relevant quantum reference frames, which in turn depends on non-degenerate relative separations, sharp localisation, and sufficiently large macro-fractions.

[1] F. Giacomini, E. Castro-Ruiz, and C. Brukner, “Quantum mechanics and the covariance of physical laws in quantum reference frames”, Nat. Commun. 10, 494 (2019)   

The Nakano-Nishijima-Gell-Mann Formula From Galois Fields

If world has a finite compact space (I₁₂₀: Poincare Dodecahedron) [1] and discrete coordinates [2,3], what happens? In this case, the problem of infinities in gravity and in the standard model might be avoided. To avoid this problem, quantum gravity theories such as the superstring theory or the loop quantum gravity are developing, but neither of those theories have been completed. We reconstruct the Nakano-Nishijima-Gell-Mann (NNG) formula by using a discrete Galois field without using continuous coordinate. When we reconstruct new theories with a Galois field, these new theories must satisfy fundamental conservation law related to unitary, Lorentz, and gauge invariance.
Here, we reexamine previous model [2] using isospin I_z. Consequently, instead of the NNG formula, we obtained the alternate formula Q = 2(n + I_z), where Q is charge number and n is multi-valuedness in Galois field. These results may be a starting point to develop a theory without many problems of infinity.

1) J.-P. Luminet et al.: Nature 425 (2003) 593.
2) H. R. Coish: Phys.Rev. 114 (1959) 383.
3) Y. Nambu, Field Theory of Galois Fields, In I. A. Batalin (ed), Quantum Field Theory and Quantum Statistics, Vol. 1, p. 625. IOP Publishing, 1987, also in Broken Symmetry, World Scientific Pub. Co. Inc., 1995.   

Quantum Fields from Quantum Cellular Automata? A No-Go Theorem and a Path Forward

A quantum walk (QW) is a unitary analogue of a classical random walk, and a quantum cellular automaton (QCA) is a unitary analogue of a classical cellular automaton. QWs can be treated as the one-particle sector of a QCA. Since quantum walks on the body-centered cubic lattice give rise to relativistic wave equations in the long wavelength limit, it is natural to seek for QCAs that give rise to quantum field theories in a similar limit. We show that this can be done in one spatial dimension, with the QCA being naturally described in terms of creation and annihilation operators that create or destroy particle locally, evolve straightforwardly under the QCA unitary, and obey the usual anticommutation relations (ACR). However, generalizing this construction to two or more spatial dimensions fails: the requirements on the creation and annihilation operators are inconsistent with a local QCA. For a QCA to give rise to a quantum field theory in the long-wavelength limit, one must give up at least one of the desired properties. A likely choice is to give up the requirement that the creation and annihilation operators create and destroy localized states.   

Quantum mechanics and the covariance of physical laws in quantum reference frames

In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? In my talk, I will introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a “superposition of coordinate transformations”. With this method, it is possible to describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, and to the possibility of defining the rest frame of a quantum system. This last feature allows us to find an operational definition of the relativistic spin of a Dirac particle, and to devise a "relativistic Stern-Gerlach experiment".   

Quantum Mechanics of a Single Photon and the question of its Localizability in Space

We use the dynamical Maxwell equations to determine the space of state vectors of a single photon. We endow this space with a relativistically invariant positive-definite inner product to make it into a Hilbert space. We identify the Hamiltonian operator with the generator of time translations, construct momentum and helicity operators, and introduce a chirality (direction-of-time) operator. Next, we construct a position operator that has commuting components. These also commute with the helicity and chirality operators. We obtain the eigenstates of the position operator, which we identify with the localized states of the photon, and use them to determine photon’s position wave function. The position wave function for the localized states has a delta-function singularity at a single point in space, but their electromagnetic fields diverge on a particular plane containing this point. This behavior turns out to be related to an implicit freedom in the choice of the position operator. Each choice for the position operator determines the plane at which the electromagnetic fields of a given localized state diverge.   

Quantized Electromagnetic-Field Propagation in a General Non-Local and Non-Stationary Dispersive and Absorbing Medium

We develop dynamical descriptions for the propagation of quantized electromagnetic fields, in the presence of environmental interactions. These descriptions are systematically and self-consistently developed in the complimentary Schrödinger and Heisenberg pictures. An open-systems (non-equilibrium) quantum-electrodynamics and quantum-statistical description is thereby provided for electromagnetic-field propagation in a general non-local and non-stationary dispersive and absorbing optical medium, including a fundamental microscopic treatment of decoherence and relaxation processes due to environmental collisional and electromagnetic interactions. Particular interest is centered on entangled states and other non-classical states of electromagnetic fields, which may be created by non-linear electromagnetic interactions and detected by the measurement of various electromagnetic-field correlation functions. Accordingly, we develop dynamical descriptions based on general forms of electromagnetic-field correlation functions involving both the electric-field and the magnetic-field components of the electromagnetic field, which are treated on an equal footing.   

Probability arises from entropy in axiomatic information thermodynamics

Axiomatic information thermodynamics (Entropy 2018, 20(4), 237) is a formal mathematical system that describes thermodynamic processes including those in which information is acquired, transformed or discarded. A central concept in the theory is the "eidostate", which is a simple array of possible states, including both thermodynamic macrostates and information records. Well-motivated axioms lead to an entropy function that describes the irreversibility of eidostate processes, and this function in turn leads to a unique probability measure for the states and records in a uniform eidostate. Does this development shed light on the kinds of reasoning available to physical agents that observe and intervene in the quantum world?


  

Emergence of the Born rule in quantum optics

The Born rule provides a fundamental connection between theory and observation in quantum mechanics, yet its origin remains a mystery. We consider this problem within the context of quantum optics using only classical physics and the assumption of a quantum electrodynamic vacuum that is real rather than virtual. The connection to observation is made via classical intensity threshold detectors that are used as a simple, deterministic model of single-photon detection. By following standard experimental conventions of data analysis on discrete detection events, it can be shown that this model is capable of reproducing several observed phenomena thought to be uniquely quantum in nature.   

Interaction-Free Energy Transfer

The interaction-free measurement scheme of Elitzur and Vaidman enables a probe particle to detect the presence of an object blocking one arm of a Mach-Zender (MZ) interferometer without the probe ever locally interacting with that object. THE MZ is tuned so that without the block, the dark port detector never fires. The presence of the block changes the interference pattern in the MZ, allowing the dark port to fire, in which case the block is detected while the probe traveled the unblocked arm. We consider a quantum block in the ground state of a confining potential, a superposition of inside and outside one arm of the MZ. When the dark port detector fires, the quantum block is localized inside the arm, which increases its energy. This energy is delivered by the probe even though it traveled the unblocked arm to reach the dark port, and we thus obtain interaction-free energy transfer. This effect obtains even though the coupling Hamiltonian is local.   

Diffraction-based Interaction-Free Measurements

We consider the problem of determining if a single-slit contains a bomb in its middle region using a single photon test. We notice that the quantum case allows the bomb to significantly increase the probability for the photon to reach certain detection positions, in particular the dark bands of the single-slit pattern. The bomb's presence can be inferred without exploding it when the photon lands at one of these positions. We thus find a diffractive extension to the interaction-free measurement protocol of Elitzur and Vaidman. In addition, we consider the role of time in interaction-free measurements. We find that, if the bomb is thought of as a measuring device which checks periodically if the photon is in its space, then it can only find the particle (and "explode") if its measurement rate is finite. The alternative is the Zeno limit, where the bomb acts like an infinite potential barrier. Critics of interaction-free measurement may find it interesting that the normal, non-Zeno bomb can be passed through without setting it off.   

Quantum mysteries for anybody: Solved

In 1981, Mermin published a now famous paper titled, "Bringing home the atomic world: Quantum mysteries for anybody." Therein, he presented the 'Mermin device' that illustrates the conundrum of entanglement per the spin singlet state for the "general reader." He then challenged the "physicist reader" to explain the way the device works "in terms meaningful to a general reader struggling with the dilemma raised by the device." I will show how conservation per no preferred reference frame (NPRF) answers that challenge, but still leaves a mystery for those who seek dynamical explanation via hidden variables or 'causal influences'. Since NPRF also underwrites the postulates of special relativity, we see a common theme between relativistic and non-relativistic modern physics.