Autonomous Ticking Clocks from Axiomatic Principles

There are many different types of time keeping devices. We use the phrase ticking clock to describe those which -- simply put -- "tick" at approximately regular intervals. Various important results have been derived for ticking clocks, and more are in the pipeline. It is thus important to understand the underlying models on which these results are founded. The aim of this paper is to introduce a new ticking clock model from axiomatic principles that overcomes concerns in the community about the physicality of the assumptions made in previous models. The ticking clock model in [arXiv:1806.00491] achieves high accuracy, yet lacks the autonomy of the less accurate model in [10.1103/PhysRevX.7.031022]. Importantly, the model we introduce here achieves the best of both models: it retains the autonomy of [10.1103/PhysRevX.7.031022] while allowing for the high accuracies of [arXiv:1806.00491]. What is more, [10.1103/PhysRevX.7.031022] is revealed to be a special case of the new ticking clock model.Comment: 14 + 14 page

Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames

Following the introduction of the task of reference frame error correction, we show how, by using reference frame alignment with clocks, one can add a continuous Abelian group of transversal logical gates to any error-correcting code. With this we further explore a way of circumventing the no-go theorem of Eastin and Knill, which states that if local errors are correctable, the group of transversal gates must be of finite order. We are able to do this by introducing a small error on the decoding procedure that decreases with the dimension of the frames used. Furthermore, we show that there is a direct relationship between how small this error can be and how accurate quantum clocks can be: the more accurate the clock, the smaller the error; and the no-go theorem would be violated if time could be measured perfectly in quantum mechanics. The asymptotic scaling of the error is studied under a number of scenarios of reference frames and error models. The scheme is also extended to errors at unknown locations, and we show how to achieve this by simple majority voting related error correction schemes on the reference frames. In the Outlook, we discuss our results in relation to the AdS/CFT correspondence and the Page-Wooters mechanism.Comment: 10+35 pages. Also see related work uploaded to the arXiv on the same day; arXiv:1902.0771

Dynamical error bounds for continuum discretisation via Gauss quadrature rules, -- a Lieb-Robinson bound approach

Instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discrete set of modes. We derive analytical error bounds on expectation values of system observables that have been time evolved under such discretised Hamiltonians. These bounds take on the form of a function of time and the number of discrete modes, where the discrete modes are chosen according to Gauss quadrature rules. The derivation makes use of tools from the field of Lieb-Robinson bounds and the theory of orthonormal polynominals.Comment: 12 pages + 14 pages of proofs and appendices, Journal of Mathematical Physics, Vol.57, Issue 2 (2016) http://scitation.aip.org/content/aip/journal/jmp/57/2/10.1063/1.494043

The maximum efficiency of nano heat engines depends on more than temperature

Sadi Carnot's theorem regarding the maximum efficiency of heat engines is considered to be of fundamental importance in thermodynamics. This theorem famously states that the maximum efficiency depends only on the temperature of the heat baths used by the engine, but not on the specific structure of baths. Here, we show that when the heat baths are finite in size, and when the engine operates in the quantum nanoregime, a revision to this statement is required. We show that one may still achieve the Carnot efficiency, when certain conditions on the bath structure are satisfied; however if that is not the case, then the maximum achievable efficiency can reduce to a value which is strictly less than Carnot. We derive the maximum efficiency for the case when one of the baths is composed of qubits. Furthermore, we show that the maximum efficiency is determined by either the standard second law of thermodynamics, analogously to the macroscopic case, or by the non increase of the max relative entropy, which is a quantity previously associated with the single shot regime in many quantum protocols. This relative entropic quantity emerges as a consequence of additional constraints, called generalized free energies, that govern thermodynamical transitions in the nanoregime. Our findings imply that in order to maximize efficiency, further considerations in choosing bath Hamiltonians should be made, when explicitly constructing quantum heat engines in the future. This understanding of thermodynamics has implications for nanoscale engineering aiming to construct small thermal machines.Comment: Main text 14 pages. Appendix 60 pages. Accepted in Journal Quantu

Universal quantum modifications to general relativistic time dilation in delocalised clocks

The theory of relativity associates a proper time with each moving object via its world line. In quantum theory however, such well-defined trajectories are forbidden. After introducing a general characterisation of quantum clocks, we demonstrate that, in the weak-field, low-velocity limit, all "good" quantum clocks experience time dilation as dictated by general relativity when their state of motion is classical (i.e. Gaussian). For nonclassical states of motion, on the other hand, we find that quantum interference effects may give rise to a significant discrepancy between the proper time and the time measured by the clock. The universality of this discrepancy implies that it is not simply a systematic error, but rather a quantum modification to the proper time itself. We also show how the clock's delocalisation leads to a larger uncertainty in the time it measures -- a consequence of the unavoidable entanglement between the clock time and its center-of-mass degrees of freedom. We demonstrate how this lost precision can be recovered by performing a measurement of the clock's state of motion alongside its time reading.Comment: 7 + 10 pages. V3: accepted versio

A general framework for consistent logical reasoning in Wigner's friend scenarios: subjective perspectives of agents within a single quantum circuit

It is natural to expect a complete physical theory to have the ability to consistently model agents as physical systems of the theory. In [Nat. Comms. 9, 3711 (2018)], Frauchiger and Renner (FR) claim to show that when agents in quantum theory reason about each other's knowledge in a certain Wigner's friend scenario, they arrive at a logical contradiction. In light of this, Renner often poses the challenge: provide a set of reasoning rules that can be used to program quantum computers that may act as agents, which are (a) logically consistent (b) generalise to arbitrary Wigner's friend scenarios (c) efficiently programmable and (d) consistent with the temporal order of the protocol. Here we develop a general framework where we show that every logical Wigner's friend scenario (LWFS) can be mapped to a single temporally ordered quantum circuit, which allows agents in any LWFS to reason in a way that meets all four criteria of the challenge. Importantly, our framework achieves this general resolution without modifying classical logic or unitary quantum evolution or the Born rule, while allowing agents' perspectives to be fundamentally subjective. We analyse the FR protocol in detail, showing how the apparent paradox is resolved there. We show that apparent logical contradictions in any LWFS only arise when ignoring the choice of Heisenberg cut in scenarios where this choice does matter, and taking this dependence into account will always resolve the apparent paradox. Our results establish that universal applicability of quantum theory does not pose any threat to multi-agent logical reasoning and we discuss the implications of these results for FR's no-go theorem. Moreover, our formalism suggests the possibility of a truly relational and operational description of Wigner's friend scenarios that is consistent with quantum theory as well as probability theory applied to measurement outcomes.Comment: 33 + 14 pages, 10 figure