Indefinite Causal Structure

The quantum SWITCH is the simplest example of indefinite causal structure. Technically, it is a higher-order transformation that takes two physical processes A and B in input and combines them in a coherent superposition of two alternative orders, AB and BA. In the past decade, the quantum SWITCH has been the object of active research, both theoretically and experimentally. In this talk, I will review the state of the art, and outline two new applications to quantum Shannon theory and quantum metrology.

I will present an extension of the recent theory of quantum causal models to cyclic causal structures. This offers a novel causal perspective on processes beyond those corresponding to standard circuits, such as processes with dynamical causal order and causally nonseparable processes, including processes violating causal inequalities. As an application, I will use the algebraic structure of process operators that is induced by the causal structure to prove that all unitarily extendible bipartite processes are causally separable, i.e., their unitary extensions are variations of the quantum SWITCH. Remarkably, the latter implies that all unitarily extendible tripartite quantum processes have realizations on time-delocalized systems within standard quantum mechanics. This includes, in particular, classical processes violating causal inequalities, which admit simple implementations! I will explain what the violation of causal inequalities implies for the variables of interest in these implementations. The answer is given again by the theory of cyclic causal models. Based on joint works with Jonathan Barrett, Cyril Branciard, Robin Lorenz, and Julian Wechs.

When studying (definite or indefinite) causal orderings of processes, it is often useful to consider higher-order processes, i.e. processes which take other processes as their input. However, as a recent no-go result of Guerin et al indicates, our naive first-order notions of "composition" of processes become ill-defined at higher-order. Unlike state spaces, there are multiple non-equivalent notions of "joint system" for process spaces and many different ways one might attempt to plug processes together, with only some giving well-defined (i.e. normalised) processes as outputs. While this starts to look a bit like the Wild West, I'll show in this talk that we can get quite a bit of mileage from considering just two kinds of joint systems: a "non-signalling" tensor product, and a (de Morgan dual) "signalling" product. The interaction between these two products has in fact been well-understood by logicians since the 1980s in a very different disguise: multiplicative linear logic. Using this connection, I'll show how a set of "contractibility" criteria due to Danos and Regnier give a relatively simple, dimension-independent technique for determining whether an arbitrary plugging of higher-order processes is well-defined.

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? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a “superposition of coordinate transformations”. We 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, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.

The lesson of general relativity is background independence: a physical theory should not be formulated in terms of external structures. This motivates a relational approach to quantum dynamics, which is necessary for a quantum theory of gravity. Using a covariant POVM to define a time observable, I will introduce the so-called trinity of relational quantum dynamics comprised of three distinct formulations of the same relational quantum theory: evolving constants of motion, the Page-Wootters formalism, and a symmetry reduction procedure. The equivalence between these formulations yields a temporal frame change map that transforms between the dynamics seen by different clocks. This map will be used to illustrate a temporal nonlocality effect that results in a superposition of time evolutions from the perspective of a clock indicating a superposition of different times. Then, a time-nonlocal modification to the Schrödinger equation will be shown to manifest when a system is coupled to the clock that is tracking its evolution. Such clock-system interactions should be expected at some scale when the gravitational interaction between them is taken into account. Finally, I will examine relativistic particles with internal degrees of freedom that constitute a clock that tracks their proper time. By evaluating the conditional probability associated with two such clocks reading different proper times, I will show that these clocks exhibit a novel quantum time dilation effect when moving in a superposition of different momenta.

We are building an experiment in which a levitated 1 µm diamond containing a nitrogen vacancy (NV) centre would be put into a spatial quantum superposition [1-3]. This would be able to test theories of spontaneous wavefunction collapse [4]. We have helped theory collaborators to propose how to do this experiment [5-9], as well as a much more experimentally ambitious extension which would test if gravity permits a quantum superposition [10]. There are related proposals from other groups [11-13]. [1] A. T. M. A. Rahman, A. C. Frangeskou, M. S. Kim, S. Bose, G. W. Morley & P. F. Barker, Sci. Rep. 6, 21633 (2016). [2] A. T. M. A. Rahman, A. C. Frangeskou, P. F. Barker & G. W. Morley, Rev. Sci. Instrum. 89, 023109 (2018). [3] A. C. Frangeskou, A. T. M. A. Rahman, L. Gines, S. Mandal, O. A. Williams, P. F. Barker & G. W. Morley, NJP 20, 043016 (2018). [4] A. Bassi, K. Lochan, S. Satin, T. P. Singh & H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013). [5] S. Bose & G. W. Morley, arXiv:1810.07045 (2018). [6] M. Scala, M. S. Kim, G. W. Morley, P. F. Barker & S. Bose, PRL 111, 180403 (2013). [7] C. Wan, M. Scala, G. W. Morley, A. T. M. A. Rahman, H. Ulbricht, J. Bateman, P. F. Barker, S. Bose & M. S. Kim, PRL 117, 143003 (2016). [8] R. J. Marshman, A. Mazumdar, G. W. Morley, P. F. Barker, H. Steven & S. Bose, arXiv:1807.10830 (2018). [9] J. S. Pedernales, G. W. Morley & M. B. Plenio, arXiv:1906.00835 (2019). [10] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim & G. Milburn, PRL 119, 240401 (2017). [11] Z.-q. Yin, T. Li, X. Zhang & L. M. Duan, PRA 88, 033614 (2013). [12] A. Albrecht, A. Retzker & M. B. Plenio, PRA 90, 033834 (2014). [13] C. Marletto & V. Vedral, PRL 119, 240402 (2017).

I will discuss how the standard frameworks for operational theories involve a scrambling of causal and inferential concepts. I will then present a new framework for operational theories which separates out the inferential and the causal aspects of a given physical theory. Generalized probabilistic theories and operational probabilistic theories are recovered within our framework when one ignores some of these distinctions. We then make similar refinements to the traditional notion of ontological theories, and discuss how our framework revises the standard notions of ontological representations and of generalized noncontextuality.