Reconstructing quantum theory from diagrammatic postulates

Abstract

A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann. We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.

The arrow of time in operational formulations of quantum theory

“…This approach has lead to a wealth of insights, both of theoretical and technological value [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In particular, the information-theoretic reconstructions of quantum theory [21][22][23][24][25][26][27][28][29][30] derive the formal Hilbert space structure of quantum theory from simple physical principles (much like Einstein's two postulates of special relativity allowed to re-derive the Lorentz transformations [31,32]). With the notable exception of Ding Jia's [29], the reconstructions either start by considering a space of theories that is intrinsically time-oriented [21-23, 25-27, 29] or introduce the time orientation explicitly as a postulate [24,28,30] ("no signaling from the future").…”

Routed quantum circuits

“…Any theory with these features accepts sound and intuitive diagrammatic representations of its maps in terms of circuits [53,5]. These concepts originate from category theory, a mathematical theory which has been at the centre of a recent reformalisation of quantum theory [3,5,48,59]. Our point here is not to present them in depth, but to provide the reader with an intuition of the simple structures that they express.…”

“…First, given the strong compositional flavour of the structures at hand, it would need to be appropriately compositional (i.e., emphasising the sequential and parallel composition operations on channels and their structures). Second, drawing lessons from recent successful re-expressions of quantum theory [45,46,47,48,3], this extension would have to form a process theory [5], focusing on dynamical processes (i.e., channels) and obtaining states as specific cases of the latter: the emphasis on compositional structure is more natural in process theories. Third, it should yield intuitive, higherlevel (i.e.…”

Process Tomography in General Physical Theories

“…The origin of this favourable situation is a property, known as Local Tomography, whereby the states of every composite system can be uniquely identified by performing local measurements on the components [16,17,[19][20][21][22][23][24][25][26][27]. In quantum foundations, Local Tomography has often been taken as an axiom for the characterization of standard quantum theory (on complex Hilbert spaces) [21,25,[28][29][30][31][32][33][34] and as a principle for the construction of new physical theories [22][23][24]. In locally tomographic theories, the action of a process on its input system uniquely determines the action of the process on every composite system [24].…”

“…Note that, in turn, every finite set of auxiliary systems {C i } k i=1 can be replaced without loss of generality by a single auxiliary system R, for example by merging all the systems of the finite set into a composite system. Indeed, most of the frameworks for general probabilistic theories include a notion of system composition, denoted by ⊗ and corresponding to a generalization of the tensor product in quantum theory [21][22][23][24]27,28,34,[42][43][44][45][46][47][48]. As long as the number of components is finite, the composite system is well defined in all these frameworks.…”