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Quantum from Principles

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Quantum Theory: Informational Foundations and Foils

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 181))

Abstract

Quantum theory was discovered in an adventurous way, under the urge to solve puzzles—like the spectrum of the blackbody radiation—that haunted the physics community at the beginning of the 20th century. It soon became clear, though, that quantum theory was not just a theory of specific physical systems, but rather a new language of universal applicability. Can this language be reconstructed from first principles? Can we arrive at it from logical reasoning, instead of ad hoc guesswork? A positive answer was provided in Phys Rev A, 81:062348, 2010 [34], Phys Rev A, 84:012311, 2011 [26], where we put forward six principles that identify quantum theory uniquely in a broad class of theories. We first defined a class of “theories of information”, constructed as extensions of probability theory in which events can be connected into networks. In this framework, we formulated the six principles as rules governing the control and the accessibility of information. Directly from these rules, we reconstructed a number of quantum information features, and eventually, the whole Hilbert space framework. In short, our principles characterize quantum theory as the theory of information that allows for maximal control of randomness.

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Notes

  1. 1.

    This was also the title of one influential conference, held in May 2000 at the Université de Montréal [22], which kickstarted the new wave of quantum axiomatizations.

  2. 2.

    More precisely, “nothing that the theory cares to describe”.

  3. 3.

    Per se, the mathematical formalism does not force us to interpret the order of sequential composition as an order in time. Nevertheless, composition in time is the reference situation that we will have in mind when phrasing our axioms.

  4. 4.

    Note that the summation is well-defined thanks to the vector space structure of \(\mathsf {Transf}_{\mathbb R}(\mathrm {A}\rightarrow \mathrm{B})\).

  5. 5.

    In previous works, we used different names for transformations that do not allow for side information: in Refs. [26, 34] they were called atomic, while in the popularized version of Ref. [49] they were called fine-grained. We apologize with our readers for the changes of terminology, due to an ongoing search for the word that best captures this operational concept. In this chapter, we adopted the word pure, because (i) this term is the standard one in the case of states and (ii) using the same term for transformations should hopefully ease the reading. Still, a warning is in order: when the set of transformations \(\mathsf {Transf}(\mathrm {A}\rightarrow \mathrm{B})\) is convex, the pure transformations \(\mathsf {Pur}\mathsf {Transf}(\mathrm {A}\rightarrow \mathrm{B})\) may not coincide with the extreme points of \(\mathsf {Transf}(\mathrm {A}\rightarrow \mathrm{B})\). For example, in quantum theory the identity effect \( I_\mathrm {A}\) is an extreme point of the set of effects, but is not pure in the sense of our definition because it can be decomposed e.g. as \(I_\mathrm {A}= \sum _{n=1}^{d_\mathrm {A}} \, P_n\), where the effects \( \{P_n = |n\rangle \langle n|~|~ n=1,\dots , \, d_\mathrm {A}\}\) represent a projective measurement on some orthonormal basis \(\{ |n\rangle ~|~ n= 1,\dots \, , d_\mathrm {A}\}\).

  6. 6.

    Note that, in principle, our definition of “internal transformations” may not include all the transformations in the interior of the cone, because the \(\lambda \, \mathcal {F}\) and \(\mathcal {G}\) may fail to coexist in a test. However, this annoying discrepancy disappears under the mild assumption that the set of transformations is convex. Later, we will justify this assumption on the basis of the Causality axiom.

  7. 7.

    It turns out that the second condition is automatically satisfied if the theory satisfies the Causality axiom—see the next section.

  8. 8.

    We differentiate the names in order to highlight the different roles of these principles in our reconstruction. Mathematically, there is no difference between axioms, postulates, background assumptions, and requirements in the OPT framework (all of them are “axioms”). The point of using different names is just to provide a more intuitive picture.

  9. 9.

    Recall that we are assuming that the state spaces are finite-dimensional.

  10. 10.

    The expression is due to John Smolin, see e.g. the lecture notes [58].

  11. 11.

    It is worth stressing that Schrödinger’s paper was not just about the existence of entangled states, but also about how entanglement interacted with the reversible dynamics and with the process of measurement (cf. the notion of steering, which made its first appearance in the very same paper).

  12. 12.

    We recall that a face of a convex set C is a convex subset \(F\subseteq C\) satisfying the condition that, for every \(x \in F\), if x is a non-trivial convex combination of \(x_1\) and \(x_2\) with \(x_1,x_2 \in C\), then \(x_1\) and \(x_2\) belong to F.

  13. 13.

    We call an effect of system \(\mathrm {A}\) normalized iff there exists an effect a state \(\rho \) such that \((a|\rho ) =1\).

  14. 14.

    In the original work [26], we also required that projections be pure. However, in the context of our axioms, purity is implied by the two conditions in the present definition. This follows from the fact that i) one can construct a pure projection, and ii) it is possible to prove that projections are unique. A sketch of proof is the following: First, one can prove that for every pure state \(\alpha \in F\) one must have \((\alpha ^\dag | \, \Pi _F = (\alpha ^\dag |\) (this follows from the definition and from Proposition 14). As a consequence, one also has \((a_F | \, \Pi _F = (a_F|\). This implies that, for every state \(\rho \in \mathsf {St}(\mathrm {A})\), the unnormalized state \(\Pi _F \, | \rho )\) is proportional to a state in F. Now, for two projections \(\Pi _F\) and \(\Pi _F'\) one must have

    figure a

    for every pure state \(\alpha \in F\). Since the states \(\Pi _F\, |\rho )\) and \(\Pi _F' \, |\rho )\) are proportional to states in F and \(\alpha \in F\) is a generic pure state, Ideal Compression implies \(\Pi _F\, |\rho ) = \Pi _F' \, |\rho )\), or equivalently, \(\Pi _F = \Pi _F'\), because the state \(\rho \) is generic.

  15. 15.

    In general, the dimension of the convex set \(C_\mathrm {A}\) is given by \(D_\mathrm {A}-1\).

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Acknowledgments

The work is supported by the Templeton Foundation under the project ID# 43796 A Quantum-Digital Universe, by the Foundational Questions Institute through the large grant The fundamental principles of information dynamics (FQXi-RFP3-1325), by the 1000 Youth Fellowship Program of China, and by the National Natural Science Foundation of China through Grants 11450110096 and 11350110207. GC acknowledges the hospitality of the Simons Center for the Theory of Computation and of Perimeter Institute for Theoretical Physics. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.

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Authors and Affiliations

  1. Department of Computer Science, The University of Hong Kong, Pokfulam road, Hong Kong, People’s Republic of China

    Giulio Chiribella

  2. QUIT Group, Dipartimento di Fisica, Università di Pavia, via Bassi 6, 27100, Pavia, Italy

    Giacomo Mauro D’Ariano & Paolo Perinotti

  3. INFN Sezione di Pavia, via Bassi 6, 27100, Pavia, Italy

    Giacomo Mauro D’Ariano & Paolo Perinotti

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  1. Giulio Chiribella
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  2. Giacomo Mauro D’Ariano
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Correspondence to Giulio Chiribella .

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Editors and Affiliations

  1. Department of Computer Science, The University of Hong Kong, Hong Kong, China

    Giulio Chiribella

  2. Perimeter Inst Theoretical Physics , Waterloo, Ontario, Canada

    Robert W. Spekkens

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Chiribella, G., D’Ariano, G.M., Perinotti, P. (2016). Quantum from Principles. In: Chiribella, G., Spekkens, R. (eds) Quantum Theory: Informational Foundations and Foils. Fundamental Theories of Physics, vol 181. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7303-4_6

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