Candidates for Principles of Quantumness

          @misc{ pirsa_PIRSA:09080014,
            doi = {10.48660/09080014},
            url = {https://pirsa.org/09080014},
            author = {},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Candidates for Principles of Quantumness},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {aug},
            note = {PIRSA:09080014 see, \url{https://pirsa.org}}
          }
          

Abstract

Quantum Mechanics (QM) is a beautiful simple mathematical structure--- Hilbert spaces and operator algebras---with an unprecedented predicting power in the whole physical domain. However, after more than a century from its birth, we still don't have a "principle" from which to derive the mathematical framework. The situation is similar to that of Lorentz transformations before the advent of the relativity principle. The invariance of the physical law with the reference system and the existence of a limiting velocity, are not just physical principles: they are mandatory operational principles without which one cannot do experimental Physics. And it is a very seductive idea to think that QM could be derived from some other principle of such epistemological kind, which is either indispensable or crucial in dramatically reducing the experimental complexity. Indeed, the large part of the formal structure of QM is a set of formal tools for describing the process of gathering information in any experiment, independently on the particular physics involved. It is mainly a kind of "information theory", a theory about our knowledge of physical entities rather than about the entities themselves. If we strip off such informational part from the theory, what would be left should be a "principle of the quantumness" from which QM should be derived. In my talk I will analyze the consequences of two possible candidates for the principle of quantumness: 1) PFAITH: the existence of a pure bipartite state by which we can calibrate all local tests and prepare all bipartite states by local tests; 2) PURIFY: the existence of a purification for all states. We will consider the two postulates within the general context of probabilistic theories---also called test-theories. Within test-theories we will introduce the notion of "time-cascade" of tests, which entails the identifications "events=transformations" and "evolution=conditioning", and derive the general matrix-algebra representation of such theories, with particular focus on theories that satisfy the "local discriminability principle". Some of the concepts will be illustrated in some specific test-theories, including the usual cases of classical and quantum mechanics, the extended versions of the PR boxes, the so-called "spin-factors", and quantum mechanics on a real (instead of complex) Hilbert spaces. After the brief tutorial on test-theories, I will analyze all the consequences of the two candidate postulates. We will see how postulate PFAITH implies the "local observability principle" and the tensor-product structure for the linear spaces of states and effects, along with a remarkable list of additional features that are typically quantum, including purification for some states, the impossibility of bit commitment, and many others. We will see how the postulate is not satisfied by classical mechanics, and a stronger version of the postulate also exclude theories where we cannot have teleportation, e.g. PR-boxes. Finally we will analyze the consequences of postulate PURIFY, and show how it is equivalent to the possibility of dilating any probabilistic transformation on a system to a deterministic invertible transformation on the system interacting with an ancilla. Therefore PURIFY is equivalent to the general principle that "every transformation can be in-principle inverted, having sufficient control on the environment". Using a simple diagrammatic representation we will see how PURIFY implies general theorems as: 1) deterministic full teleportation; 2) inverting a transformation upon an input state (i.e. error-correction) is equivalent to the fact that environment and reference remain uncorrelated; 3) inverting some transformations by reading the environment; etc. We will see that some non-quantum theories (e.g. QM on real Hilbert spaces) still satisfy PURIFY. Finally I will address the problem on how to prove that a test-theory is quantum. One would need to show that also the "effects" of the theory---not just the transformations---make a matrix algebra. A way of deriving the "multiplication" of effects is to identify them with atomic events. This can be done assuming the atomicity of evolution in conjunction with the Choi-Jamiolkowski isomorphism. Suggested readings: 1. arXiv:0807.4383, to appear in "Philosophy of Quantum Information and Entanglement", Eds A. Bokulich and G. Jaeger (Cambridge University Press, Cambridge UK, in press) 2. G. Chiribella, G. M. D'Ariano, and P. Perinotti (in preparation) 3. G. M. D'Ariano, A. Tosini (in preparation