Reconstructions of Quantum Theory
Quantum theory is one of our most successful physical theories, but also one of the most mysterious ones: why are detector click probabilities in nature described by abstract mathematical structures like Hilbert spaces, complex numbers, and operators? This question of “why the quantum” has more than just philosophical significance -- there are important reasons for addressing this question in a rigorous mathematical way:
- We need to construct consistent modifications of quantum theory (QT) that we can in principle test against QT in experiments. If we simply modify some of the quantum postulates in an ad hoc way (e.g. by adding nonlinear terms to the Schrödinger equation), then we typically do not obtain a consistent theory. But if we derive quantum theory from principles, then we can weaken or modify the principles, and work out mathematically what set of alternative consistent theories appears as solutions. These alternative theories will also have implications for quantum computing, since they can serve as theoretical models of computation that can be contrasted in their computational power to quantum mechanics.
- Perhaps a reformulation of QT in terms of simple physical principles can be helpful in the search for a theory of quantum gravity. Or, somewhat less ambitiously, such a reformulation may help to illuminate how the structure of quantum theory is related to the structure of spacetime, by taking a broader perspective beyond operator algebras.
- It has proven tremendously important in the history of physics to derive ad hoc equations from first principles. A paradigmatic example is given by the Lorentz transformations: initially, they were discovered in an ad hoc way as symmetries of the Maxwell equations. But Einstein has shown that they can be understood as simple consequences of two principles: the relativity principle and the light postulate. This has enormously improved our understanding, and paved the way to some of the subsequent development of relativity. Perhaps we can profit in a similar way from a principled derivation of quantum theory.
We have contributed to this research program in a number of different ways, and have found several successful reconstructions of the (finite-dimensional) quantum formalism from simple physical, or broadly information-theoretic, principles. In one such work , later improved in , we have derived quantum theory from the following postulates:
- Continouous reversibility: In any system, for any pair of pure states, continuous reversible time evolution can bring one state to the other.
- Tomographic locality: The state of any composite system is completely characterized by measurements on its individual components and their correlations.
- Existence of an information unit: There is a type of system (“generalized bit”) such that the state of any system can be reversibly encoded in a sufficiently large number of such bits. Moreover, state tomography for the bit is possible, and these bits can interact.
- No simultaneous encoding (aka “Zeilinger’s Principle”): If a generalized bit is used to perfectly encode one classical bit, it cannot simultaneously encode any further information.
To show that quantum theory follows from these postulates, the first step is to derive that the generalized bit corresponds exactly to the quantum bit, which can be represented as a three-dimensional ball (the Bloch ball). The picture above (from ) shows how this is achieved step by step: first, the state space of the generalized bit can be any convex set of any dimension d. No simultaneous encoding shows that there can be no “flat pieces in its boundary”, since these could be used to encode additional information into a state – hence it must have a droplet-like shape. Continuous reversibility shows that the state space must be very symmetric – that is, due to group representation theory, an ellipsoid, which can be reparametrized as a ball. Then, interaction between pairs of d-dimensional ball state spaces turns out to be possible only if d=3, as for the quantum bit, which can be shown with quite some effort . Finally, the only consistent way to combine qubit state spaces is in a way that is equivalent to standard quantum theory’s state space over many qubits, as shown in , and we are done.
Note that at no point is it assumed that there are wave functions, operators or complex numbers – instead, those arise as consequences of the postulates. And we get all other ingredients and predictions of abstract finite-dimensional quantum: unitary transformations, uncertainty relations, the Schrödinger equation (but not the choice of Hamiltonian or Lagrangian), Tsirelson’s bound on Bell correlations and more.
Once we have postulates that give us quantum theory (QT), we can start to relax those postulates, and in this way try to construct “quantum theory’s closest cousins” – theories that predict measurement outcome probabilities in a way that is not related to operator algebras or Hilbert spaces, but that are still in some sense physically close to QT. This turns out to be very difficult to do for the postulates above; but in , we have derived QT from four other principles that may be more suitable for this goal (because they do mention composite systems which are much harder to analyze):
- Classical decomposability: Every state of a physical system can be represented as a probabilistic mixture of perfectly distinguishable pure states.
- Strong symmetry: Every set of perfectly distinguishable pure states (of a given cardinality) can be reversibly transformed into any other such set (of the same cardinality).
- No higher-order interference: The interference patterns between mutually exclusive “paths” in an experiment is exactly the sum of the patterns which would be observed in all two-path subexperiments, corrected for overlaps.
- Observability of energy: There is non-trivial continuous reversible time evolution, and the generator of every such evolution can be associated to an observable (“energy”) which is a conserved quantity.
The insight that QT only allows for “second-order”, but not for “higher-order interference” is due to Sorkin , and this has initiated several recent experimental tests of this property (e.g. ). It would significantly improve the experimental situation if one would not only test for general violations of the quantum predictions, but if one actually had a viable alternative theory that could predict a certain concrete alternative behavior in those experiments. In principle, we can obtain such a theory by using the reconstruction above: Simply drop postulate (3), and work out the resulting set of theories that appear as solutions. This is one of the problems that we are currently thinking about, but it seems to be a mathematically extremely difficult problem.
Apart from the goals listed above, what does all this tell us about the nature of the quantum world? In , we argue that this tells us that QT can be understood as a “principle theory of information” (referring to Einstein’s distinction between principle and constructive theories). While the information-theoretic reconstructions of QT are to a large extent agnostic regarding the question how to interpret the quantum state, they do seem to tell us that we can understand the full quantum formalism as a simple “theory of probability” – namely, of quantum states expressing our information, knowledge, or belief about future observations (conditioned on our choices of which observations to make).
In other words: taking an informational view on the quantum state as a starting point can lead us more or less directly to a derivation of QT’s formalism. No other interpretations (like many-worlds or Bohmian mechanics) can currently claim something like this. The question now is how far this reasoning can be pushed: can we start with a concrete “informational” interpretation of QT – concretely, QBism – and derive the full quantum formalism from its basic premises? Our group is part of an international team (funded by the Foundational Questions Institute) that is currently working on this question:
 Ll. Masanes and M. P. Müller, A derivation of quantum theory from physical requirements, New J. Phys. 13, 063001 (2011). arXiv:1004.1483
 Ll. Masanes, M. P. Müller, R. Augusiak, and D. Pérez-García, Existence of an information unit as a postulate of quantum theory, Proc. Natl. Acad. Sci. USA 110(41), 16373 (2013). arXiv:1208.0493
 Ll. Masanes, M. P. Müller, R. Augusiak, and D. Pérez-García, Entanglement and the three-dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014). arXiv:1111.4060
 G. de la Torre, Ll. Masanes, A. J. Short, and M. P. Müller, Deriving quantum theory from its local structure and reversibility, Phys. Rev. Lett. 109, 090403 (2012). arXiv:1110.5482
 H. Barnum, M. P. Müller, and C. Ududec, Higher-order interference and single-system postulates characterizing quantum theory, New J. Phys. 16, 123029 (2014). arXiv:1403.4147
 A. Koberinski and M. P. Müller, Quantum theory as a principle theory: insights from an information-theoretic construction, to appear in S. Fletcher and M. Cuffaro (ed.), “Physical perspectives on computation, computational perspectives on physics”, Cambridge University Press, 2017. arXiv:1707.05602
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