Howard Barnum, Los Alamos National Laboratories
Jordan algebras and spectrality as tools for axiomatic characterizations of quantum theory
The normalized-state spaces of finite-dimensional Jordan algebras constitute a relatively narrow class of convex sets that includes the finite-dimensional quantum mechanical and classical state spaces. Several beautiful mathematical characterizations of Jordan statespaces exist, notably Koecher's characterization as the bases of homogeneous self-dual cones, and Alfsen and Shultz's characterization based on the notion of spectral convex sets plus additional axioms.  I will review the notion of spectral convex set and the Alfsen-Shultz characterization and discuss how these mathematical characterizationsof Jordan state spaces might be useful in developing accounts of quantum theory based on more operational principles, for example ones concerning information processing.  If time permits, I will present joint work with Cozmin Ududec in which we define analogues of multiple-slit experiments in systems described by spectral convex state spaces, and obtain results on Sorkin's notion of higher-level interference in this setting.  For example, we show that, like the finite-dimensional quantum systems which are a special case, Jordan state spaces exhibit only lowest-order (I_2 in Sorkin's hierarchy) interference.

Jonathan Barrett, University of Cambridge
A de Finetti theorem from the no-signalling principle

Suggested reading:
J.Barrett and M.Leifer, arXiv:0712.2265
J.Barrett arXiv:quant-ph/0508211
C.M.Caves, C.A.Fuchs and R.Schack, arXiv:quant-ph/0104088

Caslav Brukner, University of Vienna
Quantum Mechanics as a Theory of Systems with Limited Information Content
I will consider physical theories which describe systems with limited information content. This limit is not due observer's ignorance about some “hidden” properties of the system - the view that would have to be confronted with Bell's theorem - but is of fundamental nature. I will show how the mathematical structure of these theories can be reconstructed from a set of reasonable axioms about probabilities for measurement outcomes. Among others these include the “locality” assumption according to which the global state of a composite system is completely determined by correlations between local measurements. I will demonstrate that quantum mechanics is the only theory from the set in which composite systems can be in entangled (non-separable) states. Within Hardy's approach this feature allows to single out quantum theory from other probabilistic theories without a need to assume the “simplicity” axiom.

1. Borivoje Dakic, Caslav Brukner (in preparation)
2. Caslav Brukner, Anton Zeilinger, Information Invariance and Quantum Probabilities, arXiv:0905.0653
3. Tomasz Paterek, Borivoje Dakic, Caslav Brukner, Theories of systems with limited information content, arXiv:0804.1423

Ariel Caticha, University at Albany
Quantum Theory from Entropic Inference
Non-relativistic quantum theory is derived from information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the configuration space is a statistical manifold with a natural information metric. The dynamics then follows from a principle of inference, the method of Maximum Entropy: entropic dynamics is an instance of law without law. The concept of time is introduced as a convenient device to keep track of the accumulation of changes. The resulting formalism is close to Nelson's stochastic mechanics. The statistical manifold is a dynamical entity: its (information) geometry determines the evolution of the probability distribution which, in its turn, reacts back and determines the evolution of the geometry. As in General Relativity there is a kind of equivalence principle in that “fictitious” forces – in this case diffusive “osmotic” forces – turn out to be “real”. This equivalence of quantum and statistical fluctuations – or of quantum and classical probabilities – leads to a natural explanation of the equality of inertial and “osmotic” masses and allows explaining quantum theory as a sophisticated example of entropic inference. Mass and the phase of the wave function are explained as features of purely statistical origin.

Recommended Reading:
arXiv:0907.4335  "From Entropic Dynamics to Quantum Theory" (2009)

Bob Coecke, Oxford Comlab
Interaction axiomatics for quantum phenomena.
In our approach, rather than aiming to recover the 'Hilbert space model' which underpins the orthodox quantum mechanical formalism, we start from a general `pre-operational' framework, and verify how much additional structure we need to be able to describe a range of quantum phenomena. This also enables us to investigate which mathematical models, including more abstract categorical ones, enable one to model quantum theory.  Till now, all of our axioms only refer to the particular nature of how compound quantum systems interact, rather that to the particular structure of state-spaces.  This is in sharp contrast with other approaches of this kind which aim to recover quantum theory out of a much broader class of theories.  A more abstract quantum mechanical model has other many advantages.  It elucidates which are the key ingredients that make `the Hilbert space model' work.  Since it relies on monoidal categories, it comes with a high-level diagrammatic description (which we think of as `the' mathematical formalism).  It moreover removes the dependency on continuous underlying mathematical structures, paving the way for discrete combinatorial models, which might blend better with the other ingredients required for a theory of quantum gravity.

Giacomo Mauro D'Ariano, University of Pavia
Candidates for Principles of Quantumness
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

Daniel Fivel, University of Maryland
Reconstructing Quantum Mechanics from the Behavior of Partially Polarized Mixtures
It will be shown that the conventional (i.e. real or complex Hilbert space) model of quantum mechanics  can be deduced from the indistinguishability of the simplest types of statistical mixtures. The result does not have the  low dimension exclusion of the  quantum logic approach.

Note: This will be a simplification of the result obtained by me arXiv:hep-th/9405042 many years ago. The relevant portion of the paper is the first 12 pages.

Chris Fuchs, Perimeter Institute
Quantum-Bayesian Coherence (or, My Favorite Convex Set)
In a quantum-Bayesian delineation of quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state.  (A quantum system has potentially as many quantum states as there are agents considering it.)  But what then is the role of the rule?  In this paper, we argue that it should be seen as an  empirical addition to Bayesian reasoning itself.  Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutch-book coherence.  It is a rule that takes into account how one should assign probabilities to the outcomes of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the full state-space structure of quantum mechanics.  It seems to go some way.

Here are three papers associated with the approach:

http://arxiv.org/abs/0906.2187

http://arxiv.org/abs/quant-ph/0205039

http://arxiv.org/abs/quant-ph/0404156

Philip Goyal, Perimeter Institute
Why Quantum Theory is Complex
Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. We show that it is possible to derive the complex nature of the quantum formalism directly from the assumption that a pair of real numbers is associated to each sequence of measurement outcomes, and that the probability of this sequence is a real-valued function of this number pair. By making use of elementary symmetry and consistency conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.

Reference: arXiv:0907.0909 (http://arxiv.org/abs/0907.0909)

Alexei Grinbaum, CEA, Paris
The quantum logical reconstruction from Rovelli's axioms and its limits
What belongs to quantum theory is no more than what is needed for its derivation. Keeping to this maxim, we record a paradigmatic shift in the foundations of quantum mechanics, where the focus has recently moved from interpreting to reconstructing quantum theory. We present a quantum logical derivation based on Rovelli's information-theoretic axioms. Its strengths and weaknesses will be studied in the light of recent developments, focusing on the subsystems rule, continuity assumptions, and the definition of observer.

Publications:

• "Reconstruction of quantum theory," British Journal for the Philosophy of Science, 58, 2007, pp. 387-408.
• "Information-theoretic principle entails orthomodularity of a lattice," Foundations of Physics Letters 18 (6), 2005, pp. 563-572.
• "Elements of information-theoretic derivation of the formalism of quantum theory", International Journal of Quantum Information 1(3), 2003, pp. 289-300.

Michael Hall, Australian National University
Exact uncertainty, quantum mechanics and beyond
The fact that quantum mechanics admits exact uncertainty relations is used to motivate an ‘exact uncertainty’ approach to obtaining the Schrödinger equation.  In this approach it is assumed that an ensemble of classical particles is subject to momentum fluctuations, with the strength of the fluctuations determined by the classical probability density [1].  The approach may be applied to any classical system for which the Hamiltonian is quadratic with respect to the momentum, including all physical particles and fields [2].

The approach is based on a general formalism that describes physical ensembles via a probability density P on configuration space, together with a canonically conjugate quantity S [3].  Quantum and classical ensembles are particular cases of interest, but one can also ask more general questions within this formalism, such as (i) Can one consistently describe interactions between quantum and classical systems? and (ii) Can one obtain local nonlinear modifications of quantum mechanics?  These questions will be briefly discussed, with respect to measurement interactions and spin-1/2 systems respectively.

1. M.J.W. Hall and M. Reginatto, “Schroedinger equation from an exact uncertainty principle”, J. Phys. A 35 (2002) 3289 (http://lanl.arxiv.org/abs/quant-ph/0102069).
2. M.J.W. Hall, “Exact uncertainty approach in quantum mechanics and quantum gravity”,  Gen. Relativ. Gravit. 37 (2005) 1505 (http://lanl.arxiv.org/abs/gr-qc/0408098).
3. M.J.W. Hall and M. Reginatto, “Interacting classical and quantum systems”, Phys. Rev. A 72 (2005) 062109 (http://lanl.arxiv.org/abs/quant-ph/0509134).

Lucien Hardy, Perimeter Institute
Operational structures and Natural Postulates for Quantum Theory
In this talk we provide four postulates that are more natural than the usual postulates of QT.  The postulates require the predefinition of two integers, K and N, for a system. K is the number of probabilities that must be listed to specify the state.  N is the maximum number of states that can be distinguished in a single shot measurement and consequently log N is the information carrying capacity. The postulates are

P1  Information:  Systems having, or constrained to have, a given information carrying capacity have the same properties.

P2  Composites:  For a composite system, AB, we have N_AB=N_A N_B and  K_AB=K_A K_B.

P3 Continuity:  There exists a continuous reversible transformation between any two pure states.

P4 Simplicity: For each N, K takes the smallest value consistent with the other postulates.

Note that P2 is equivalent to requiring that information carrying capacity be additive and that the state of a composite system can be determined by measurements on the components alone (local tomography is possible).

We can prove a reconstruction theorem:   the standard formalism of QT (for finite N) follows from these postulates. This includes the properties that quantum states can be represented by density operators on a complex Hilbert space, evolution is given by completely positive maps (of which unitary evolution is a special case), and that composite systems are formed using the tensor product.  We derive the Born rule (or, equivalently, the trace rule) for calculating probabilities.  If the single word “continuous” is dropped from P3 the postulates are consistent with both Classical Probability Theory and Quantum Theory.  
In this talk we will place particular emphasis on laying the operational foundations for such postulates. 

Then we will provide some highlights of the proof.  Finally we will speculate on what needs to be changed for a theory of quantum gravity. 

Inge Helland, University of Oslo
Steps Towards a Unified Basis.
A new foundation of quantum mechanics for systems symmetric under a compact symmetry group is proposed. This is given by a link to classical statistics and coupled to the concept of a statistical parameter. A vector \phi of parameters is called an inaccessible c-variable if experiments can be provided for each single parameter, but no experiment can be provided for \phi. This is related to the concept of complementarity in quantum mechanics, but more generally to contrafactual parameters. Using these concepts and some weak assumptions, the Hilbert space of quantum mechanics is constructed. The complete set of axioms for time-independent quantum mechanics is provided by proving Born`s formula under weak assumptions.

1. Helland, I.S. (2006) Extended statistical modeling under symmetry: the link towards quantum mechanics. Ann. Statistics 34, 1,
   42-77.
2. Helland, I.S. (2008) Quantum Mechanics from Focusing and Symmetry. Found. Phys. 38, 818-842.
   Reference: arXiv 0801.2026. (http://arxiv.org/abs/0801.2026)

Gerd Niestegge, Munich, Germany
A reconstruction of quantum mechanics from quantum logics with unique conditional probabilities
The starting point of the reconstruction process is a very simple quantum logical structure on which probability measures (states) and conditional probabilities are defined. This is a generalization of Kolmogorov's measure-theoretic approach to probability theory. In the general framework, the conditional probabilities need neither exist nor be uniquely determined if they exist. Postulating their existence and uniqueness becomes the major step in the reconstruction process. A certain new mathematical structure can then be derived, and examples immediately reveal that probability conditionalization is identical with the Lüders - von Neumann measurement process. Some further postulates bring us to Jordan algebras, and the consideration of composite systems finally shows why these algebras must be the self-adjoint parts of von Neumann algbras such that they can be represented as linear operators on Hilbert spaces over the complex numbers. This is why the approach gets ahead of other ones that are not able to justify the need for the complex Hilbert space or the Jordan operator algebras. The mathematical structure of quantum mechanics can thus be reconstructed from a few probabilistic basic principles and becomes a non-Boolean extension of classical probability theory. Its link to physics is that probability conditionalization in this structure is identical with the Lüders - von Neumann measurement process.

Papers which could serve as useful preparatory reading:

•  [2008a ] "An approach to quantum mechanics via conditional probabilities," Found. Phys. 38, 241–256
•  [2008b ] "A representation of quantum measurement in order-unit spaces," Found. Phys. 38, 783–795

Jochen Rau, Goethe University, Frankfurt
Demarcating probability theories by their degree of agent-dependency
Recent advances in quantum computation and quantum information theory have led to revived interest in, and cross-fertilisation with, foundational issues of quantum theory. In particular, it has become apparent that quantum theory may be interpreted as but a variant of the classical theory of probability and information. While the two theories may at first sight appear widely different, they actually share a substantial core of common properties; and their divergence can be reduced to a single attribute only, their respective degree of agent-dependency. I propose a mathematical description for this ?degree of agent-dependency? and show how assuming different values allows one to derive the classical and the quantum case from their common core. Finally, I explore ? and eventually dismiss ? the possibility that beyond quantum theory there might be other variants of classical probability theory that are relevant to physics.

Reference for preparatory reading:
http://arxiv.org/abs/0710.2119

Marcel Reginatto, Physikalisch-Technische Bundesanstalt, Braunschweig, Germany
Exact uncertainty, bosonic fields, and interacting classical-quantum systems
The quantum equations for bosonic fields may be derived using an 'exact uncertainty' approach [1]. This method of quantization can be applied to fields with Hamiltonian functionals that are quadratic in the momentum density, such as the electromagnetic and gravitational fields.  The approach, when applied to gravity [2], may be described as a Hamilton-Jacobi quantization of the gravitational field. It differs from previous approaches that take the classical Hamilton-Jacobi equation as their starting point in that it incorporates some new elements, in particular the use of a formalism of ensembles on configuration space and the postulate of an exact uncertainty relation. These provide the fundamental elements needed for the transition to the quantum theory. The formalism of ensembles on configuration space is general enough to describe classical, quantum, and interacting classical-quantum systems in a consistent way. This is of some relevance to gravity: although there are many physical arguments in favour of a quantum theory of gravity, it appears that the justification for such a theory does not follow from logical arguments alone [3]. It is therefore of interest to consider the coupling of quantum fields to a classical gravitational field. This leads to a theory that is fundamentally different from standard semiclassical gravity.

1. Michael J W Hall, Kailash Kumar and Marcel Reginatto, Bosonic field equations from an exact uncertainty principle, J. Phys. A 36 (2003) 9779-9794 (http://arxiv.org/abs/hep-th/0307259).
2. M. Reginatto, Exact Uncertainty Principle and Quantization: Implications for the Gravitational Field, Proceedings of DICE2004 in: Braz. J. Phys. 35 (2005) 476-480 (http://arxiv.org/abs/gr-qc/0501030).
3. Mark Albers, Claus Kiefer and Marcel Reginatto, Measurement analysis and quantum gravity, Phys. Rev. D 78 (2008) 064051 (http://arxiv.org/abs/0802.1978).

Lee Smolin, Perimeter Institute
Reconstructing quantum theory from Brownian motion: an assessment

Rob Spekkens, Perimeter Institute
The power of epistemic restrictions in reconstructing quantum theory: from trits to qutrits
A significant part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over classical states that can be prepared.  (Such a restriction is termed “epistemic” because it implies a fundamental limit on the amount of knowledge that any observer can have about the classical state.)  I will support this claim in the particular case of a theory of many classical 3-state systems (trits) where if a particular kind of epistemic restriction is assumed -- one that appeals to the symplectic structure of the classical state space -- it is possible to reproduce the operational predictions of the stabilizer formalism for qutrits.  The latter is an interesting subset of the full quantum theory of qutrits, a discrete analogue of Gaussian quantum mechanics. This is joint work with Olaf Schreiber.

Christopher Timpson, Oxford
Quantum Bayesian: Pros and Cons
The Quantum Bayesianism of Caves, Fuchs and Schack presents a distinctive starting point from which to attack the problem of axiomatising - or re-constructing - quantum theory. However, many have had the doubt that this starting point is itself already too radical. In this talk I will briefly introduce the position (it will be familiar to most, no doubt) and describe what I take to be its philosophical standpoint. More importantly, I shall seek to defend it from some bad objections, before going on to level some more substantive challenges.

The background paper is: 0804.2047 on the arXiv.

William Wootters, Williams College
Quantum Mechanics as a Real-Vector-Space Theory with a Universal Auxiliary Rebit
In a 1960 paper, E. C. G. Stueckelberg showed how one can obtain the familiar complex-vector-space structure of quantum mechanics by starting with a real-vector-space theory and imposing a superselection rule.  In this talk I interpret Stueckelberg’s construction in terms of a single auxiliary real-vector-space binary object—a universal rebit, or “ubit."  The superselection rule appears as a limitation on our ability to measure the ubit or to use it in state transformations.  This interpretation raises the following questions: (i) What is the ubit?  (ii) Could the superselection rule emerge naturally as a result of decoherence?  (iii) If so, could one hope to see experimentally any effects of imperfect decoherence?

Background reading:

E. C. G. Stueckelberg, "Quantum Theory in Real Hilbert Space," Helv. Phys. Acta 33, 727 (1960).

P. Goyal, "From Information Geometry to Quantum Theory," arXiv:0805.2770.

C. M. Caves, C. A. Fuchs, and P. Rungta, "Entanglement of Formation of an Arbitrary State of Two Rebits," arXiv:quant-ph/0009063.

Alexander Wilce, Sesquahanna University
Four and a Half Axioms for Quantum Mechanics
I will discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. These require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent up to the action of a compact group of symmetries, and that every state be the marginal of a bipartite state perfectly correlating two measurements. This much yields a mathematical representation of measurements, states and symmetries that is already very suggestive of quantum mechanics. One final postulate (a  simple minimization principle, still in need of a clear interpretation) forces the theory's state space to be that of a formally real Jordan algebra.