Reconstruction of Quantum Theory by Lucien Hardy

[marcus]

Lucien Hardy is a prominent foundations of QM guy at Perimeter and he just came out with this paper
http://arxiv.org/abs/1104.2066
Reformulating and Reconstructing Quantum Theory
Lucien Hardy
159 pages. Many pictures
(Submitted on 11 Apr 2011)
"We provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following:
[Axiom 1] Operations correspond to operators.
[Axiom 2] Every complete set of positive operators corresponds to a complete set of operations.

The following operational postulates are shown to be equivalent to these mathematical axioms:
[P1] Definiteness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state.
[P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components.
[P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components.
[P4] Compound permutatability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system.
[P5] Preparability. Filters are non-mixing and non-flattening.

Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilities follows. A more detailed abstract is provided in the paper."
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I don't know much about the subject, but I suspect others may be interested and want to check it out.

Being a non-standard approach to Quantum Theory, I thought it might belong in Beyond forum. But if somebody wants to move it to QM forum that's fine too.

 

[marcus]

A video lecture covering the same research:
http://pirsa.org/11040118/
Reconstructing quantum theory from reasonable postulates.
Speaker(s): Lucien Hardy
Abstract: I will give a new set of operational postulates from which quantum theory can be reconstructed. These are

(1) Definiteness: There is a one-to-one correspondence between the set of pure states and the set of maximal effects such that we get probability one for a pure state followed the corresponding maximal effect.
(2) Information locality: If we perform maximal measurements on the components of a composite system then we effect a maximal measurement on the composite.
(3) Tomographic locality: The state of a composite system can be determined by making measurements on its components.
(4) Compound permutatability: There exists a compound reversible transformation effecting any given permutation of the states in some maximal distinguishable set of states.
(5) Preparability: Filters are non-mixing and non-flattening.

I will explain these postulates and indicate how some key steps in the reconstruction work. These postulates (see arXiv:1104.2066) are deeper than the postulates I gave ten years ago (in quant-ph/0101012).
Date: 19/04/2011 - 2:00 pm

 

[andrebourbaki]

Hardy's reconstruction is fairly sloppy about what he means by
normal Quantum Mechanics.
There are two issues. the less important issue is that
he identifies mixed states as the states, whereas this is controversial
In his field, Quantum Information Theory, his view is the
consensus view, but not outside that field.

More importantly the word 'probability' used in the classic
axioms (Dirac, von Neumann, etc.) of Quantum Mechanics
has always been assumed to mean normal mathematical
probability...but in my examination of Hardy's earlier paper,
(and nothing much has changed in this regard since he
seems to be oblivious of this issue, which is indeed irrelevant
for practical purposes of Quantum Computers as far as anyone
can tell)
http://arxiv.org/abs/quant-ph/0606038
on the Quantum Mechanics Axioms and Measurement,
see also www.mast.queensu.ca/~jjohnson/HilbertSixth.pdf
and
http://arxiv.org/abs/quant-ph/0502124
I carefully show that he would need an entirely new, and idiosyncratic,
mathematical theory of probability (a commutative probability theory
yet one that was not Kolmogorovian) to model his axioms. This implies that
in fact they are far from equivalent to the usual theory, he has sneaked
in quite a different physical content.

Hardy's point of view and research program are accepted mainstream within
the QIT field. But he does not have axiomatics as his real purpose, from a
foundational or axiomatic point of view, he is just too sloppy.

Anyway, to those of us not in QIT, it would seem odd to have a theory
of measurement that merely relied on axiomatic juggling and did not
rely on a physical analysis of the actual measurement apparatuses such
as bubble chamber, Geiger counters, etc.