Abstract
The main idea in the topos formulation of quantum theory it to have logic as a fundamental concept and try and derive other concepts in terms of it. This is what is done in the case of probabilities.
In fact, in the topos approach, probabilities acquire a logical interpretation rather than a relative frequency interpretation. In this setting probabilities are described in terms of truth values.
This implies that for the concept of probabilities to be well defined one does not require the notions of external observer, observed system and measurement. All that is required is an internal logic in terms of which truth values can be assigned. Probabilities, are then derived by these truth values. Thus truth values are seen as fundamental concepts while probabilities as derived concepts.
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Notes
- 1.
Note that \(\tilde{V}=\bigcup_{i} V_{i}\) such that V i ∩U⊆W.
- 2.
Note that we will not include the value r=0. The reason, as it will be explained later on, is to avoid obtaining situations in which all propositions are totally true with probability zero.
- 3.
Recall that given the Alexandrov topology on \(\mathcal{V}(\mathcal{H})\) then \(\mathit{Sh}(\mathcal{V}(\mathcal{H}))\simeq \mathit{Sets}^{\mathcal{V}(\mathcal{H})}\). So, in the following we will alternate freely between sheaves and presheaves.
- 4.
Note that here we have added a suffix org to indicate original, since we will now change the formulation of truth objects.
- 5.
It should be noted at this point that the correspondence between measures and state is present also in the context of classical physics. In fact in that case, as we have seen in previous chapters, a pure state (i.e. a point) is identified with the Dirac measure, while a general state is simply a probability measure on the state space. Such a probability measure assigns a number in the interval [0,1] called the weight and that in a sense tells you the ‘size’ of the measurable set.
- 6.
In our usual sample of \(\mathbb {C}^{4}\) this is the case for \(\delta^{o}(\hat {P}_{1})_{V_{\hat{P_{2}}}}\) and \(\delta^{o}(\hat{P}_{3})_{V_{\hat{P_{2}}}}\).
- 7.
Gleason Theorem tells us that the only possible probability measures on Hilbert spaces of dimension at least 3 are measures of the form μ(P)=tr(ρP), where ρ is a positive semidefinite self-adjoint operator of unit trace. This theorem was extended to a von-Neumann algebra \(\mathcal{N}\) in [73] where the author shows that, provided \(\mathcal{N}\) contains no direct summand of type I 2, then every finitely additive probability measure on \(\mathcal{P}(\mathcal{N})\) can be uniquely extended to a state on \(\mathcal{N}\).
Given a Hilbert space \(\mathcal{H}\), the general form of Gleason theorem is as follows:
Theorem 15.3 Assume that \(\mathrm{dim}((\mathcal{H})\geq3)\) and let μ be a σ-additive probability measure on \(P(\mathcal{B}(\mathcal{H}))\) then the following three statements hold:
-
1.
μ is completely additive.
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2.
μ has support.
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3.
There exists a positive operator \(x\in \mathcal{B}(\mathcal{H})\) of trace class, such that tr(x)=1 and μ(e)=tr(xe) for \(e\in P(\mathcal{B}(\mathcal{H}))\).
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1.
- 8.
Note that \(\underline{\varSigma}_{|\downarrow\!\! V}\) indicates the sheaf \(\underline{\varSigma}\) defined on the lower set ↓V.
References
A. Doering, C.J. Isham, Classical and Quantum Probabilities as Truth Values (2011). arXiv:1102.2213v1
A. Doering, Quantum states and measures on the spectral presheaf (2008). arXiv:0809.4847v1 [quant-ph]
A. Doering, Topos quantum logic and mixed states. arXiv:1004.3561 [quant-ph]
A.M. Gleason, Measures on the closed subspaces of a Hilbert space. J. Math. Phys. 6, 885 (1957)
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Flori, C. (2013). Probabilities in Topos Quantum Theory. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_15
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