Proof. Using successively two times properties (186) and (4), we obtain : $\forall \sigma \in \mathfrak{S}^{(0_1)}, \forall t \in \mathfrak{T}^{(0_2)}, \tilde{\mathfrak{e}}_{f^{(21)}(t)}^{(0_1)}(\sigma) = \tilde{\mathfrak{e}}_{\tilde{t}}^{(0_2)}(\tilde{f}_{(12)}(\sigma)) = \overline{\tilde{\mathfrak{e}}_{\tilde{t}}^{(0_2)}(\tilde{f}_{(12)}(\sigma))} = \overline{\tilde{\mathfrak{e}}_{f^{(21)}(t)}^{(0_1)}(\sigma)} = \overline{\tilde{\mathfrak{e}}_{f^{(21)}(t)}^{(0_1)}(\sigma)}$.

Property (208) is a direct consequence of (153) and (207) : $\forall \sigma \in \mathfrak{S}^{(0_1)}, \forall t \in \mathfrak{T}^{(0_2)}, \tilde{\mathfrak{e}}_{\tilde{t}}^{(0_1)}(\tilde{f}_{(12)}(\sigma^*)) = \tilde{\mathfrak{e}}_{f^{(21)}(t)}^{(0_1)}(\sigma^*) = \overline{\tilde{\mathfrak{e}}_{f^{(21)}(t)}^{(0_1)}(\sigma)} = \overline{\tilde{\mathfrak{e}}_{f^{(21)}(t)}^{(0_1)}(\sigma)} = \tilde{\mathfrak{e}}_{\tilde{t}}^{(0_1)}(\tilde{f}_{(12)}(\sigma)) = \tilde{\mathfrak{e}}_{\tilde{t}}^{(0_1)}((\tilde{f}_{(12)}(\sigma))^*). \quad \square$

Before studying the preservation of orthogonality, we will firstly note some elementary results. First of all, we have

$$ \forall t_{(\Sigma, \Sigma^*)} \in \tilde{\mathfrak{T}}_{\text{disc}}^{(0_2)}, \quad f^{(21)}(t_{(\Sigma, \Sigma^*)}) \in \tilde{\mathfrak{T}}_{\text{disc}}^{(0_1)}. \qquad (209) $$

Indeed, let us introduce $\kappa, \kappa' \in \mathfrak{S}^{(0_1)}$ as follows

$$ \kappa := \prod_{\mathfrak{S}} \{ \sigma \mid \tilde{\mathfrak{e}}_{f^{(21)}(t_{(\Sigma, \Sigma^*)})}^{(0_1)}(\sigma) = \mathbf{Y} \} \quad \text{and} \quad \kappa' := \prod_{\mathfrak{S}} \{ \sigma \mid \tilde{\mathfrak{e}}_{f^{(21)}(t_{(\Sigma, \Sigma^*)})}^{(0_1)}(\sigma) = \mathbf{N} \} \qquad (210) $$

Using the surjectivity of $\tilde{f}_{(12)}$ (Lemma 43) and the fact that $\tilde{f}_{(12)}$ preserves infima (Lemma 42), we obtain

$$ \tilde{f}_{(12)}(\kappa) = \Sigma \quad \text{and} \quad \tilde{f}_{(12)}(\kappa') = \Sigma^*. \qquad (211) $$

$t_{(\Sigma, \Sigma^*)} \in \tilde{\mathfrak{T}}_{\text{disc}}^{(0_2)}$ means that $(\Sigma \bowtie_{\mathfrak{S}} \Sigma^*)$ and $\tilde{\Sigma} \Sigma^*_{\mathfrak{S}}$. Using the bijectivity of $\tilde{f}_{(12)}$ (Lemma 43 and property (189)) and its order-preserving property (Lemma 42), with the explicit definition of the relations $\bowtie_{\mathfrak{S}}$ and $\tilde{\Sigma}^*$, we deduce that $\kappa \bowtie_{\mathfrak{S}} \kappa'$ and $\tilde{\kappa} \kappa'_{\mathfrak{S}}$. As a result, $f^{(21)}(t_{(\Sigma, \Sigma^*)}) \in \tilde{\mathfrak{T}}_{\text{disc}}^{(0_1)}$. Moreover,

$$ \kappa = \tilde{f}_{(12)}^{-1}(\Sigma) \quad \text{and} \quad \kappa' = \tilde{f}_{(12)}^{-1}(\Sigma^*) = \kappa^*. \qquad (212) $$

As a result, we have

$$ f^{(21)}(t_{(\Sigma, \Sigma^*)}) = t_{(\tilde{f}_{(12)}^{-1}(\Sigma) \tilde{f}_{(12)}^{-1}(\Sigma^*))}. \qquad (213) $$

In the following, we will impose the following property of preservation of schemes :

$$ f^{(21)}(\mathfrak{U}^{(0_2)}) = \mathfrak{U}^{(0_1)}. \qquad (214) $$

We now check the following fundamental property

**Theorem 17. [Preservation of the orthogonality of states]**

$$ \forall (\tilde{f}_{(12)}, f^{(21)}) \in \text{Sym} \left[ (\mathfrak{S}^{(0_1)}, \mathfrak{T}^{(0_1)}, \tilde{\mathfrak{e}}^{(0_1)}) \rightarrow (\mathfrak{S}^{(0_2)}, \mathfrak{T}^{(0_2)}, \tilde{\mathfrak{e}}^{(0_2)}) \right], \forall \sigma_1, \sigma_2 \in \mathfrak{S}^{(0_1)}, \sigma_1 \perp \sigma_2 \Rightarrow \tilde{f}_{(12)}(\sigma_1) \perp \tilde{f}_{(12)}(\sigma_2). \quad (215) $$

<figure><img src="image_74.png" /></figure>

Proof.

$$ \begin{align*} \sigma_1 \perp \sigma_2 &\Leftrightarrow \exists t \in \mathfrak{U}^{(0_1)} \mid (\tilde{\mathfrak{e}}_t(\sigma_1) = \mathbf{Y} \quad \text{and} \quad \tilde{\mathfrak{e}}_t(\sigma_2) = \mathbf{Y}) \\ &\Leftrightarrow \exists u \in \mathfrak{U}^{(0_2)} \mid (\tilde{\mathfrak{e}}_{f^{(21)}(u)}(\sigma_1) = \mathbf{Y} \quad \text{and} \quad \tilde{\mathfrak{e}}_{f^{(21)}(u)}(\sigma_2) = \mathbf{Y}) \quad \text{from eq. (190) and property (214)} \\ &\Leftrightarrow \exists u \in \mathfrak{U}^{(0_2)} \mid (\tilde{\mathfrak{e}}_u(f_{(12)}(\sigma_1)) = \mathbf{Y} \quad \text{and} \quad \tilde{\mathfrak{e}}_u(f_{(12)}(\sigma_2)) = \mathbf{Y}) \quad \text{from eq. (186) and Lemma 44} \\ &\Leftrightarrow \tilde{f}_{(12)}(\sigma_1) \perp \tilde{f}_{(12)}(\sigma_2). \tag{216} \end{align*} $$

<figure><img src="image_75.png" /></figure>
**Lemma 45.**

$$ \varphi_L : \mathscr{C}(\mathfrak{S}_{\text{pure}}) \underset{\mathfrak{c}}{\longrightarrow} \mathfrak{S} \underset{\mathfrak{c}}{\sqcap} \mathfrak{c} \qquad \varphi_D : \mathfrak{S} \underset{\mathfrak{c}}{\longrightarrow} \mathscr{C}(\mathfrak{S}_{\text{pure}}) \qquad (217) $$

satisfy the following Galois adjunction relation

$$ \forall \mathfrak{c} \in \mathscr{C}(\mathfrak{S}_{\text{pure}}), \forall \sigma \in \mathfrak{S}, \quad (\varphi_L(\mathfrak{c}) \subseteq_{\mathfrak{S}} \sigma) \Leftrightarrow (\mathfrak{c} \subseteq \varphi_D(\sigma)), \qquad (218) $$

and $\varphi_L \circ \varphi_D = \text{id}_{\mathfrak{S}}$.

<figure><img src="image_76.png" /></figure>

**Notion 54.** Using the elements of a symmetry $(\tilde{f}_{(12)}, f^{(21)}) \in \text{Sym}[(\mathfrak{S}^{(0_1)}, \mathfrak{T}^{(0_1)}, \tilde{\mathfrak{e}}^{(0_1)}) \to (\mathfrak{S}^{(0_2)}, \mathfrak{T}^{(0_2)}, \tilde{\mathfrak{e}}^{(0_2)})]$, we define the following maps

$$ \begin{array}{rcl} \tilde{f}_{(12)} : & \mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_1)}) & \longrightarrow \mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_2)}) \\ & \mathfrak{c} & \mapsto \bigvee \{\tilde{f}_{(12)}(\sigma) \mid \sigma \in \mathfrak{c}\} \end{array} \qquad \begin{array}{rcl} \underline{f}^{(21)} : & \mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_2)}) & \longrightarrow \mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_1)}) \\ & \mathfrak{c} & \mapsto (\varphi_D^{(0_1)} \circ f_{(21)}^{\downarrow} \circ \varphi_L^{(0_2)})(\mathfrak{c}) \end{array} \qquad (219) $$

Note that $f_{(21)}^{\downarrow}$ has been defined in equation (205).

**Lemma 46.** $(\tilde{f}_{(12)}, \underline{f}^{(21)})$ forms a Galois connection.

<figure><img src="image_77.png" /></figure>

*Proof.* For any $\mathfrak{c}_1 \in \mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_1)})$ and $\mathfrak{c}_2 \in \mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_2)})$, we have

$$ \begin{align} (\tilde{f}_{(12)}(\mathfrak{c}_1) \subseteq \mathfrak{c}_2) &\Leftrightarrow (\forall \sigma \in \mathfrak{c}_1, \varphi_L^{(0_2)}(\mathfrak{c}_2) \subseteq_{\mathfrak{S}_{(0_2)}} \tilde{f}_{(12)}(\sigma)) \nonumber \\ &\Leftrightarrow (\forall \sigma \in \mathfrak{c}_1, \tilde{\mathfrak{e}}_{\mathfrak{t}_{(\mathfrak{c}_2, (\mathfrak{c}_2))}}^{(0_2)}(\tilde{f}_{(12)}(\sigma)) = \mathbf{Y}) \nonumber \\ &\Leftrightarrow (\forall \sigma \in \mathfrak{c}_1, \tilde{\mathfrak{e}}_{f^{(21)}(\mathfrak{t}_{(\mathfrak{c}_2, (\mathfrak{c}_2))}}^{(0_1)}(\sigma) = \mathbf{Y}) \nonumber \\ &\Leftrightarrow (\forall \sigma \in \mathfrak{c}_1, \{\sigma\} \subseteq \left\{ \prod_{\mathfrak{S}^{(0_1)} \ni \sigma'} \mathfrak{S}_{\text{pure}}^{(0_1)} | \tilde{\mathfrak{e}}_{f^{(21)}(\mathfrak{t}_{(\mathfrak{c}_2, (\mathfrak{c}_2))}}^{(0_1)}(\sigma) = \mathbf{Y} \right\}) \nonumber \\ &\Leftrightarrow (\forall \sigma \in \mathfrak{c}_1, \{\sigma\} \subseteq (\varphi_D^{(0_1)} \circ f_{(21)}^{\downarrow} \circ \varphi_L^{(0_2)})(\mathfrak{c}_2)) \nonumber \\ &\Leftrightarrow (\mathfrak{c}_1 \subseteq \underline{f}^{(21)}(\mathfrak{c}_2)) \tag{220} \end{align} $$

<figure><img src="image_78.png" /></figure>

**Theorem 18.** $\tilde{f}_{(12)}$ is an injective map from $\mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_1)})$ to $\mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_2)})$ preserving suprema, mapping atoms to atoms and preserving orthogonality (i.e. $\forall \mathfrak{c} \in \mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_1)}), \tilde{f}_{(12)}(\mathfrak{c}^{\perp}) = (\tilde{f}_{(12)}(\mathfrak{c})^{\perp})$). It is then an ortho-morphism of Hilbert lattice (see [67, Definition 5.12 and Definition 5.17]).

<figure><img src="image_79.png" /></figure>

*Proof.* $\tilde{f}_{(12)}$ preserves atoms as an extension of $\tilde{f}_{(12)}$ from $\mathfrak{S}_{\text{pure}}^{(0_1)}$ to $\mathscr{C}(\mathfrak{S}_{\text{pure}}^{(0_1)})$. It is injective and preserves suprema as a left Galois adjunct. It preserves the orthogonality relation according to Theorem 17.

<figure><img src="image_80.png" /></figure>
# 5 Conclusion

We aim to develop a new axiomatic approach to quantum theory and this article is designed as a first decisive step for this axiomatic program. A precise semantic description of the space of preparations and of the associated 'mixed states' of the system was formulated. This semantic formalism is based on a Chu space construction involving the set of preparations, the set of yes/no tests and an evaluation map with a three-valued target space. The values taken by this map are associated with counterfactual statements of the observer for a given yes/no test and a given prepared sample. The three values are interpreted in a possibilistic perspective, i.e., as 'certainly yes', 'certainly no' and 'maybe'. This domain structure on the target space led to an 'informational' interpretation of the set of preparations. The space of preparations was equipped with a notion of 'mixtures', expressed in terms of the meet operation on this poset. From natural requirements about the inductive definition of states, it appeared that this Inf semi-lattice was also a pointed directed-complete partial order. Then, an 'Information Principle' was introduced in the form of two topological requirements on the space of states. Although new in its form, this principle is very standard in different quantum axiomatic programs. This basic set of axioms was shown to be sufficient to constrain the structure of the space of states to be a 'projective domain'. The space of pure states was then basically identified in terms of maximal elements of this domain.

Then, the relation between yes/no tests and states was studied from two perspectives: using the notion of Chu duality and using the notion of 'measurement'. Adopting the first perspective, the notion of 'properties' of the system was defined and a 'property-state' was identified for any property, as in Piron's construction. The second perspective emphasizes the recursive aspect of preparation processes. In order to identify a subclass of measurement operations corresponding to minimally disturbing measurements, it appeared necessary to clarify the notion of 'compatibility between measurements'. The compatibility between two measurements was defined in terms of the existence of preparations that simultaneously exhibited the two corresponding properties as actual. This notion was used to define 'ideal first-kind measurements' and to characterize them as 'minimally disturbing measurements'. We finally proved the existence of a 'bi-extensional Chu₃ duality' between the space of minimally disturbing yes/no tests and the space of states.

The simultaneous ideal first-kind measurements of compatible properties was then studied and 'Specker's principle' was proved. Using this result, we obtained a 'coherent domain' structure on the space of 'descriptions' formalizing the families of compatible properties used to define a state of the system. An orthogonality relation was then defined on the space of states in terms of the class of discriminating yes/no tests. We achieve the characterization of the domain structure on the space of states by requiring the existence of a scheme of discriminating yes/no tests necessary to the construction of an orthogonality relation on the space of states. In generic examples, this last constraint is equivalent to the existence of an ortho-complementation on the projective domain defining the space of states.

Using the properties of the domain structure established on the space of states, we deduced that the set of ortho-closed subsets of pure states, equipped with the induced orthogonality relation, inherits a structure of Hilbert lattice. This is the first part of our reconstruction theorem.

In Section 4, we explored the properties of Chu morphisms with respect to previous notions. As a central result, a sub-algebra of the algebra of Chu morphisms, corresponding to 'symmetries' of the system, was defined. These symmetries appear to preserve the class of minimally disturbing yes/no tests and the orthogonality of states. The link between these Chu symmetries and the morphisms of Hilbert lattice, defined on the space of ortho-closed subsets of pure states, is finally emphasized. This is the second part of our reconstruction theorem.

Despite its self-contained character and the fact that it achieves to produce a reconstruction theorem from very basic premises for quantum theory, this paper requires more developments to be plainly satisfactory. First of all, it appears necessary to study the ways to recover a probabilistic formalism from a possibilistic one. Secondly, it appears necessary to pursue the construction to the case of compound systems by studying the tensor products structures on the domains appearing in the present paper. Endly, it seems unavoidable to study extensively the category of 'projective domains' extracted from our set of axioms. These problems will be attacked in our forthcoming papers.
References

[1] Abramsky, S. Big toy models. *Synthese*, 186(3):697–718, Jun 2012.

[2] Abramsky, S. Coalgebras, Chu Spaces, and Representations of Physical Systems. *Journal of Philosophical Logic*, 42(3):551–574, 2013.

[3] Abramsky, S., Coecke, B. Categorical Quantum Mechanics. In Engesser, K., Gabbay, D.M., Lehmann, D., editor, *Handbook of Quantum Logic and Quantum Structures*, pages 261 – 323. Elsevier, Amsterdam, 2009.

[4] Abramsky, S., Heunen, C. *Operational theories and categorical quantum mechanics*, page 88–122. Lecture Notes in Logic. Cambridge University Press, 2016.

[5] Abramsky, S., Jung, A. *Handbook of Logic in Computer Science - Vol 3*, chapter Domain Theory, pages 1–168. Oxford University Press, Inc., New York, NY, USA, 1994.

[6] Aerts, D. Construction of the tensor product for the lattices of properties of physical entities. *Journal of mathematical physics*, 25(5):1434–1441, 1984.

[7] Aerts, D., Valckenborgh, F. Failure of standard quantum mechanics for the description of compound quantum entities. *International Journal of Theoretical Physics*, 43(1):251–264, 2004.

[8] Aerts, D., Van Steirteghem, B. Quantum Axiomatics and a Theorem of M. P. Solèr. *International Journal of Theoretical Physics*, 39(3):497–502, 2000.

[9] Baltag, A., Smets, S. Complete axiomatizations for quantum actions. *International Journal of Theoretical Physics*, 44(12):2267–2282, Dec 2005.

[10] Baltag, A., Smets, S. LQP: the dynamic logic of quantum information. *Mathematical Structures in Computer Science*, 16(3):491–525, 2006.

[11] Baltag, A., Smets, S. A dynamic-logical perspective on quantum behavior. *Studia Logica*, 89(2):187–211, Jul 2008.

[12] Barr, M. *-Autonomous Categories and Linear Logic. *Mathematical Structures in Computer Science*, 1:159–178, 1991.

[13] Barrett, J. Information processing in generalized probabilistic theories. *Phys. Rev. A*, 75:032304, Mar 2007.

[14] Bergfeld, J.M., Kishida, K., Sack, J., Zhong, S. Duality for the Logic of Quantum Actions. *Studia Logica*, 103(4):781–805, Aug 2015.

[15] Birkhoff, G., Von Neumann, J. The logic of quantum mechanics. *Annals of Mathematics*, 37:823–843, 01 1936.

[16] Bordalo, G.H., Rodrigues, E. Complements in modular and semimodular lattices. *Portugaliae Mathematica*, 55:373–380, 1998.

[17] Cabello, A. Specker's fundamental principle of quantum mechanics, 2012.

[18] Cassinelli, G., Beltrametti, E. G. Ideal, first-kind measurements in a proposition-state structure. *Communications in Mathematical Physics*, 40(1):7–13, Feb 1975.

[19] Chiribella, G., Chiribella, D'A., Mauro, Perinotti, P. Informational derivation of quantum theory. *Phys. Rev. A*, 84:012311, Jul 2011.
[20] Coecke, B., Moore, D. J., Smets, S. *Logic of Dynamics and Dynamics of Logic: Some Paradigm Examples*, pages 527–555. Springer Netherlands, Dordrecht, 2004.

[21] Coecke, B., Moore, D. J., Wilce, A. *Operational Quantum Logic: An Overview*, pages 1–36. Springer Netherlands, Dordrecht, 2000.

[22] Coecke, B., Moore, D.J., Stubbe, I. Quantaloids describing causation and propagation of physical properties. *Foundations of Physics Letters*, 14(2):133–145, Apr 2001.

[23] Coecke, B., Smets, S. A logical description for perfect measurements. *International Journal of Theoretical Physics*, 39(3):595–603, Mar 2000.

[24] Coecke, B., Smets, S. The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes. *International Journal of Theoretical Physics*, 43(7):1705–1736, Aug 2004.

[25] Crawley, P., Dilworth, R.P. Algebraic theory of lattices. Prentice-Hall Englewood Cliffs, N.J, 1973.

[26] Dacey, J.R. Orthomodular spaces, Ph.D. Thesis, University of Massachusetts Amherst (1968).

[27] Einstein, A., Podolsky, B., Rosen, N. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? *Phys. Rev.*, 47:777–780, May 1935.

[28] Faure, C-A., Froelicher, A. Projective Geometries and Projective Lattices. In *Modern Projective Geometry*, Mathematics and Its Applications, pages 25–53. Springer, Dordrecht, 2000.

[29] Foulis, D. J., Randall, C. H. *Empirical logic and tensor products*. Bibliographisches Inst, Germany, 1981.

[30] Foulis, D., Piron, C., Randall, C. Realism, operationalism, and quantum mechanics. *Foundations of Physics*, 13(8):813–841, Aug 1983.

[31] Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., Scott, D. S. *Continuous Lattices and Domains*. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2003.

[32] Girard, J.-Y. Linear logic. *Theoretical Computer Science*, 50(1):1–101, 1987.

[33] Girard, J.-Y., Taylor, P., Lafont, Y. *Proofs and Types*. Cambridge University Press, New York, NY, USA, 1989.

[34] Gleason, A. M. *Measures on the Closed Subspaces of a Hilbert Space*, pages 123–133. Springer Netherlands, Dordrecht, 1975.

[35] Hardegree, G. M. Reichenbach and the logic of quantum mechanics. *Synthese*, 35(1):3–40, 1997.

[36] Hardy, L. Quantum theory from five reasonable axioms. 2001.

[37] Hardy, L. Reformulating and Reconstructing Quantum Theory. 2011.

[38] Hardy, L. Reconstructing quantum theory. *Fundam. Theor. Phys.*, 181:223–248, 2016.

[39] Hoehn, P.A. Toolbox for reconstructing quantum theory from rules on information acquisition. *Quantum*, 1:38, dec 2017.

[40] Hoehn, P.A., Wever, C.S.P. Quantum theory from questions. *Phys. Rev. A*, 95:012102, Jan 2017.

[41] Holland, S. S. Orthomodularity in infinite dimensions : a theorem of M. Soler. *Bull. Am. Math. Soc.*, 32(math.RA/9504224):205–234, 1995.
[42] Janotta, P., Hinrichsen, H. Generalized probability theories: what determines the structure of quantum theory? *Journal of Physics A: Mathematical and Theoretical*, 47(32):323001, jul 2014.

[43] Kochen, S., Specker, E. The problem of hidden variables in quantum mechanics. *Indiana Univ. Math. J.*, 17:59–87, 1968.

[44] Kraus, K., Böhm, A., Dollard, J.D., Wootters, W.H., editor. *States, Effects, and Operations - Fundamental Notions of Quantum Theory*, volume 190 of *Lecture Notes in Physics*, Berlin Springer Verlag, 1983.

[45] Ludwig, G. Quantum theory as a theory of interactions between macroscopic systems which can be described objectively. *Erkenntnis*, 16(3):359–387, Nov 1981.

[46] Ludwig, G. *Foundations of Quantum Mechanics*, volume 1 of *Theoretical and Mathematical Physics*. Springer-Verlag, Berlin Heidelberg, 1983. Original German edition published in one volume as Band 70 of the series: Grundlehren der mathematischen Wissenschaften.

[47] Ludwig, G., Summers, S.J. An Axiomatic Basis for Quantum Mechanics Volume 1: Derivation of Hilbert Space Structure and Volume 2: Quantum Mechanics and Macrosystems. *Physics Today*, 41:72, 1988.

[48] MacKey, G. W. *The Mathematical Foundations of Quantum Mechanics: a Lecture*. Mathematical physics monograph series. Benjamin, New York, NY, 1963. This book has also been published by Dover in 1963.

[49] Moore, D. J. Categories of representations of physical systems. *Helvetica Physica Acta*, 68(7-8):658–678, 1995.

[50] Moore, D. J. On State Spaces and Property Lattices. *Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics*, 30(1):61–83, 1999.

[51] Mueller, M.P., Masanes, L. Information-theoretic postulates for quantum theory. *Fundam. Theor. Phys.*, 181:139–170, 2016.

[52] Pauli, W. Die allgemeinen Prinzipien der Wellenmechanik. *Handbuch der Physik*, 5:1–168, 1958.

[53] Peres, A. *Quantum theory: concepts and methods*. Dordrecht; Boston: Kluwer Academic, 1993. "Reprinted with corrections" [1995?]-Verso t.p. paperback reprint.

[54] Piron, C. Survey of general quantum physics. *Foundations of Physics*, 2(4):287–314, Oct 1972.

[55] Piron, C. *On the Foundations of Quantum Physics*, pages 105–116. Springer Netherlands, Dordrecht, 1976.

[56] Piron, C. *A First Lecture on Quantum Mechanics*, pages 69–87. Springer Netherlands, Dordrecht, 1977.

[57] Piron, C. Ideal measurement and probability in quantum mechanics. *Erkenntnis*, 16(3):397–401, Nov 1981.

[58] Pokorny, F., Zhang, C., Higgins, G., Cabello, A., Kleinmann, M., Hennrich, M. Tracking the Dynamics of an Ideal Quantum Measurement. *Phys. Rev. Lett.*, 124:080401, Feb 2020.

[59] Pratt, V. R. Chu spaces: automata with quantum aspects. In *Proceedings Workshop on Physics and Computation. PhysComp '94*, pages 186–195, 1994.

[60] Pratt, V.R. Chu Spaces, 1999.
[61] Pratt, V.R. Chu spaces from the representational viewpoint. *Annals of Pure and Applied Logic*, 96(1):319 – 333, 1999.

[62] Randall, C., Foulis, D. Tensor products of quantum logics do not exist. *Notices Amer. Math. Soc*, 26(6), 1979.

[63] Reichenbach, H. *The Logico-Algebraic Approach to Quantum Mechanics*, volume vol 5a of The University of Western Ontario Series in Philosophy of Science (A Series of Books on Philosophy of Science, Methodology, and Epistemology, chapter Three-Valued Logic and the Interpretation of Quantum Mechanics, pages 53–97. Dordrecht, springer edition, 1975.

[64] Rovelli, C. Relational quantum mechanics. *International Journal of Theoretical Physics*, 35(8):1637–1678, Aug 1996.

[65] Seevinck, M.P. from E. Specker: "The logic of non-simultaneously decidable propositions" (1960). Translation of 'Die Logik Der Nicht Gleichzeitig Entscheidbarer Aussagen' by Ernst Specker, Di-alectica, vol. 14, 239 - 246 (1960).

[66] Spekkens, R.W. Evidence for the epistemic view of quantum states: A toy theory. *Phys. Rev. A*, 75:032110, Mar 2007.

[67] Stubbe, I., van Steirteghem, B. Propositional systems, Hilbert lattices and generalized hilbert spaces. In Engesser, K., Gabbay D.M., Lehmann, D., editor, *Handbook of Quantum Logic and Quantum Structures*, pages 477 – 523. Elsevier Science B.V., Amsterdam, 2007.

[68] Vetterlein, T. Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces. *International Journal of Theoretical Physics*, 60(2):727–738, 2021.

[69] Wilce, A. *Test Spaces and Orthoalgebras*, pages 81–114. Springer Netherlands, Dordrecht, 2000.

[70] Wilce, A. *Test Spaces* In Engesser, K., Gabbay D.M., Lehmann, D., editor, *Handbook of Quantum Logic and Quantum Structures*, pages 443–549. Elsevier Science B.V., Amsterdam, 2009.

[71] Zeilinger, A. A foundational principle for quantum mechanics. *Foundations of Physics*, 29(4):631–643, Apr 1999.

[72] Zhong, S. Correspondence Between Kripke Frames and Projective Geometries. *Studia Logica*, 106(1):167–189, 2018.

[73] Zhong, S. *Quantum States: An Analysis via the Orthogonality Relation*. *Synthese*, 2021.