Since its birth quantum mechanics has inspired new logical ideas. The most important “logical revolution” of the theory concerns a divergence between the concepts of maximal information and logically complete information. This is a natural consequence of Heisenberg’s uncertainty principle: any quantum pure state represents a maximal information that cannot be consistently extended to a richer knowledge about the physical system under consideration. At the same time, such information is logically incomplete, since it cannot decide all relevant properties of the system in question. Birkhoff and von Neumann’s quantum logic represents a natural logical abstraction from the algebraic structure of all events (that may occur to a quantum system <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10699_2025_9988_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">S</mi> </math></EquationSource> </InlineEquation>), mathematically represented as projection operators of the Hilbert space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10699_2025_9988_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\mathcal {H}}^{\textbf{S}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold-script">H</mi> </mrow> <mi mathvariant="bold">S</mi> </msup> </math></EquationSource> </InlineEquation> associated to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10699_2025_9988_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">S</mi> </math></EquationSource> </InlineEquation>. Projections of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10699_2025_9988_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\mathcal {H}}^{\textbf{S}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold-script">H</mi> </mrow> <mi mathvariant="bold">S</mi> </msup> </math></EquationSource> </InlineEquation> represent sharp quantum events that satisfy the logical non-contradiction principle. Following Ludwig, the sharp concept of quantum event can be generalized to the unsharp concept of quantum effect, that gives rise to possible violations of the non-contradiction principle. The concepts of observable and of compatibility between observables have different behaviors in the framework of sharp and of unsharp quantum theory. Canonical examples of observables (say, position and momentum), that are incompatible in sharp quantum theory, turn out to be jointly measurable in the case of unsharp quantum theory. Interestingly enough, a famous thought experiment (proposed by Heisenberg) about the possibility of an approximate simultaneous measurement of position and momentum can be described and justified in the formal language of unsharp quantum theory.

Sharp and Unsharp Quantum Incompatibilities. A Comparison and Some Foundational Questions

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发表日期：

2025年7月10日

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Roberto Beneduci,
Maria Luisa Dalla Chiara,
Roberto Giuntini,
Giuseppe Sergioli

摘要

Since its birth quantum mechanics has inspired new logical ideas. The most important “logical revolution” of the theory concerns a divergence between the concepts of maximal information and logically complete information. This is a natural consequence of Heisenberg’s uncertainty principle: any quantum pure state represents a maximal information that cannot be consistently extended to a richer knowledge about the physical system under consideration. At the same time, such information is logically incomplete, since it cannot decide all relevant properties of the system in question. Birkhoff and von Neumann’s quantum logic represents a natural logical abstraction from the algebraic structure of all events (that may occur to a quantum system $\textbf{S}$ $S$ ), mathematically represented as projection operators of the Hilbert space $\varvec{\mathcal {H}}^{\textbf{S}}$ $H^{S}$ associated to $\textbf{S}$ $S$ . Projections of $\varvec{\mathcal {H}}^{\textbf{S}}$ $H^{S}$ represent sharp quantum events that satisfy the logical non-contradiction principle. Following Ludwig, the sharp concept of quantum event can be generalized to the unsharp concept of quantum effect, that gives rise to possible violations of the non-contradiction principle. The concepts of observable and of compatibility between observables have different behaviors in the framework of sharp and of unsharp quantum theory. Canonical examples of observables (say, position and momentum), that are incompatible in sharp quantum theory, turn out to be jointly measurable in the case of unsharp quantum theory. Interestingly enough, a famous thought experiment (proposed by Heisenberg) about the possibility of an approximate simultaneous measurement of position and momentum can be described and justified in the formal language of unsharp quantum theory.