On a quantum particle in the unit interval [0, 1], perform a position measurement with imprecision 1/n and then a quantum measurement of the projection <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|\phi \rangle \langle \phi |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>ϕ</mi> <mo stretchy="false">⟩</mo> <mo stretchy="false">⟨</mo> <mi>ϕ</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> with some arbitrary but fixed normalized <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>. Call the outcomes <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X \in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Y \in \{0,1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Y</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that in the limit <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> corresponding to perfect precision for X, the probability of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Y=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Y</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> tends to 0 for every <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>. Since there is no density matrix, pure or mixed, which upon measurement of any <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$|\phi \rangle \langle \phi |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>ϕ</mi> <mo stretchy="false">⟩</mo> <mo stretchy="false">⟨</mo> <mi>ϕ</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> yields outcome 1 with probability 0, our result suggests that a novel type of quantum state beyond Hilbert space is necessary to describe a quantum particle after a perfect position measurement.

Quantum Zeno-like paradox for position measurements: a particle precisely found in space is nowhere to be found in Hilbert space

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2026年7月30日

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Xabier Oianguren-Asua,
Roderich Tumulka

摘要

On a quantum particle in the unit interval [0, 1], perform a position measurement with imprecision 1/n and then a quantum measurement of the projection $|\phi \rangle \langle \phi |$ $| ϕ ⟩ ⟨ ϕ |$ with some arbitrary but fixed normalized $\phi $ $ϕ$ . Call the outcomes $X \in [0,1]$ $X \in [0, 1]$ and $Y \in \{0,1\}$ $Y \in {0, 1}$ . We show that in the limit $n\rightarrow \infty $ $n \to \infty$ corresponding to perfect precision for X, the probability of $Y=1$ $Y = 1$ tends to 0 for every $\phi $ $ϕ$ . Since there is no density matrix, pure or mixed, which upon measurement of any $|\phi \rangle \langle \phi |$ $| ϕ ⟩ ⟨ ϕ |$ yields outcome 1 with probability 0, our result suggests that a novel type of quantum state beyond Hilbert space is necessary to describe a quantum particle after a perfect position measurement.