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

Abstract 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 $$|\phi \rangle \langle \phi |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>⟩</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> with some arbitrary but fixed normalized $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> . Call the outcomes $$X \in [0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> and $$Y \in \{0,1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . We show that in the limit $$n\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> corresponding to perfect precision for X , the probability of $$Y=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> tends to 0 for every $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> . Since there is no density matrix, pure or mixed, which upon measurement of any $$|\phi \rangle \langle \phi |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>⟩</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> 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.