Table 2 Measurements carried out by the agents

From: Quantum theory cannot consistently describe the use of itself

Agent

Value

Measured system

Measurement completed at

Relevant vectors of measurement basis

Heisenberg projectors used for reasoning via (Q)

\({\bar{\rm F}}\)

r

R

n:01

\(\left| {{\rm{heads}}} \right\rangle _{\rm{R}}\quad\left| {{\rm{tails}}} \right\rangle _{\rm{R}}\)

\(\begin{array}{l}\pi _{w = {\rm{ok}}}^{n:10} = \left[ {(U_{{\rm{S}} \to {\rm{L}}}^{10 \to 20})^\dagger \left| {{\rm{ok}}} \right\rangle _{\rm{L}}} \right]\left[ \cdot \right]^\dagger \\ \hskip -40pt \pi _{w ={\rm{fail}}}^{n:10} = 1 - \pi _{w = {\rm{ok}}}^{n:10}\end{array}\)

F

z

S

n:11

\(\left| \downarrow \right\rangle _{\rm{S}}\quad\left| \uparrow \right\rangle _{\rm{S}}\)

\(\begin{array}{l}\pi _{z = - \frac{1}{2}}^{n:10} = \left| \downarrow \right\rangle \left\langle \downarrow \right|_{\rm{S}}\\ \pi _{z = + \frac{1}{2}}^{n:10} = \left| \uparrow \right\rangle \left\langle \uparrow \right|_{\rm{S}}\end{array}\)

\({\bar{\rm W}}\)

\(\bar w\)

\({\bar{\rm L}}\)

n:21

\(\left| {\overline {{\rm{ok}}} } \right\rangle _{{\bar{\rm L}}}=\sqrt {1/2} \left( {\left| {{\bar{\rm h}}} \right\rangle _{{\bar{\rm L}}} - \left| {{\bar{\rm t}}} \right\rangle _{{\bar{\rm L}}}} \right)\)

\(\begin{array}{l}\pi _{(\bar w ,z) = (\overline {{\rm{ok,}}} - {\textstyle{1 \over 2}})}^{n:00} = \left[ {(U_{{\rm{R}} \to {\overline{\rm{L}S}}}^{00 \to 10})^\dagger \left| {\overline {{\rm{ok}}} } \right\rangle _{{\bar{\rm L}}}\left| \downarrow \right\rangle _{\rm{S}}} \right]\left[ \cdot \right]^\dagger \\ \pi _{(\bar w ,z) \ne (\overline {{\rm{ok,}}} - {\textstyle{1 \over 2}})}^{n:00} = 1 - \pi _{(\bar w ,z) = (\overline {{\rm{ok,}}} - {\textstyle{1 \over 2}})}^{n:00}\end{array}\)

W

w

L

n:31

\(\left| {{\rm{ok}}} \right\rangle _{\rm{L}}=\sqrt {1/2} \left( {\left| { - \frac{1}{2}} \right\rangle _{\rm{L}} - \left| {{\rm{ + }}\frac{1}{2}} \right\rangle _{\rm{L}}} \right)\)

\(\pi _{(\bar w ,w ) = (\overline {{\rm{ok,}}} {\rm{ok}})}^{n:00} = \left[ {(U_{{\rm{R}} \to {\overline{\rm{L}S}}}^{00 \to 10})^\dagger (U_{{\rm{S}} \to {\rm{L}}}^{10 \to 20})^\dagger \left| {\overline {{\rm{ok}}} } \right\rangle _{{\bar{\rm L}}}\left| {{\rm{ok}}} \right\rangle _{\rm{L}}} \right]\left[ \cdot \right]^\dagger\)

  1. Each of the four agents observes a value, defined as the outcome of a measurement on a particular system at a particular time. The measurement basis vectors \(\left| {\overline {{\rm{ok}}} } \right\rangle _{{\bar{\rm L}}}\) and \(\left| {{\rm{ok}}} \right\rangle _{{{\rm L}}}\) shown in the last two rows are expressed in terms of states, such as \(\left| { - \frac{1}{2}} \right\rangle _{\rm{L}}\) and \(\left| { + \frac{1}{2}} \right\rangle _{\rm{L}}\), which are defined in Table 1. The last column shows the measurement operators that the agents insert into statement A(ii) when reasoning according to Assumption (Q). These operators are given in the Heisenberg picture, referring to the system’s state at a particular time, which is specied by a superscript. The bracket \(\left[ \cdot \right]^\dagger\) stands for the adjoint of the preceding expression
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