Qm Measurement And Instruments — Examples

Adopt P1, P2 and P3 for the declared system and assume Countable Choice, as in Declared ideal measurements and instruments of the framework. Declare a two-level system whose assigned Hilbert space is $H={\mathbb{C}}^2$ with the standard inner product and orthonormal basis $(e_{+},e_{-})$ (The state space of the framework: preparations, pure rays and density operators, Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space), interpreted as the two eigenstates of the $z$-component of a spin-half degree of freedom. Use SI units: $\mathrm{\hslash }>0$ is the adopted reduced Planck constant, with action dimension $M{L}^2{T}^{-1}$ and unit joule second ($\mathrm{J}\mathrm{s}$); the matrices $J_i$ and the angles are dimensionless. Thus $S_i$, $S_n$ and their outcome values have angular-momentum dimension $M{L}^2{T}^{-1}$ and are read in $\mathrm{J}\mathrm{s}$. No numerical value of $\mathrm{\hslash }$ is used. By the spin assignment of P6 with $s=\frac{1}{2}$, the spin operators are $S_i=\mathrm{\hslash }{J}_i$ with $J_i=\frac{1}{2}{\sigma }_i$ for the Pauli matrices in this basis (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii), Finite spin matrices clause (e)).

For a unit vector $n=(\sin \theta ,0,\cos \theta )$, $0\le \theta \le \pi$, put $u(\theta ):=\cos \frac{\theta }{2}{e}_{+}+\sin \frac{\theta }{2}{e}_{-},v(\theta ):=-\sin \frac{\theta }{2}{e}_{+}+\cos \frac{\theta }{2}{e}_{-}\mathrm{.}$ Then $u(\theta )$ and $v(\theta )$ are orthonormal and $S_n:=n\cdot \mathbf{S}=\frac{\mathrm{\hslash }}{2}(\sin \theta {\sigma }_x+\cos \theta {\sigma }_z),S_{n}u(\theta )=\frac{\mathrm{\hslash }}{2}u(\theta ),S_{n}v(\theta )=-\frac{\mathrm{\hslash }}{2}v(\theta )\mathrm{.}$ Declare the ideal filter $F(n)$ to be the ideal sharp measurement with the two outcomes $+$ and $-$, whose outcome projections are $T_{u(\theta ),u(\theta )}$ and $T_{v(\theta ),v(\theta )}$, the spectral projections of $S_n$ at $\pm \mathrm{\hslash }\mathrm{/}2$ (Declared ideal measurements and instruments of the framework clause (c), Rank-one operator calculus).

The example considers three ideal filters in series, $F_1=F(e_z)$ (so $\theta =0$), $F_2=F(n)$ with axis $n$ at angle $\theta$, and $F_3=F(m)$ with axis $m=(\sin \phi ,0,\cos \phi )$, and the initial preparation ${\rho }_0=T_{e_{+},e_{+}}$, the state of a spin-half system with $S_z=+\mathrm{\hslash }\mathrm{/}2$. Then:

(i) Through $F_1$ the outcome $+$ has probability $1$ and the conditional state is again ${\rho }_0$.

(ii) Through $F_2$, conditioned on having passed $F_1$, $\mathbb{P}(+)={\cos }^2\frac{\theta }{2},\mathbb{P}(-)={\sin }^2\frac{\theta }{2},$ and, if ${\cos }^2(\theta \mathrm{/}2)>0$, after the outcome $+$ the conditional state is the pure state of $u(\theta )$.

(iii) If ${\cos }^2(\theta \mathrm{/}2)>0$, through $F_3$, conditioned on having passed $F_2$ with the outcome $+$, the outcome $+$ has probability $\mathbb{P}(+)=\mid ⟨u(\phi ),u(\theta )⟩{\mid }^2={\cos }^2\frac{\phi -\theta }{2}\mathrm{.}$ For all $0\le \theta \le \pi$, including the zero-probability first branch, the joint probability of the outcomes $+,+,+$ is ${\cos }^2(\theta \mathrm{/}2){\cos }^2((\phi -\theta )\mathrm{/}2)$.

(iv) In particular, for $\theta =\pi \mathrm{/}2$ and $\phi =\pi$, the model probability of outcome $+$ at $F_2$ is $\frac{1}{2}$, and conditioned on that outcome the model probability of outcome $+$ at $F_3$ is also $\frac{1}{2}$, giving joint probability $\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$ for passing all three declared filters. If $F_2$ is omitted, $F_3$ has model probability ${\cos }^2(\pi \mathrm{/}2)=0$ of outcome $+$ on the state emerging from $F_1$. Thus the intervening declared measurement changes the later model probabilities; these are not measured transmission fractions.

The computation is a model computation for declared ideal filters and the declared spin assignment: it does not derive the spatial Stern-Gerlach deflection or the inhomogeneous-field geometry, records no detector counts, resolution, background or uncertainty, and asserts nothing about the historical silver-atom experiments or about any particular apparatus.

• Idealization. The declared filters model only the spin-component outcomes and conditional updates. This example supplies no spatial field, beam-deflection or detector-response model; those are separate material and are not claimed here (Scope and empirical status of the measurement-and-instruments page).
• Order dependence is model-internal. The statement that inserting $F_2$ changes the later statistics is a consequence of the Lueders update for the declared filters; it is not a claim that every physical filter disturbs every subsequent measurement in this way, and it is not a claim about atoms in the absence of a declared preparation and detection model.
• No empirical content. The number $\frac{1}{4}$ is an exact model probability for the declared ideal filters, not a measured fraction; no calibration, efficiency, background, alignment or sampling model is supplied.

Adopt the following finite polarization assignment and measurement rules as additional model assumptions for this example, using the declared-instrument vocabulary of Declared ideal measurements and instruments of the framework. These are stipulated for the finite polarization space, beyond the massive-matter sector of P1; they are not consequences of its massive-particle assignment. Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the mathematical suppliers. Fix a propagation label $k_0\in {\mathbb{R}}^3\setminus \{0\}$ in ${\mathrm{m}}^{-1}$ relative to fixed Euclidean spatial coordinates, and a real orthonormal transverse pair $v_x,v_y\in {\mathbb{R}}^3$ with $k_0\cdot v_x=k_0\cdot v_y=0$. Identify $H={\mathbb{C}}^2$, with its standard inner product and orthonormal basis $(e_x,e_y)$, unitarily with $\mathbb{C}{v}_x\oplus \mathbb{C}{v}_y$ by $e_x\mapsto v_x$, $e_y\mapsto v_y$. Its state space is $D(H)$ and a unit ray $[\psi ]$ represents $T_{\psi ,\psi }$ for $\mathrm{\parallel }\psi \mathrm{\parallel }=1$ (The state space: density operators, pure and mixed states, and purity, Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space). The identification of these unit rays as polarization preparations is the adopted finite assignment, motivated by the source account in Binney and Skinner, Section 1.3.6, printed pages 14-15. It supplies no photon number or radiation dynamics. Angles are dimensionless, read in radians. For a real angle $\alpha$ put $e_{\alpha }:=\cos \alpha e_x+\sin \alpha e_y,e_{\alpha }^{\perp }:=-\sin \alpha e_x+\cos \alpha e_y\mathrm{.}$ The ideal analyser $A(\alpha )$ is the declared ideal sharp measurement with the two outcomes transmit and block, whose projections are $T_{e_{\alpha },e_{\alpha }}$ and $T_{e_{\alpha }^{\perp },e_{\alpha }^{\perp }}$ respectively, with outcome and update rules stipulated in [F2] below (Declared ideal measurements and instruments of the framework clause (c), Rank-one operator calculus).

The example takes the preparation $[e_x]$, the chain of analysers $A(0)$ with $\alpha =0^{\circ }$, $A(\pi \mathrm{/}4)$ with $\alpha ={45}^{\circ }$ and $A(\pi \mathrm{/}2)$ with $\alpha ={90}^{\circ }$, and computes:

(i) The analyser $A(\alpha )$ transmits the preparation $[e_{\beta }]$ with probability $\mathbb{P}(\text{transmit})=\mid ⟨e_{\alpha },e_{\beta }⟩{\mid }^2={\cos }^2(\alpha -\beta ),$ and, when ${\cos }^2(\alpha -\beta )>0$, after transmission the conditional state is the pure state of $e_{\alpha }$.

(ii) Consequently the chain $0^{\circ },{45}^{\circ },{90}^{\circ }$ transmits with joint probability ${\cos }^2(0-0)\cdot {\cos }^2(\frac{\pi }{4}-0)\cdot {\cos }^2(\frac{\pi }{2}-\frac{\pi }{4})=1\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4},$ whereas the chain $0^{\circ },{90}^{\circ }$ alone transmits with probability ${\cos }^2(\pi \mathrm{/}2)=0$.

(iii) Removing the middle analyser therefore changes the transmission of the chain from $\frac{1}{4}$ to $0$ within the declared model: the intervening declared measurement changes the state, and the later analyser sees the state selected by the earlier one, not the original preparation.

(iv) Model comparison only. A declared classical amplitude model for a transverse field in the same plane, in which the analyser transmits the component along $e_{\alpha }$ and the transmitted intensity is stipulated to be proportional to the squared modulus of that component, gives the same ratio ${\cos }^2(\alpha -\beta )$ for a preparation along $e_{\beta }$. The agreement of these two model computations does not establish that either describes nature, and it does not by itself establish field quantization (What the worked instruments of the companion page establish); no classical Maxwell theorem is invoked, and the field model is a stipulated finite-dimensional comparison. No transmission counts, detector efficiencies, backgrounds or sampling statistics are reported.

• Idealisation. Each analyser is represented by the declared ideal sharp measurement of clause (c) of Declared ideal measurements and instruments of the framework: the blocked component is not available for later experiments and the transmitted component leaves the analyser in the corresponding pure state. A physical Polaroid or polarizing beam splitter has its own absorption, reflection, efficiency and background behaviour, which this example does not model.
• No radiation model. The two-level space rather than a quantized radiation mode is the declared model; no photon number, mode function, field quantization or Maxwell dynamics is asserted, and no position or arrival-time information is described.
• Sampling. The numbers $1,\frac{1}{2},\frac{1}{4}$ are single-trial model probabilities. To describe a run of $N\ge 1$ trials one must additionally specify a joint sampling law. For example, independent Bernoulli variables $X_1,\dots ,X_N\in \{0,1\}$ with $\mathbb{P}(X_j=1)=q$ encode identical transmit trials at a stated probability $q$; their frequency is $f_N=N^{-1}{\sum }_j{X}_j$. Even for this sampling model a particular finite frequency need not be close to $q$: at $q=1\mathrm{/}4$ the all-blocked and all-transmitted runs have probabilities $(3\mathrm{/}4{)}^N>0$ and $(1\mathrm{/}4{)}^N>0$. Identical single-trial probabilities alone specify no joint sampling law. No trial data or inference is reported here.

Adopt the following finite detector state and instrument assignments as additional assumptions of this example, using the preparation and instrument formulas of P1-P3 on this explicitly declared effective model; their massive-matter formulation alone does not supply this extension. Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis of the mathematical suppliers. Fix a propagation label $k_0\in {\mathbb{R}}^3\setminus \{0\}$ in ${\mathrm{m}}^{-1}$ relative to fixed Euclidean spatial coordinates, and a real orthonormal transverse pair $v_1,v_2$ with $k_0\cdot v_1=k_0\cdot v_2=0$. As an additional finite detector-model assumption, assign $H=\mathbb{C}{e}_0\oplus V\cong {\mathbb{C}}^3$, where $V=\mathbb{C}{v}_1\oplus \mathbb{C}{v}_2$, with orthonormal coordinates $(e_0,e_1,e_2)$, $e_j\leftrightarrow v_j$ for $j=1,2$, and state space $D(H)$ (The state space: density operators, pure and mixed states, and purity, Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space). A unit vector $\xi$ represents the preparation $T_{\xi ,\xi }$. The level $e_0$ is the declared absorbed/no-photon marker, orthogonal to the polarization plane $V=\mathrm{span}(e_1,e_2)$; it is a bookkeeping level rather than an assigned Fock vacuum. The extension to $H$ and the instrument rule below are adopted assumptions of this effective detector model, not consequences of the massive-matter state assignment or a full radiation dynamics. Angles and all amplitude coordinates are dimensionless, with angles read in radians. Fix a unit vector $v=\cos \gamma e_1+\sin \gamma e_2,v^{\perp }:=-\sin \gamma e_1+\cos \gamma e_2,$ so that $(e_0,v,v^{\perp })$ is an orthonormal basis, and fix a detection parameter $p\in (0,1)$, a dimensionless model parameter. Declare the two-outcome instrument with Kraus family (Declared ideal measurements and instruments of the framework clause (d)) $M_{\mathrm{d}\mathrm{e}\mathrm{t}}:=\sqrt{p}{T}_{e_0,v},M_{\mathrm{n}\mathrm{o}}:=T_{e_0,e_0}+\sqrt{1-p}{T}_{v,v}+T_{v^{\perp },v^{\perp }},$ where $T_{x,y}z=⟨z,y⟩x$ (Rank-one operator calculus), with outcomes $\mathrm{d}\mathrm{e}\mathrm{t}$ and $\mathrm{n}\mathrm{o}$, effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}=M_{\mathrm{d}\mathrm{e}\mathrm{t}}^{\ast }{M}_{\mathrm{d}\mathrm{e}\mathrm{t}}$, $F_{\mathrm{n}\mathrm{o}}=M_{\mathrm{n}\mathrm{o}}^{\ast }{M}_{\mathrm{n}\mathrm{o}}$, probabilities $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=\mathrm{tr}(\rho F_{\mathrm{d}\mathrm{e}\mathrm{t}})$, $p_{\mathrm{n}\mathrm{o}}=\mathrm{tr}(\rho F_{\mathrm{n}\mathrm{o}})$ and updates $\rho \mapsto M_a\rho M_a^{\ast }\mathrm{/}{p}_a$ for positive probabilities. The example considers preparations $\rho =T_{\psi ,\psi }$ with $\psi =\alpha v+\beta v^{\perp },\mathrm{\mid }\alpha {\mathrm{\mid }}^2+\mathrm{\mid }\beta {\mathrm{\mid }}^2=1,$ so that the preparation lies in the polarization plane, and computes:

(i) The family is normalized, $M_{\mathrm{d}\mathrm{e}\mathrm{t}}^{\ast }{M}_{\mathrm{d}\mathrm{e}\mathrm{t}}+M_{\mathrm{n}\mathrm{o}}^{\ast }{M}_{\mathrm{n}\mathrm{o}}=I$, with effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}=p{T}_{v,v}$ and $F_{\mathrm{n}\mathrm{o}}=T_{e_0,e_0}+(1-p)T_{v,v}+T_{v^{\perp },v^{\perp }}$; hence $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=p\mathrm{\mid }\alpha {\mathrm{\mid }}^2,p_{\mathrm{n}\mathrm{o}}=1-p\mathrm{\mid }\alpha {\mathrm{\mid }}^2\mathrm{.}$

(ii) When $\alpha \mathrm{\ne }0$, conditioned on the outcome $\mathrm{d}\mathrm{e}\mathrm{t}$, the conditional state is $T_{e_0,e_0}$: the branch carries the absorbed/no-photon record and leaves no outgoing state in the polarization plane. In particular the model does not produce an outgoing one-photon polarization state for the detected branch.

(iii) Conditioned on the outcome $\mathrm{n}\mathrm{o}$, the conditional state is the pure state of the normalized vector $\frac{\sqrt{1-p}\alpha v+\beta v^{\perp }}{\sqrt{(1-p)\mathrm{\mid }\alpha {\mathrm{\mid }}^2+\mathrm{\mid }\beta {\mathrm{\mid }}^2}},$ that is, the $v$-component is amplitude-suppressed by $\sqrt{1-p}$ rather than projected away. For the fully detectable preparation $\psi =v$ ($\alpha =1$) and $p=\frac{1}{2}$ one has $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=p_{\mathrm{n}\mathrm{o}}=\frac{1}{2}$ and the no-detection conditional state is again $T_{v,v}$; a repetition of the instrument on that branch again detects with probability $\frac{1}{2}$, so the no-detection outcome is not repeatable.

(iv) When $\alpha \mathrm{\ne }0$, after the detection branch the state is $T_{e_0,e_0}$, and a repetition gives $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=\mathrm{tr}(T_{e_0,e_0}{F}_{\mathrm{d}\mathrm{e}\mathrm{t}})=p\mathrm{\mid }⟨e_0,v⟩{\mathrm{\mid }}^2=0$: the detection outcome is not repeatable either. The effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}$ and $F_{\mathrm{n}\mathrm{o}}$ are not projections for $0<p<1$, so the instrument is not a declared ideal sharp measurement; it is a declared general instrument whose update is part of its declaration (Finite effects and instruments: normalization and non-determination of the update clause (d), Declared ideal measurements and instruments of the framework clauses (d)-(f)).

No real detector, efficiency, dark count, dead time, photon number or measured rate is described; the computation is a model computation in the declared three-level space.

• Interpretation of the marker. The level $e_0$ is a bookkeeping record of the branch in which the model detector has absorbed the excitation of the polarization mode. The example is honest about being a declared three-level idealisation: it asserts no photon-number content, no outgoing vacuum field, no detector dynamics and no real absorption mechanism.
• Why this is not a projection. The effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}$ and $F_{\mathrm{n}\mathrm{o}}$ sum to $I$ and have eigenvalues in $[0,1]$, so they are legitimate effects of a finite-resolution apparatus model; the principal contrast with the projective case is that the update of the no-detection branch suppresses the detected component by the amplitude $\sqrt{1-p}$ rather than projecting it away, and that neither outcome is repeatable.

Declare a three-level system with assigned Hilbert space $H={\mathbb{C}}^3$ and orthonormal basis $(e_1,e_2,e_3)$, and put $K:=\mathrm{span}(e_1,e_2),P:=T_{e_1,e_1}+T_{e_2,e_2},\psi :=\frac{e_1+e_2}{\sqrt{2}},w^{\mathrm{\prime }}:=\frac{e_1-e_2}{\sqrt{2}},$ with $T_{x,y}z=⟨z,y⟩x$ (The state space of the framework: preparations, pure rays and density operators, Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space, Rank-one operator calculus). Declare the two ideal sharp measurements $\mathcal{C}:=(P,I-P)\text{(coarse readout of }K\text{)},\mathcal{G}:=(T_{e_1,e_1},T_{e_2,e_2},T_{e_3,e_3})\text{(fine readout)},$ and consider the pure preparation $\rho =T_{\psi ,\psi }$ and the bounded self-adjoint observable, declared dimensionless with spectral unit $1$, $W:=T_{\psi ,\psi }-T_{w^{\mathrm{\prime }},w^{\mathrm{\prime }}}$ with spectral projections $T_{\psi ,\psi }$ at $+1$, $T_{w^{\mathrm{\prime }},w^{\mathrm{\prime }}}$ at $-1$ and $I-T_{\psi ,\psi }-T_{w^{\mathrm{\prime }},w^{\mathrm{\prime }}}$ at $0$ (Declared ideal measurements and instruments of the framework, Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely). Then:

(i) In the coarse measurement $\mathcal{C}$ the outcome $K$ has probability $1$, and the conditional state is again $\rho =T_{\psi ,\psi }$: the coarse readout leaves the coherent superposition inside the degenerate subspace intact.

(ii) In the fine measurement $\mathcal{G}$ the outcomes $e_1$ and $e_2$ have probabilities $\frac{1}{2}$ each (and $e_3$ has probability $0$), with conditional states $T_{e_1,e_1}$ and $T_{e_2,e_2}$: the fine readout selects a basis state.

(iii) For the observable $W$ the two declared measurements give different model predictions: after the coarse outcome the value $+1$ has probability $1$; the mixture $\frac{1}{2}{T}_{e_1,e_1}+\frac{1}{2}{T}_{e_2,e_2}$ (the discarded ensemble of the fine measurement) assigns probability $\frac{1}{2}$ to $+1$ and $\frac{1}{2}$ to $-1$. Hence the coarse Lueders update is not equivalent to selecting one eigenvector of the degenerate subspace with probability $\frac{1}{2}$ and forgetting the other: that replacement destroys the coherence of $\psi$ and changes the statistics of $W$.

(iv) The fine measurement refines the coarse outcome structure: grouping outcomes $e_1,e_2$ gives $K$, and outcome $e_3$ gives $K^{\perp }$, since $P=T_{e_1,e_1}+T_{e_2,e_2}$ and $I-P=T_{e_3,e_3}$. It is nevertheless a different declared measurement with more detailed outcomes. The coarse outcome "the system is in $K$" does not itself report whether the fine outcome was $e_1$ or $e_2$; replacing the coarse Lueders update by the discarded fine ensemble is a different model, with the observable consequence exhibited in (iii). No apparatus is asserted to realize either declared measurement, and no empirical comparison is made.