Qm Measurement And Instruments

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a nonzero complex Hilbert space (Hilbert space), let $P$ be an orthogonal projection on $H$ (The Hilbert orthogonal projection onto a closed subspace) and let $\rho$ be a density operator with $p:=\mathrm{tr}(\rho P)>0$ (Density operators and scalar spectral probability clause (a)). Define the selective update ${\rho }_P:=\frac{P\rho P}{p}\mathrm{.}$ Then:

(a) $P\rho P$ is a positive trace-class operator with $\mathrm{tr}(P\rho P)=p$, so ${\rho }_P$ is a density operator;

(b) $\mathrm{tr}({\rho }_{P}P)=1$ and $P{\rho }_{P}P={\rho }_P=P{\rho }_P={\rho }_{P}P;$ in particular ${\rho }_P$ commutes with $P$ and is fixed by compression with $P$;

(c) if $Q$ is an orthogonal projection with $QP=0$, then $\mathrm{tr}({\rho }_{P}Q)=0$ and $Q{\rho }_{P}Q=0$;

(d) if $\rho =T_{u,u}$ for a unit vector $u$ with $Pu\mathrm{\ne }0$, where $T_{u,u}x=⟨x,u⟩u$ (Rank-one operator calculus), then ${\rho }_P=T_{Pu,Pu}\mathrm{/}\mathrm{\parallel }Pu{\mathrm{\parallel }}^2$, the pure state of the normalized vector $Pu$; the value depends only on the ray of $Pu$, meaning its equivalence class of nonzero vectors under $x\sim y$ if $y=zx$ for some $z\in \mathbb{C}\setminus \{0\}$.

No physical interpretation of the update is asserted.

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a nonzero complex Hilbert space (Hilbert space), let $(P_a{)}_{a\in A}$ be a finite or countably infinite family of pairwise orthogonal orthogonal projections on $H$ (The Hilbert orthogonal projection onto a closed subspace) whose strong sum is the identity, ${\sum }_{a\in A}{P}_{a}x=x(x\in H),$ let $\rho$ be a density operator (Density operators and scalar spectral probability clause (a)), and put $p_a:=\mathrm{tr}(\rho P_a)$. Define the non-selective Lueders map $\mathrm{\Phi }(\rho ):={\sum }_{a\in A}{P}_a\rho P_a\mathrm{.}$ Then:

(a) the series converges in trace norm; $\mathrm{\Phi }(\rho )$ is a positive trace-class operator with $\mathrm{tr}\mathrm{\Phi }(\rho )=1$, hence a density operator, and $\mathrm{\Phi }(\rho )={\sum }_a{P}_a\mathrm{\Phi }(\rho )P_a$;

(b) for every $b$, $P_b\mathrm{\Phi }(\rho )=\mathrm{\Phi }(\rho )P_b=P_b\mathrm{\Phi }(\rho )P_b=P_b\rho P_b$; in particular $\mathrm{\Phi }(\rho )$ commutes with every $P_b$;

(c) discarded-outcome ensemble. $\mathrm{\Phi }(\rho )={\sum }_{a:p_a>0}{p}_a{\rho }_a$, where ${\rho }_a=P_a\rho P_a\mathrm{/}{p}_a$ is the selective update of The selective Lueders update by an orthogonal projection; the series converges in trace norm;

(d) if $B\in \mathcal{B}(H)$ is bounded and $P_{a}B=B{P}_a$ for every $a$, then $\mathrm{tr}(\mathrm{\Phi }(\rho )B)=\mathrm{tr}(\rho B)$;

(e) if $\psi =\alpha u+\beta v$ with $u,v$ orthonormal, $u\in \mathrm{ran}{P}_1$, $v\in \mathrm{ran}{P}_2$ for distinct labels $1\mathrm{\ne }2$, and $\rho =T_{\psi ,\psi }$, then $\mathrm{\Phi }(\rho )=\mathrm{\mid }\alpha {\mathrm{\mid }}^2{T}_{u,u}+\mathrm{\mid }\beta {\mathrm{\mid }}^2{T}_{v,v}$, and $\mathrm{tr}(\mathrm{\Phi }(\rho )(T_{u,v}+T_{v,u}))=0,\mathrm{tr}(\rho (T_{u,v}+T_{v,u}))=2\mathrm{Re}(\alpha \bar{\beta });$

(f) if $\rho =T_{\psi ,\psi }$ for a nonzero $\psi$ then $\mathrm{\Phi }(\rho )={\sum }_{a:P_a\psi \mathrm{\ne }0}\mathrm{\parallel }{P}_a\psi {\mathrm{\parallel }}^2{T}_{\nu (P_a\psi ),\nu (P_a\psi )}$ where $\nu :H\setminus \{0\}\to H$ is the normalization map $\nu (x):=x\mathrm{/}\mathrm{\parallel }x\mathrm{\parallel }$, the series converging in trace norm.

No physical interpretation of the map is asserted.

Adopt the state postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1), the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) and the ideal conditional-measurement postulate Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes (P3) for the declared system, whose state space is $D(H)$ for the assigned nonzero separable complex Hilbert space $H$ (The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity). Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis of the cited mathematics. Here the projective outcome law is supplied directly by P3(ii), and the finite-effect law by P2(iv); no spectral PVM is constructed from an assigned self-adjoint observable. The additional Axiom of Choice required by P2(ii) for that construction is therefore not used here. The following vocabulary is used by this page.

(a) Declared ideal sharp measurement. A declared ideal sharp measurement of the system is a finite or countably infinite family $(P_a{)}_{a\in A}$ of bounded orthogonal projections on $H$ (The Hilbert orthogonal projection onto a closed subspace) that are mutually orthogonal, $P_a{P}_b=0$ for $a\mathrm{\ne }b$, and whose strong sum is the identity, ${\sum }_{a\in A}{P}_{a}x=x(x\in H)\mathrm{.}$ The index $a$ is the outcome label and the family $(P_a{)}_{a\in A}$ is the outcome structure of the declared measurement; it is part of the declared apparatus model, not a quantity computed from the state. This is clause (i) of P3.

(b) The Lueders instrument of a declared ideal sharp measurement. For a preparation $\rho$ the outcome probabilities are $p_a:=\mathrm{tr}(\rho P_a)\ge 0,{\sum }_a{p}_a=1,$ and for a pure preparation of the unit ray $[\psi ]$ they are $p_a=\mathrm{\parallel }{P}_a\psi {\mathrm{\parallel }}^2$ (clause (ii) of P3). If $p_a>0$ the conditional state after the outcome $a$ is the selective Lueders update ${\rho }_a:=\frac{P_a\rho P_a}{p_a};$ it is a state with $\mathrm{tr}({\rho }_a{P}_a)=1$ and $P_a{\rho }_a{P}_a={\rho }_a$ (The selective Lueders update by an orthogonal projection, clause (iii) of P3). If the outcomes are recorded but not selected, the discarded-outcome state is $\mathrm{\Phi }(\rho ):={\sum }_{a\in A}{P}_a\rho P_a,$ the trace-norm convergent series of The non-selective Lueders map and the discarded-outcome ensemble; it is a state with $\mathrm{\Phi }(\rho )={\sum }_{a:p_a>0}{p}_a{\rho }_a$ and $\mathrm{tr}(\mathrm{\Phi }(\rho )P_a)=p_a$ (clause (iv) of P3).

(c) Declared ideal filter. A declared ideal filter is a declared ideal sharp measurement with exactly two outcomes $a=1,2$: the filter passes the component in the range of $P_1$ and blocks the component in the range of $P_2=I-P_1$. Repeated use of the filter in series means the sequential application of the conditional updates of clause (b).

(d) Declared instrument. A declared finite or countable instrument for an outcome set $A$ consists of a finite or countable family $(M_{a,r}{)}_{\begin{array}{c}a\in A\\ r\in R_a\end{array}}$ of bounded operators on $H$ with ${\sum }_{a\in A}{\sum }_{r\in R_a}{M}_{a,r}^{\ast }{M}_{a,r}=I$ in the strong sense (Countable Kraus normalization and trace-norm convergence), the effects $F_a:={\sum }_{r\in R_a}{M}_{a,r}^{\ast }{M}_{a,r},$ where each subseries converges strongly, independently of its enumeration. An effect on general $H$ is a bounded self-adjoint operator $F$ with $0\le ⟨Fx,x⟩\le \mathrm{\parallel }x{\mathrm{\parallel }}^2$ for all $x\in H$; the order $F\le G$ means $⟨(G-F)x,x⟩\ge 0$ for all $x$. Here $⟨F_{a}x,x⟩={\sum }_r\mathrm{\parallel }{M}_{a,r}x{\mathrm{\parallel }}^2\le \mathrm{\parallel }x{\mathrm{\parallel }}^2$, so $0\le F_a\le I$. In particular the positive partial sums $S_N$ of any subseries are contractions and have convergent quadratic forms. For a positive tail $D=S_N-S_m$, expanding $⟨D(x+ty),x+ty⟩\ge 0$ with $t=-⟨Dx,y⟩\mathrm{/}⟨Dy,y⟩$ when $⟨Dy,y⟩>0$, and varying $t\in \mathbb{C}$ when $⟨Dy,y⟩=0$, gives $\mathrm{\mid }⟨Dx,y⟩{\mathrm{\mid }}^2\le ⟨Dx,x⟩⟨Dy,y⟩$. Since $0\le D\le I$, taking $y=Dx$ yields $\mathrm{\parallel }Dx{\mathrm{\parallel }}^4\le ⟨Dx,x⟩⟨D(Dx),Dx⟩\le ⟨Dx,x⟩\mathrm{\parallel }Dx{\mathrm{\parallel }}^2$, hence $\mathrm{\parallel }Dx{\mathrm{\parallel }}^2\le ⟨Dx,x⟩$ (also when $Dx=0$). The partial sums therefore converge on every $x$ and their limit defines $F_a$. The outcome probabilities are $p_a:=\mathrm{tr}(\rho F_a)$. For finite $A$ this is clause (iv) of P2, even if an outcome has countably many Kraus operators. For countably infinite $A$ outside the projective case of (a)-(b), we explicitly adopt this same probability law as an additional assumption of the declared apparatus model; P2(iv) does not state that extension. The Kraus-normalization lemma establishes that these numbers are nonnegative and sum to one, not that they describe physical outcomes. The model also explicitly stipulates the conditional updates $\rho ⟼\frac{1}{p_a}{\sum }_{r\in R_a}{M}_{a,r}\rho M_{a,r}^{\ast }(p_a>0),$ and the discarded-outcome map $\rho \mapsto {\sum }_{a,r}{M}_{a,r}\rho M_{a,r}^{\ast }$. These update series converge in trace norm by the Kraus-normalization lemma, with grouping permitted by absolute trace-norm convergence for positive inputs. Their interpretation as post-measurement states is part of the instrument declaration; P3 supplies that interpretation only for its projective Lueders case. The family $(M_{a,r})$, called the Kraus family of the instrument, is part of the declared apparatus model. The effects $(F_a)$ alone do not determine the conditional updates: distinct normalized Kraus families can share the same effects while defining different updates (Finite effects and instruments: normalization and non-determination of the update clause (d); clause (v) of P3). More generally, on a specified measurable space $(X,\mathrm{\Sigma })$, a declared abstract instrument is a map $J:\mathrm{\Sigma }\to \mathcal{B}({\mathcal{S}}_1(H))$ into bounded complex-linear maps of trace-class operators satisfying: each $J_B$ is completely positive, meaning every finite matrix amplification preserves positive trace-class block operators; for disjoint $B_n$ with union $B$, $J_B(T)={\sum }_n{J}_{B_n}(T)$ in trace norm for every trace-class $T$; and $J_X$ preserves trace. Its outcome effect $F_B$ is the unique effect with $\mathrm{tr}(J_B(T))=\mathrm{tr}(T{F}_B)$, its probability is $p(B)=\mathrm{tr}(J_B(\rho ))$, and its conditional state is $J_B(\rho )\mathrm{/}p(B)$ only when $p(B)>0$. The interpretation of these formulas as an outcome law and a post-measurement state is likewise an explicitly adopted assumption of this abstract apparatus model. Abstract trace-class instruments: effects and Heisenberg duality establishes the effect duality, countably additive normalization and well-definedness of the positive-probability conditional state, not this physical interpretation. The Kraus case above is a concrete instance on $(A,2^A)$, with $J_B(T)={\sum }_{a\in B,r}{M}_{a,r}T{M}_{a,r}^{\ast }$ in trace norm. This declaration does not assert that every abstract instrument has a Kraus representation or that a laboratory apparatus realizes it.

(e) Declaration discipline. A model of an apparatus that is used to predict post-measurement states must declare both an outcome structure (effects) and an update rule; the outcome law of P2 alone does not supply the update, and the update is not a mathematical consequence of the effects. The special case in which the Kraus family consists of the projections $P_a$ themselves (with a single $r$ per outcome) is the Lueders instrument of clauses (a)-(b). The claim that a concrete laboratory device realizes a particular declared measurement, filter or instrument is a physical identification with uncertainty under the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), not a consequence of this definition.

(f) Limits. Declared ideal sharp measurements are idealizations of apparatus that are idealized as repeatable and projection-valued; the framework does not assert that every apparatus, or any particular apparatus, is of this form (clause (v) of P3). The following are outside the declared class and require their own models: a measurement of an exact value of a quantity whose spectrum is continuous (no normalization of a delta symbol is intended; clauses (iii) of P2 and (v) of P3); a photon-absorption or other destructive detector, whose declared instrument need not leave an outgoing state of the same form (clause (v) of P3); and a unitary system-apparatus interaction, which by itself produces conditional correlations without selecting a unique recorded outcome or a preferred outcome family (clause (v) of P3).

(g) Status. Clauses (a)-(c) read the projective rules of P3, and the finite-outcome law in (d) reads P2(iv). Clause (d) additionally stipulates the countably infinite nonprojective outcome law, generalized conditional and discarded-outcome updates, and abstract-instrument law and updates as assumptions of each declared apparatus model, as permitted by P3(v)'s requirement for separate models outside its projective class. These are not consequences of P1-P3 alone. Clauses (e)-(f) fix the declaration discipline and limits used by the page's later items. The cited mathematical items (The selective Lueders update by an orthogonal projection, The non-selective Lueders map and the discarded-outcome ensemble, Finite effects and instruments: normalization and non-determination of the update, Countable Kraus normalization and trace-norm convergence, Abstract trace-class instruments: effects and Heisenberg duality) are conditional mathematics about the displayed formulas; no physical process, outcome uniqueness or empirical fact is deduced here.

Adopt the state postulate P1, the observable and probability postulate P2 and the ideal conditional-measurement postulate P3 of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes), and assume the Axiom of Countable Choice, the hypothesis of the cited mathematics. Let $(P_a{)}_{a\in A}$ be a declared ideal sharp measurement of a system with state space $D(H)$ (Declared ideal measurements and instruments of the framework, The state space of the framework: preparations, pure rays and density operators) and let $\rho$ be a state. Then:

(a) Outcome law. For every outcome label $a$ the probability is $p_a=\mathrm{tr}(\rho P_a)$, with $p_a\ge 0$ and ${\sum }_a{p}_a=1$; for the pure state of a unit vector $\psi$ it is $p_a=\mathrm{\parallel }{P}_a\psi {\mathrm{\parallel }}^2$.

(b) Conditional state and repeatability. If $p_a>0$, the conditional state after the outcome $a$ is the state ${\rho }_a=P_a\rho P_a\mathrm{/}{p}_a$, with $\mathrm{tr}({\rho }_a{P}_a)=1$ and $\mathrm{tr}({\rho }_a{P}_b)=0$ for $b\mathrm{\ne }a$. Immediate repetition of the same declared measurement therefore returns the same outcome with certainty: $\mathbb{P}(\text{second outcome }b\mid \text{first outcome }a)={\delta }_{ab}\mathrm{.}$

(c) Discarded outcomes. The discarded-outcome state $\mathrm{\Phi }(\rho )={\sum }_a{P}_a\rho P_a$ is a state and coincides with the outcome-weighted mixture of the conditional states, $\mathrm{\Phi }(\rho )={\sum }_{a:p_a>0}{p}_a{\rho }_a,$ the series converging in trace norm; moreover $\mathrm{tr}(\mathrm{\Phi }(\rho )P_a)=p_a$ for every $a$ and $\mathrm{\Phi }(\rho )={\sum }_a{P}_a\mathrm{\Phi }(\rho )P_a$. The conditional-vs-discarded distinction is therefore exactly the distinction between one selected branch ${\rho }_a$ and the mixture of all branches.

(d) Compatible bounded quantities. If $B$ is a bounded operator with $P_{a}B=B{P}_a$ for every $a$, then $\mathrm{tr}(\mathrm{\Phi }(\rho )B)=\mathrm{tr}(\rho B)$. In particular, if the declared measurement is the spectral measurement of a bounded assigned sharp observable $A={\sum }_a{a}_a{P}_a$ (a finite or norm-convergent series with real $a_a$), then $A$ commutes with every $P_a$ and the mean of $A$ is unchanged by discarding the outcomes.

(e) Loss of coherence. If $\psi =\alpha u+\beta v$ with $u,v$ orthonormal, $u\in \mathrm{ran}{P}_1$, $v\in \mathrm{ran}{P}_2$ for distinct labels $1\mathrm{\ne }2$, and $\rho =T_{\psi ,\psi }$, then $\mathrm{\Phi }(\rho )=\mathrm{\mid }\alpha {\mathrm{\mid }}^2{T}_{u,u}+\mathrm{\mid }\beta {\mathrm{\mid }}^2{T}_{v,v},\mathrm{tr}(\mathrm{\Phi }(\rho )(T_{u,v}+T_{v,u}))=0,$ whereas for the coherent preparation $\mathrm{tr}(\rho (T_{u,v}+T_{v,u}))=2\mathrm{Re}(\alpha \bar{\beta })$: the discarded-outcome ensemble is the classical mixture of the branch states and carries no coherence between the ranges of $P_1$ and $P_2$.

(f) Scope. Clauses (a)-(e) are conditional on the adopted P3 and on the mathematical items cited; they do not assert that any laboratory apparatus realizes a declared ideal sharp measurement, do not select a unique recorded outcome, and do not describe the disturbance produced by an apparatus outside the declared class (clause (v) of P3). The identification of a device with a declared measurement is a tested identification with uncertainty under P7.

Adopt the state postulate P1, the observable and probability postulate P2 and the ideal conditional-measurement postulate P3 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes), and assume the Axiom of Countable Choice and the Axiom of Choice, the hypotheses of the cited spectral mathematics. Let $A$ be a bounded assigned sharp observable of a system with state space $D(H)$ (Assigned sharp observables and the outcome law of the framework), with spectral projection-valued measure $E_A$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator), and let $\lambda$ be an eigenvalue of $A$ with eigenspace $K=\ker (A-\lambda I)$ and $P:=E_A(\{\lambda \})$, the orthogonal projection of $H$ onto $K$. Declare the two-outcome ideal sharp measurement $(P,I-P)$, the $\lambda$-readout of $A$. Let $\psi \in K$ with $\mathrm{\parallel }\psi \mathrm{\parallel }=1$ and let $\rho =T_{\psi ,\psi }$ be the pure state of $\psi$. Then:

(a) Certainty and trivial update. $p_{\lambda }=\mathrm{tr}(\rho P)=1$, and the conditional state after the outcome $\lambda$ is ${\rho }_{\lambda }=\rho$: the $\lambda$-readout leaves this preparation unchanged.

(b) No disturbance inside the eigenspace. For every assigned sharp observable $C$ and every Borel set $S\subseteq \mathbb{R}$, the outcome law of $C$ computed in the conditional state equals its law computed in $\rho$ (Assigned sharp observables and the outcome law of the framework clause (b)); in particular the $\lambda$-readout induces no decoherence or disturbance within the eigenspace.

(c) Degenerate case: coherence is retained. If $\dim K\ge 2$ and $\psi =\alpha u+\beta v$ with $u,v$ orthonormal vectors of $K$ and $\alpha ,\beta \in \mathbb{C}\setminus \{0\}$, then the conditional state is the coherent superposition $T_{\psi ,\psi }$, not the mixture $\mathrm{\mid }\alpha {\mathrm{\mid }}^2{T}_{u,u}+\mathrm{\mid }\beta {\mathrm{\mid }}^2{T}_{v,v}$: for the bounded assigned sharp observable $W:=T_{w,w}-T_{w^{\mathrm{\prime }},w^{\mathrm{\prime }}},w:=\frac{u+v}{\sqrt{2}},w^{\mathrm{\prime }}=\frac{u-v}{\sqrt{2}},$ with $W$ declared dimensionless (spectral unit $1$), the conditional state assigns probability $1$ to the value $+1$ of $W$ whenever $\alpha =\beta =1\mathrm{/}\sqrt{2}$, i.e. whenever $\psi =w$, while the mixture $\mathrm{\mid }\alpha {\mathrm{\mid }}^2{T}_{u,u}+\mathrm{\mid }\beta {\mathrm{\mid }}^2{T}_{v,v}$ assigns probability $1\mathrm{/}2$ to each of the values $+1$ and $-1$ of $W$.

(d) Scope. The update used here is the Lueders update with the eigenspace projection $P$ itself, as adopted in P3; no physical apparatus is asserted to perform the $\lambda$-readout without disturbance, and a declared apparatus that resolves the eigenspace further (a finer family of projections inside $K$) is a different declared measurement, not the coarse $\lambda$-readout.