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3.3. Non-projective state update: atomic instruments

In general, the statistical properties of any measurement are characterized by

the output probability distribution ${\textstyle Pr\{{\text{x}}=x\parallel \rho \}}$ , the probability distribution of the output ${\textstyle x}$ of the measurement in the input state ${\textstyle \rho }$ ;
the quantum state reduction ${\textstyle \rho \rightarrow \rho _{(X=x)}}$ ,the state change from the input state ${\textstyle \rho }$ to the output state ${\textstyle \rho \rightarrow \rho _{(X=x)}}$ conditional upon the outcome ${\textstyle {\text{X}}=x}$ of the measurement.

In von Neumann’s formulation, the statistical properties of any measurement of an observable is uniquely determined by Born’s rule (5) and the projection postulate (6), and they are represented by the map (9), an instrument of von Neumann type. However, von Neumann’s formulation does not reflect the fact that the same observable $A$ represented by the Hermitian operator ${\hat {A}}$ in ${\textstyle {\mathcal {H}}}$ can be measured in many ways.8 Formally, such measurement-schemes are represented by quantum instruments.

Now, we consider the simplest quantum instruments of non von Neumann type, known as atomic instruments. We start with recollection of the notion of POVM (probability operator valued measure); we restrict considerations to POVMs with a discrete domain of definition ${\textstyle X=\{x_{1}....,x_{N}.....\}}$ . POVM is a map ${\textstyle x\rightarrow {\hat {D}}(x)}$ such that for each ${\textstyle x\in X}$ , ${\hat {D}}(x)$ is a positive contractive Hermitian operator (called effect) (i.e., ${\textstyle {\hat {D}}(x)^{*}={\hat {D}}(x),0\leq \langle \psi |{\hat {D}}(x)\psi \rangle \leq 1}$ or any ${\textstyle \psi \in {\mathcal {H}}}$ ), and the normalization condition

${\textstyle \sum _{x}{\hat {D}}(x)=I}$

holds, where ${\textstyle I}$ is the unit operator. It is assumed that for any measurement, the output probability distribution ${\textstyle Pr\{{\text{x}}=x||\rho \}}$ is given by

{\textstyle Pr\{{\text{x}}=x||\rho \}=Tr[{\hat {D}}(x)\rho ]}

(10)

where ${\textstyle {\hat {D}}(x)}$ is a POVM. For atomic instruments, it is assumed that effects are represented concretely in the form

{\textstyle {\hat {D}}(x)={\hat {V}}(x)^{*}{\hat {V}}(x)}

(11)

where ${\textstyle {V}(x)}$ is a linear operator in ${\textstyle {\mathcal {H}}}$ . Hence, the normalization condition has the form ${\textstyle \sum _{x}V(x)^{*}V(x)=I}$ .9 The Born rule can be written similarly to (5):

{\textstyle Pr\{{\text{x}}=x||\rho \}=Tr[{V}(x)\rho {V}^{*}(x)]}

(12)

It is assumed that the post-measurement state transformation is based on the map:

*

{\textstyle \rho \rightarrow {\mathcal {L_{A}(x)\rho =V(X)\rho V^{*}(x)}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (13)}

so the quantum state reduction is given by

*

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \rho\rightarrow\rho_{(\text{x}=x)}=\frac{\mathcal{L}_A(x) \rho}{Tr[\mathcal{L}_A(x)\rho]}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (14)}

The map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\rightarrow\mathcal{L_A(x)}} given by (13) is an atomic quantum instrument. We remark that the Born rule (12) can be written in the form

*

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle Pr\{\text{x}=x||\rho\}=Tr [\Im_A(x)\rho]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (15)} f

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} be a Hermitian operator in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{H}} . Consider a POVM Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \hat{D}=\biggl(\hat{D}^A(x)\Biggr)} with the domain of definition given by the spectrum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} . This POVM represents a measurement of observable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} if Born’s rule holds:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle Pr\{\text{A}=x||\rho\}=Tr [\widehat{D}^A(x)\rho]=Tr[\widehat{E}^A(x)\rho]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (16)}

Thus, in principle, probabilities of outcomes are still encoded in the spectral decomposition of operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} or in other words operators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \biggl(\hat{D}^A(x)\Biggr)} should be selected in such a way that they generate the probabilities corresponding to the spectral decomposition of the symbolic representation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} of observables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , i.e.,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \biggl(\hat{D}^A(x)\Biggr)} is uniquely determined byFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \hat{D}^A(x)=\hat{E}^A(x)} . We can say that this operator carries only information about the probabilities of outcomes, in contrast to the von Neumann scheme, operator ${\hat {A}}$ does not encode the rule of the state update. For an atomic instrument, measurements of the observable $A$ has the unique output probability distribution by the Born’s rule (16), but has many different quantum state reductions depending of the decomposition of the effect ${\textstyle {\hat {D}}(x)={\hat {E}}^{A}(x)=V(x)^{*}V(x)}$ in such a way that

{\textstyle \rho \rightarrow \rho _{({\text{A}}=x)}={\frac {{V}(x)\rho V(x)^{*}}{Tr[{V}(x)\rho V(x)^{*}]}}}

(17)