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| ===3.3. Non-projective state update: atomic instruments===
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| In general, the statistical properties of any measurement are characterized by
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| # the output probability distribution <math display="inline">Pr\{\text{x}=x\parallel\rho\}</math>, the probability distribution of the output <math display="inline">x</math> of the measurement in the input state <math display="inline">\rho
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| </math>;
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| # the quantum state reduction <math display="inline">\rho\rightarrow\rho_{(X=x)}
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| </math>,the state change from the input state <math display="inline">\rho
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| </math> to the output state <math display="inline">\rho\rightarrow\rho_{(X=x)}
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| </math> conditional upon the outcome <math display="inline">\text{X}=x
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| </math> of the measurement.
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| 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 <math>A</math> represented by the Hermitian operator <math>\hat{A}</math> in <math display="inline">\mathcal{H}</math> can be measured in many ways.8 Formally, such measurement-schemes are represented by quantum instruments.
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| 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 <math display="inline">X=\{x_1....,x_N.....\}</math>. POVM is a map <math display="inline">x\rightarrow \hat{D}(x)</math> such that for each <math display="inline">x\in X</math>,<math>\hat{D}(x)</math> is a positive contractive Hermitian operator (called effect) (i.e.,<math display="inline">\hat{D}(x)^*=\hat{D}(x), 0\leq \langle\psi|\hat{D}(x)\psi\rangle\leq1</math> or any <math display="inline">\psi\in\mathcal{H}</math>), and the normalization condition
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| <math display="inline">\sum_x \hat{D}(x)=I</math>
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| holds, where <math display="inline">I</math>
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| is the unit operator. It is assumed that for any measurement, the output probability distribution <math display="inline">Pr\{\text{x}=x||\rho\}</math> is given by
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math display="inline">Pr\{\text{x}=x||\rho\}=Tr [\hat{D}(x)\rho]</math>
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| | width="33%" align="right" |<math>(10)</math>
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| |}
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| where <math display="inline"> \hat{D}(x)</math> is a POVM. For atomic instruments, it is assumed that effects are represented concretely in the form
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| {| width="80%" |
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| | width="33%" |<math display="inline"> \hat{D}(x)=\hat{V}(x)^*\hat{V}(x)</math>
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| | width="33%" align="right" |<math>(11)</math>
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| |}
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| where <math display="inline"> {V}(x)</math> is a linear operator in <math display="inline">\mathcal{H}</math>. Hence, the normalization condition has the form <math display="inline">\sum_x V(x)^*V(x)=I</math>.9 The Born rule can be written similarly to (5):
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math display="inline">Pr\{\text{x}=x||\rho\}=Tr [{V}(x)\rho{V}^*(x)]</math>
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| | width="33%" align="right" |<math>(12)</math>
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| |}
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| It is assumed that the post-measurement state transformation is based on the map:
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| {| width="80%" |
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| | width="33%" |'''<big>*</big>'''
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| | width="33%" |<math display="inline">\rho\rightarrow\mathcal{L_A(x)\rho=V(X)\rho V^*(x)}</math>
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| | width="33%" align="right" |<math>(13)</math>
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| |}
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| so the quantum state reduction is given by
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| {| width="80%" |
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| | width="33%" | '''<big>*</big>'''
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| | width="33%" |<math display="inline">\rho\rightarrow\rho_{(\text{x}=x)}=\frac{\mathcal{L}_A(x) \rho}{Tr[\mathcal{L}_A(x)\rho]}</math>
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| | width="33%" align="right" |<math>(14)</math>
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| |}
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| The map <math>x\rightarrow\mathcal{L_A(x)}</math> given by (13) is an atomic quantum instrument. We remark that the Born rule (12) can be written in the form
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| {| width="80%" |
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| | width="33%" | '''<big>*</big>'''
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| | width="33%" |<math display="inline">Pr\{\text{x}=x||\rho\}=Tr [\Im_A(x)\rho]</math>
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| | width="33%" align="right" |<math>(15)</math>f
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| |}
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| Let <math>\hat{A}</math> be a Hermitian operator in <math display="inline">\mathcal{H}</math>. Consider a POVM <math display="inline"> \hat{D}=\biggl(\hat{D}^A(x)\Biggr)</math> with the domain of definition given by the spectrum of <math>\hat{A}</math>. This POVM represents a measurement of observable <math>A</math> if Born’s rule holds:
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math display="inline">Pr\{\text{A}=x||\rho\}=Tr [\widehat{D}^A(x)\rho]=Tr[\widehat{E}^A(x)\rho]</math>
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| | width="33%" align="right" |<math>(16)</math>
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| |}
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| Thus, in principle, probabilities of outcomes are still encoded in the spectral decomposition of operator <math>\hat{A}</math> or in other words operators <math display="inline"> \biggl(\hat{D}^A(x)\Biggr)</math> should be selected in such a way that they generate the probabilities corresponding to the spectral decomposition of the symbolic representation <math>\hat{A}</math> of observables <math>A</math>, i.e.,<math display="inline"> \biggl(\hat{D}^A(x)\Biggr)</math> is uniquely determined by<math>\hat{A}</math> as <math display="inline"> \hat{D}^A(x)=\hat{E}^A(x)</math>. We can say that this operator carries only information about the probabilities of outcomes, in contrast to the von Neumann scheme, operator <math>\hat{A}</math> does not encode the rule of the state update. For an atomic instrument, measurements of the observable <math>A</math> 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 <math display="inline"> \hat{D}(x)=\hat{E}^A(x)=V(x)^*V(x)</math> in such a way that
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math display="inline">\rho\rightarrow\rho_{(\text{A}=x)}=\frac{{V}(x) \rho V(x)^*}{Tr[{V}(x)\rho V(x)^*]}</math>
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| | width="33%" align="right" |<math>(17)</math>
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| |}
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| ----
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| ===3.4. General theory (Davies–Lewis–Ozawa)=== | | ===3.4. General theory (Davies–Lewis–Ozawa)=== |