quantum instrument

Let $$

X

be a countable set describing the outcomes of a quantum measurement, and let $$

\{\mathcal{E}_x \}_{x\in X}

denote a collection of trace-non-increasing completely positive maps, such that the sum of all $$

\mathcal{E}_x

is trace-preserving, i.e. $$

\operatorname{tr}\left(\sum_x\mathcal{E}_x(\rho)\right)=\operatorname{tr}(\rho)

for all positive operators $$

\rho.

Now for describing a measurement by an instrument $$

\mathcal{I}

, the maps $$

\mathcal{E}_x

are used to model the mapping from an input state $$

\rho

to the output state of a measurement conditioned on a classical measurement outcome $$

x

. Therefore, the probability that a specific measurement outcome $$

x

occurs on a state $$

\rho

is given by{{cite book | last1=Busch | first1=Paul | last2=Lahti | first2= Pekka | last3=Pellonpää | first3=Juha-Pekka | last4=Ylinen | first4=Kari | title=Quantum measurement | publisher =Springer| date=2016 | volume=23 | pages=261--262 | isbn=978-3-319-43387-5 | doi=10.1007/978-3-319-43389-9}}

p(x|\rho)=\operatorname{tr}(\mathcal{E}_x(\rho)).

The state after a measurement with the specific outcome $$

x

is given by

\rho_x=\frac{\mathcal{E}_x(\rho)}{\operatorname{tr}(\mathcal{E}_x(\rho))}.

If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections $$

|x\rangle\langle x| \in \mathcal{B}(\mathbb{C}^

, then the action of an instrument $$

\mathcal{I}

is given by a quantum channel $$

\mathcal{I}:\mathcal{B}(\mathcal{H}_1) \rightarrow \mathcal{B}(\mathcal{H}_2)\otimes \mathcal{B}(\mathbb{C}^

with

\mathcal{I}(\rho):=

\sum_x \mathcal{E}_x

( \rho)\otimes \vert x \rangle \langle x|.

Here $$

\mathcal{H}_1

and $$

\mathcal{H}_2 \otimes \mathbb{C}^

are the Hilbert spaces corresponding to the input and the output systems of the instrument.

Just as a completely positive trace preserving (CPTP) map can always be considered as the reduction of unitary evolution on a system with an initially unentangled auxiliary, quantum instruments are the reductions of projective measurement with a conditional unitary, and also reduce to CPTP maps and POVMs when ignore measurement outcomes and state evolution, respectively. In John Smolin's terminology, this is an example of "going to the Church of the Larger Hilbert space".

Any quantum instrument on a system $\backslash mathcal\{S\}$ can be modeled as a projective measurement on $\backslash mathcal\{S\}$ and (jointly) an uncorrelated auxiliary $\backslash mathcal\{A\}$ followed by a unitary conditional on the measurement outcome.{{Cite journal |last=Ozawa |first=Masanao |year=1984 |title=Quantum measuring processes of continuous observables |url=https://doi.org/10.1063/1.526000 |journal=Journal of Mathematical Physics |volume=25 |pages=79-87}} Let $\backslash eta$ (with $\backslash eta\; >\; 0$ and $\backslash mathrm\{Tr\}\; \backslash ,\; \backslash eta\; =1$) be the normalized initial state of $\backslash mathcal\{A\}$, let $\backslash \{\backslash Pi\_i\backslash \}$ (with $\backslash Pi\_i\; =\; \backslash Pi\_i^\backslash dagger\; =\; \backslash Pi\_i^2$ and $\backslash Pi\_i\; \backslash Pi\_j\; =\; \backslash delta\_\{ij\}\; \backslash Pi\_i$) be a projective measurement on $\backslash mathcal\{SA\}$, and let $\backslash \{U\_i\backslash \}$ (with $U\_i^\backslash dagger\; =\; U\_i^\{-1\}$) be unitaries on $\backslash mathcal\{SA\}$. Then one can check that

:$\backslash mathcal\{E\}\_i\; (\backslash rho)\; :=\; \backslash mathrm\{Tr\}\_\{\backslash mathcal\{A\}\}\backslash left(U\_i\backslash Pi\_i(\backslash rho\backslash otimes\backslash eta)\backslash Pi\_i\; U\_i^\backslash dagger\backslash right)$

defines a quantum instrument. Furthermore, one can also check that any choice of quantum instrument $\backslash \{\backslash mathcal\{E\}\_i\backslash \}$ can be obtained with this construction for some choice of $\backslash eta$ and $\backslash \{U\_i\backslash \}$.

In this sense, a quantum instrument can be thought of as the reduction of a projective measurement combined with a conditional unitary.

Any quantum instrument $\backslash \{\backslash mathcal\{E\}\_i\backslash \}$ immediately induces a CPTP map, i.e., a quantum channel:

:$\backslash mathcal\{E\}\; (\backslash rho)\; :=\; \backslash sum\_i\; \backslash mathcal\{E\}\_i(\backslash rho).$

This can be thought of as the overall effect of the measurement on the quantum system if the measurement outcome is thrown away.

Any quantum instrument $\backslash \{\backslash mathcal\{E\}\_i\backslash \}$ immediately induces a positive operator-valued measurement (POVM):

:$M\_i\; :=\; \backslash sum\_a\; K\_a^\{(i)\backslash dagger\}\; K\_a^\{(i)\}$

where $K\_a^\{(i)\}$ are any choice of Kraus operators for $\backslash mathcal\{E\}\_i$,

:$\backslash mathcal\{E\}\_i\; (\backslash rho)\; =\; \backslash sum\_a\; K\_a^\{(i)\}\backslash rho\; K\_a^\{(i)\backslash dagger\}.$

The Kraus operators $K\_a^\{(i)\}$ are not uniquely determined by the CP maps $\backslash mathcal\{E\}\_i$, but the above definition of the POVM elements $M\_i$ is the same for any choice. The POVM can be thought of as the measurement of the quantum system if the information about how the system is affected by the measurement is thrown away.

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Category:Quantum mechanics