Quantum instrument

In quantum physics, a quantum instrument is a mathematical description of a quantum measurement, capturing both the classical and quantum outputs.^[1] It can be equivalently understood as a quantum channel that takes as input a quantum system and has as its output two systems: a classical system containing the outcome of the measurement and a quantum system containing the post-measurement state.^[2]

Let ${\displaystyle X}$ be a countable set describing the outcomes of a quantum measurement, and let ${\displaystyle \{{\mathcal {E}}_{x}\}_{x\in X}}$ denote a collection of trace-non-increasing completely positive maps, such that the sum of all ${\displaystyle {\mathcal {E}}_{x}}$ is trace-preserving, i.e. ${\textstyle \operatorname {tr} \left(\sum _{x}{\mathcal {E}}_{x}(\rho )\right)=\operatorname {tr} (\rho )}$ for all positive operators ${\displaystyle \rho .}$

Now for describing a measurement by an instrument ${\displaystyle {\mathcal {I}}}$, the maps ${\displaystyle {\mathcal {E}}_{x}}$ are used to model the mapping from an input state ${\displaystyle \rho }$ to the output state of a measurement conditioned on a classical measurement outcome ${\displaystyle x}$. Therefore, the probability that a specific measurement outcome ${\displaystyle x}$ occurs on a state ${\displaystyle \rho }$ is given by^[3][4] $${\displaystyle p(x|\rho )=\operatorname {tr} ({\mathcal {E}}_{x}(\rho )).}$$

The state after a measurement with the specific outcome ${\displaystyle x}$ is given by^[3][4]

$${\displaystyle \rho _{x}={\frac {{\mathcal {E}}_{x}(\rho )}{\operatorname {tr} ({\mathcal {E}}_{x}(\rho ))}}.}$$

If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections ${\textstyle |x\rangle \langle x|\in {\mathcal {B}}(\mathbb {C} ^{|X|})}$ , then the action of an instrument ${\displaystyle {\mathcal {I}}}$ is given by a quantum channel ${\displaystyle {\mathcal {I}}:{\mathcal {B}}({\mathcal {H}}_{1})\rightarrow {\mathcal {B}}({\mathcal {H}}_{2})\otimes {\mathcal {B}}(\mathbb {C} ^{|X|})}$ with^[2]

$${\displaystyle {\mathcal {I}}(\rho ):=\sum _{x}{\mathcal {E}}_{x}(\rho )\otimes \vert x\rangle \langle x|.}$$

Here ${\displaystyle {\mathcal {H}}_{1}}$ and ${\displaystyle {\mathcal {H}}_{2}\otimes \mathbb {C} ^{|X|}}$ 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.^[4] 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 ${\displaystyle {\mathcal {S}}}$ can be modeled as a projective measurement on ${\displaystyle {\mathcal {S}}}$ and (jointly) an uncorrelated auxiliary ${\displaystyle {\mathcal {A}}}$ followed by a unitary conditional on the measurement outcome.^[3][4] Let ${\displaystyle \eta }$ (with ${\displaystyle \eta >0}$ and ${\displaystyle \mathrm {Tr} \,\eta =1}$) be the normalized initial state of ${\displaystyle {\mathcal {A}}}$, let ${\displaystyle \{\Pi _{i}\}}$ (with ${\displaystyle \Pi _{i}=\Pi _{i}^{\dagger }=\Pi _{i}^{2}}$ and ${\displaystyle \Pi _{i}\Pi _{j}=\delta _{ij}\Pi _{i}}$) be a projective measurement on ${\displaystyle {\mathcal {SA}}}$, and let ${\displaystyle \{U_{i}\}}$ (with ${\displaystyle U_{i}^{\dagger }=U_{i}^{-1}}$) be unitaries on ${\displaystyle {\mathcal {SA}}}$. Then one can check that

${\displaystyle {\mathcal {E}}_{i}(\rho ):=\mathrm {Tr} _{\mathcal {A}}\left(U_{i}\Pi _{i}(\rho \otimes \eta )\Pi _{i}U_{i}^{\dagger }\right)}$

defines a quantum instrument.^[4] Furthermore, one can also check that any choice of quantum instrument ${\displaystyle \{{\mathcal {E}}_{i}\}}$ can be obtained with this construction for some choice of ${\displaystyle \eta }$ and ${\displaystyle \{U_{i}\}}$.^[4]

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 ${\displaystyle \{{\mathcal {E}}_{i}\}}$ immediately induces a CPTP map, i.e., a quantum channel:^[4]

${\displaystyle {\mathcal {E}}(\rho ):=\sum _{i}{\mathcal {E}}_{i}(\rho ).}$

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 ${\displaystyle \{{\mathcal {E}}_{i}\}}$ immediately induces a positive operator-valued measurement (POVM):

${\displaystyle M_{i}:=\sum _{a}K_{a}^{(i)\dagger }K_{a}^{(i)}}$

where ${\displaystyle K_{a}^{(i)}}$ are any choice of Kraus operators for ${\displaystyle {\mathcal {E}}_{i}}$,^[4]

${\displaystyle {\mathcal {E}}_{i}(\rho )=\sum _{a}K_{a}^{(i)}\rho K_{a}^{(i)\dagger }.}$

The Kraus operators ${\displaystyle K_{a}^{(i)}}$ are not uniquely determined by the CP maps ${\displaystyle {\mathcal {E}}_{i}}$, but the above definition of the POVM elements ${\displaystyle M_{i}}$ is the same for any choice.^[4] 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.