Quantum instrument

Mathematical abstraction of quantum measurement

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]

## Definition

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

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

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

ρ x = E x ( ρ ) tr ⁡ ( E x ( ρ ) ) . {\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  | x ⟩ ⟨ x | ∈ B ( C | X | ) {\textstyle |x\rangle \langle x|\in {\mathcal {B}}(\mathbb {C} ^{|X|})} , then the action of an instrument  I {\displaystyle {\mathcal {I}}} is given by a quantum channel  I : B ( H 1 ) -> B ( H 2 ) ⊗ B ( C | X | ) {\displaystyle {\mathcal {I}}:{\mathcal {B}}({\mathcal {H}}_{1})\rightarrow {\mathcal {B}}({\mathcal {H}}_{2})\otimes {\mathcal {B}}(\mathbb {C} ^{|X|})} with[2]

I ( ρ ) := ∑ x E x ( ρ ) ⊗ | x ⟩ ⟨ x | . {\displaystyle {\mathcal {I}}(\rho ):=\sum _{x}{\mathcal {E}}_{x}(\rho )\otimes \vert x\rangle \langle x|.}

Here  H 1 {\displaystyle {\mathcal {H}}_{1}} and  H 2 ⊗ C | X | {\displaystyle {\mathcal {H}}_{2}\otimes \mathbb {C} ^{|X|}} are the Hilbert spaces corresponding to the input and the output systems of the instrument.

## Reductions and inductions

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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.[3] In John Smolin's terminology, this is an example of "going to the Church of the Larger Hilbert space".

### As a reduction of projective measurement and conditional unitary

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Any quantum instrument on a system  S {\displaystyle {\mathcal {S}}} can be modeled as a projective measurement on  S {\displaystyle {\mathcal {S}}} and (jointly) an uncorrelated auxiliary  A {\displaystyle {\mathcal {A}}} followed by a unitary conditional on the measurement outcome.[3] Let  η {\displaystyle \eta } (with  η > 0 {\displaystyle \eta >0} and  T r η = 1 {\displaystyle \mathrm {Tr} \,\eta =1} ) be the normalized initial state of  A {\displaystyle {\mathcal {A}}} , let  { Π i } {\displaystyle \\{\Pi _{i}\\}} (with  Π i = Π i † = Π i 2 {\displaystyle \Pi _{i}=\Pi _{i}^{\dagger }=\Pi _{i}^{2}} and  Π i Π j = δ i j Π i {\displaystyle \Pi _{i}\Pi _{j}=\delta _{ij}\Pi _{i}} ) be a projective measurement on  S A {\displaystyle {\mathcal {SA}}} , and let  { U i } {\displaystyle \\{U_{i}\\}} (with  U i † = U i − 1 {\displaystyle U_{i}^{\dagger }=U_{i}^{-1}} ) be unitaries on  S A {\displaystyle {\mathcal {SA}}} . Then one can check that

E i ( ρ ) := T r A ( U i Π i ( ρ ⊗ η ) Π i U i † ) {\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.[3] Furthermore, one can also check that any choice of quantum instrument  { E i } {\displaystyle \\{{\mathcal {E}}_{i}\\}} can be obtained with this construction for some choice of  η {\displaystyle \eta } and  { U i } {\displaystyle \\{U_{i}\\}} .[3]

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

### Reduction to CPTP map

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Any quantum instrument  { E i } {\displaystyle \\{{\mathcal {E}}_{i}\\}} immediately induces a CPTP map, i.e., a quantum channel:[3]

E ( ρ ) := ∑ i E i ( ρ ) . {\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.

### Reduction to POVM

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

M i := ∑ a K a ( i ) † K a ( i ) {\displaystyle M_{i}:=\sum _{a}K_{a}^{(i)\dagger }K_{a}^{(i)}}

where  K a ( i ) {\displaystyle K_{a}^{(i)}} are any choice of Kraus operators for  E i {\displaystyle {\mathcal {E}}_{i}} ,[3]

E i ( ρ ) = ∑ a K a ( i ) ρ K a ( i ) † . {\displaystyle {\mathcal {E}}_{i}(\rho )=\sum _{a}K_{a}^{(i)}\rho K_{a}^{(i)\dagger }.}

The Kraus operators  K a ( i ) {\displaystyle K_{a}^{(i)}} are not uniquely determined by the CP maps  E i {\displaystyle {\mathcal {E}}_{i}} , but the above definition of the POVM elements  M i {\displaystyle M_{i}} is the same for any choice.[3] 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.

## References

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1. ^ Alter, Orly; Yamamoto, Yoshihisa (2001). Quantum Measurement of a Single System. New York: Wiley. doi:10.1002/9783527617128. ISBN 9780471283089.
2. ^ a b Jordan, Andrew N.; Siddiqi, Irfan A. (2024). Quantum Measurement: Theory and Practice. Cambridge University Press. ISBN 978-1009100069.
3. ^ a b c d e f g h i Busch, Paul; Lahti, Pekka; Pellonpää, Juha-Pekka; Ylinen, Kari (2016). Quantum measurement. Vol. 23\. Springer. pp. 261--262. doi:10.1007/978-3-319-43389-9. ISBN 978-3-319-43387-5.