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

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[3][4]

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

The state after a measurement with the specific outcome x is given by[3][4]

\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}^{|X|}) , 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}^{|X|}) with[2]

\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}^{|X|} are the Hilbert spaces corresponding to the input and the output systems of the instrument.

Reductions and inductions

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".

As a reduction of projective measurement and conditional unitary

Any quantum instrument on a system \mathcal{S} can be modeled as a projective measurement on \mathcal{S} and (jointly) an uncorrelated auxiliary \mathcal{A} followed by a unitary conditional on the measurement outcome.[3][4] Let \eta (with \eta > 0 and \mathrm{Tr} \, \eta =1) be the normalized initial state of \mathcal{A}, let \{\Pi_i\} (with \Pi_i = \Pi_i^\dagger = \Pi_i^2 and \Pi_i \Pi_j = \delta_{ij} \Pi_i) be a projective measurement on \mathcal{SA}, and let \{U_i\} (with U_i^\dagger = U_i^{-1}) be unitaries on \mathcal{SA}. Then one can check that

\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 \{\mathcal{E}_i\} can be obtained with this construction for some choice of \eta and \{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.

Reduction to CPTP map

Any quantum instrument \{\mathcal{E}_i\} immediately induces a CPTP map, i.e., a quantum channel:[4]

\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

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

M_i := \sum_a K_a^{(i)\dagger} K_a^{(i)}

where K_a^{(i)} are any choice of Kraus operators for \mathcal{E}_i,[4]

\mathcal{E}_i (\rho) = \sum_a K_a^{(i)}\rho K_a^{(i)\dagger}.

The Kraus operators K_a^{(i)} are not uniquely determined by the CP maps \mathcal{E}_i, but the above definition of the POVM elements 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.

References

  1. ^ Alter, Orly & Yamamoto, Yoshihisa (2001). Quantum Measurement of a Single System. New York: Wiley. doi:10.1002/9783527617128. ISBN 9780471283089.
  2. ^ Jordan, Andrew N. & Siddiqi, Irfan A. (2024). Quantum Measurement: Theory and Practice. Cambridge University Press. ISBN 978-1009100069.
  3. ^ Ozawa, Masanao (1984). "Quantum measuring processes of continuous observables". Journal of Mathematical Physics. 25: 79-87.
  4. ^ 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.