What is the best mathematical representation of a quantum system that has some classical registers and some quantum registers?

I'm asking because I'm considering any "physical" process $\pi()$ that takes no input, but that outputs both classical and quantum values, like for example a tuple $(x, \rho)$, with $x \in \{0,1\}^n$ a classical string, and $\rho$ a quantum state on $m$ qubits, and I'd like to find a kind of Kraus representation of $\pi$.

I wanted to use the usual formalism of CPTP, but it's not clear in which Hilbert space lives $(x, \rho)$... Indeed, usually, a Hilbert space is stable by superposition, but here the first "register" cannot be in superposition. Of course, I could say that $x$ is in fact $|x\rangle$ and therefore the output of $\pi()$ is strictly contained in a Hilbert space of size $2^n$, but then $\pi()$ is not simply $\sigma = \sum_i B_i 1 B_i^\dagger$, as I'd also need to "measure" $\sigma$ to extract a random $x$, and then set $\rho$ to be the matrix $\sigma$ "postselected" on $x$.

Is there some better representation for such process that makes the difference between quantum outputs and classical outputs?

  • $\begingroup$ I can also recommend Mark Wilde's great book From Classical to Quantum Shannon Theory, that you find on the arxiv: arxiv.org/abs/1106.1445 It contains large sections that concern themselves with Classical-to-Quantum channels, which should be what you're looking for. $\endgroup$ Commented Mar 12, 2020 at 20:54

1 Answer 1


CPTP map is a map from operators on one Hilbert space to operators on another Hilbert space (usually we consider only density matrices as inputs and outputs). There are no separated classical registers in this formalism.
There are some generalizations, though, like this https://en.wikipedia.org/wiki/Quantum_instrument. But in essence the idea is the same as you've described $-$ we model classical register with values $x$ as a quantum register with orthogonal states $|x\rangle$. You can model a measurement in this way, for example. If $\sum_i B_iB_i^\dagger=I$ describes a measurement, then $$\Phi(\rho) = \sum_i |i\rangle\langle i| \otimes B_i \rho B_i^\dagger = \sum_i |i\rangle\langle i| \otimes \frac{B_i \rho B_i^\dagger}{ \text{Tr}(B_i \rho B_i^\dagger)} \cdot \text{Tr}(B_i \rho B_i^\dagger)$$ is a corresponding quantum channel where the first quantum register of the output encodes the classical result $i$ and the second register contains the quantum state after the measurement.

  • $\begingroup$ Thanks a lot! Just as a side note, it seems that the paper that introduced this instrument is "An Operational Approach to Quantum Probability". $\endgroup$ Commented Apr 1, 2020 at 17:21
  • $\begingroup$ And also, I'm thinking that in order to be as general as possible, $B_i$ should apply to a map applied on $\rho$, not $\rho$ directly, and this map can itself be decomposed into a sum using the Kraus decomposition. $\endgroup$ Commented Apr 1, 2020 at 18:41

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