4
$\begingroup$

In the literature, one comes across the following situation: Alice holds two registers $X$ and $A$ and it is given that $X$ is a classical register.

What is the most general way to write down Alice's state? Is it just $\sigma_{XA} = \sum_i p_i \vert i\rangle\langle i\vert_X \otimes\rho^i_A$, with each $\rho^i_A$ being a quantum state (positive semi-definite and trace one matrix)?

Sorry that this is a yes/no question because if yes, then there is not much to add. But if not, what would be the most general way to write Alice's state?

$\endgroup$
1
$\begingroup$

Your description has X as a mixed state (a quantum state with classical uncertainty) and not a classical state. For example you can apply quantum gates to X but that shouldn’t be allowed if X was a classical state. However we can think of that mixed state as a classical state (see comments) and even use it as such.

I’m not sure if there is some notation for writing classical states rather than mixed quantum states.

$\endgroup$
2
  • 2
    $\begingroup$ eh, actually this is pretty much the way "classical-quantum states" are usually treated. I'd argue the only way to understand what a "classical state" would be in a quantum context is as such a mixture. See e.g. Eq. (2.170) in Watrous (cs.uwaterloo.ca/~watrous/TQI/TQI.pdf) $\endgroup$ – glS Feb 16 at 10:13
  • $\begingroup$ Alright, I see what you're saying. I was thinking in terms of the ZX-calculus where classical states are different from mixed states. But I do appreciate that certain mixed states are worthy of the name "classical". $\endgroup$ – user3717194 Feb 16 at 10:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.