# What is the general form of a classical-quantum state?

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?

## 1 Answer

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.

• 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)
– glS
Feb 16 '21 at 10:13
• 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". Feb 16 '21 at 10:20