# Clarification on the definition of cipherstate in the quantum one-time pad

I was reading the paper “Optimal Encryption of Quantum Bits ” (quantum one-time pad) and came across the following paragraph:

The input state, $$\rho$$, is called the message state, and the output state, $$\rho_c$$, is called the cipherstate. The protocol is secure if for every input state, $$\rho$$, the output state, $$\rho_c$$, is the totally mixed state: $$\rho_c = \sum_k p(k) U_k \rho U^*_k = \frac{1}{1/2^n} \mathbb I$$

This is while, earlier in the paper it says

The key is chosen with some probability $$p_k$$ and the input quantum state is encrypted by applying the corresponding unitary operation $$U_k$$. In the decryption stage, $$U^*_k$$ is applied to the quantum state to retrieve the original state.

I am confused about whether the authors mean that the cipherstate is the encrypted state or the decrypted state. If I interpret cipherstate as the decryption of the encrypted message in this context, then everything makes sense to me; however, usually cipherstate (or ciphertext) is different from the decrypted message. The expression $$\sum_k p(k) U_k \rho U^*_k = \frac{1}{1/2^n} \mathbb I$$ looks like what the adversary would see if they try to decrypt the message on average.

I guess you're seeing the term $$U_k\rho U_k^\star$$ and are thinking "that's got both $$U$$ and $$U^\star$$ so there's both an encryption and a decryption going on". That's not the case. It's only an encryption.
This is the difference between pure states and mixed states. If you had an initial state $$|\psi\rangle$$, you would encrypt it as $$U_k|\psi\rangle$$. However, if you write this as a density matrix, you start with $$\rho=|\psi\rangle\langle \psi|$$ and end up with the encrypted state $$U_k|\psi\rangle\langle \psi|U_k^\star$$. So, why are we using the density matrix formalism rather than pure states? Because we have a bunch of things happening with classical probabilities $$p_k$$ which are unknown from the perspective of an eavesdropper. The best way that the state can be described from that state of knowledge requires use of a density matrix.