# Quantum Bit Commitment, restricting ourselves to pure states

I have been going through the Preskill lecture notes on quantum computation, and there is a question on Quantum Bit Commitment:

Alice wants to make a prediction, either $0$ or $1$, before an event. She doesn't want Bob to know her prediction beforehand, but wants to be able to prove to him afterwards that she was right all along. To do this she prepares one of two distinguishable states represented by density operators $\rho_0$ or $\rho_1$ of a bipartite system $A\otimes B$, sending $B$ to Bob and keeping $A$ for herself. Later to unveil her prediction she sends $A$ to Bob who performs a measurement to determine if $A\otimes B$ is in $\rho_0$ or $\rho_1$. We say the protocol is binding if after commitment Alice is unable to change her prediction, and concealing if after commitment and before unveiling Bob is unable to discern the value of the bit. Show that if this protocol is concealing it is not binding.

I am asked to show that we may without loss of generality restrict ourselves to $\rho_0$, $\rho_1$ pure states, which is what I am unsure about. My guess is that we may do this because we can always regard a mixed state $\rho$ on $A\otimes B$ as a pure state $\rho'$ on some $A\otimes B\otimes C$, where $\rho=\mathrm{Tr}_C\{\rho'\}$. The tricky bit would be to show we can do this simultaneously for $\rho_0$ and $\rho_1$. To do this, if: $$\rho_0=\sum_{i=1}^np_i|\psi_i\rangle\langle\psi_i|$$ $$\rho_1=\sum_{j=1}^mq_j|\phi_j\rangle\langle\phi_j|$$ We can then define $$p'_k=\begin{cases}p_k & k\le n \\ 0 & k > n\end{cases}$$ $$|\psi'_k\rangle=\begin{cases}|\psi_k\rangle & k\le n \\ 0 & k > n\end{cases}$$ And similarly for $q'_k$, $|\phi_k'\rangle$. Then: $$\rho_0=\sum_{k=1}^{m+n}p'_k|\psi'_k\rangle\langle\psi'_k|$$ $$\rho_1=\sum_{k=1}^{m+n}q'_k|\phi'_k\rangle\langle\phi'_k|$$ Then if we pick $C$ such that $|\alpha_k\rangle\in C$ are orthonormal for $k\in\{1,\cdots,m+n\}$ we have $\rho_0'=|\Psi\rangle\langle\Psi|$ and $\rho_1'=|\Phi\rangle\langle\Phi|$ pure for: $$|\Psi\rangle=\sum_{k=1}^{m+n}\sqrt{p'_k}|\psi_k'\rangle\otimes|\alpha_k\rangle$$ $$|\Phi\rangle=\sum_{k=1}^{m+n}\sqrt{q'_k}|\phi_k'\rangle\otimes|\alpha_k\rangle$$ Does this seem correct?

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