# Probability that a quantum state is in the typical subspace of another quantum state

From the properties of the Typical subspace we already have the following theorem [1]:

Theorem (Unit Probability, see [1] page 467): Suppose that we perform a typical subspace measurement of a state $$\rho_{x^n}^{A^n}$$ ($$= \rho^{A_1}_{x_1} \otimes \cdots \otimes \rho^{A_n}_{x_n}$$), with the spectral decomposition $$\rho_x^A=\sum_{x} p_X(x)|x\rangle\left\langle\left. x\right|_A\right.$$. Then the probability that the quantum state $$\rho^{A^n}_{x^n}$$ is in the typical subspace $$T_{A^n}^{\delta, \rho_{x^n}^{A^n}}$$ approaches one as $$n$$ becomes large. That is, $$\operatorname{Tr}\left\{\Pi_{A^n}^{\delta, \rho_{x^n}^{A^n}} \rho_{x^n}^{A^n}\right\} \geq 1-\varepsilon,$$ for all $$\varepsilon \in(0,1), \delta>0$$, and sufficiently large $$n$$, where the typical subspace projector $$\Pi_{A^n}^\delta \equiv \sum_{x^n \in T_\delta^{X^n}}\left|x^n\right\rangle\left\langle\left. x^n\right|_{A^n}\right.$$, for $$\left|x^n\right\rangle$$ associated with the classical sequence $$x^n$$ via the spectral decomposition of $$\rho_x^A$$.

Now suppose from the same Hilbert space (i.e., $$\mathcal{H}_{A^n}=\mathcal{H}_{A_1} \otimes \cdots \otimes \mathcal{H}_{A_n}$$) we select any arbitrary state $$\rho_{y^n}^{A^n}$$ (may not be same as $$\rho_{x^n}^{A^n}$$), with $$\rho_y^A=\sum_{y} p_Y(y)|y\rangle\left\langle\left. y\right|_A\right.$$.

My question is: What is the probability of measuring this arbitrary state in the Typical subspace of the original state? $$\left( \operatorname{Tr}\left\{\Pi_{A^n}^{\delta, \rho_{x^n}^{A^n}} \rho_{y^n}^{A^n}\right\} \right)$$

Intuitively it should be realted somehow with the Relative Entropy and Mutual Information.

Here is my simple attempt (not sure how to proceed further) \begin{align} & \operatorname{Tr}\left\{\Pi_{A^n}^{\delta, \rho_{x^n}^{A^n}} \rho_{y^n}^{A^n}\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left|x^n\right\rangle\left\langle x^n\right| \rho_{y^n}^{A^n}\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left|x^n\right\rangle\left\langle x^n \mid x^n\right\rangle\left\langle x^n\right| \rho_{y^n}^{A^n}\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left|x^n\right\rangle\left\langle x^n\left|\rho_{y^n}^{A^n}\right| x^n\right\rangle\left\langle x^n\right|\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left\langle x^n\left|\rho_{y^n}^{A^n}\right| x^n\right\rangle\left|x^n\right\rangle\left\langle x^n\right|\right\} \\ & =\sum_{x^n \in T_\delta^{X^n}}\left\langle x^n\left|\rho_{y^n}^{A^n}\right| x^n\right\rangle \\ & \end{align}

• You'll need to add details. What is the setting, what do the $n$-sub/superscript mean? (References for the results quoted would also be good, but primarily make it self contained.) Commented 2 days ago
• @NorbertSchuch Thanks for the suggestion. I've added some context and references. A comprehensive definition of everything is challenging due to the involvement of various information-theoretic tools. However, those working in information theory are well-acquainted with these concepts and notations. Commented 2 days ago