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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) $$ \begin{align} & \operatorname{Tr}\left\{\Pi_{A^n}^{\delta, \rho_{x^n}^{A^n}} \rho_{y^n}^{A^n}\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left|x^n\right\rangle\left\langle x^n\right| \rho_{y^n}^{A^n}\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left|x^n\right\rangle\left\langle x^n \mid x^n\right\rangle\left\langle x^n\right| \rho_{y^n}^{A^n}\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left|x^n\right\rangle\left\langle x^n\left|\rho_{y^n}^{A^n}\right| x^n\right\rangle\left\langle x^n\right|\right\} \\ & =\operatorname{Tr}\left\{\sum_{x^n \in T_\delta^{X^n}}\left\langle x^n\left|\rho_{y^n}^{A^n}\right| x^n\right\rangle\left|x^n\right\rangle\left\langle x^n\right|\right\} \\ & =\sum_{x^n \in T_\delta^{X^n}}\left\langle x^n\left|\rho_{y^n}^{A^n}\right| x^n\right\rangle \\ & \end{align} $$

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  • $\begingroup$ 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.) $\endgroup$ Commented 2 days ago
  • $\begingroup$ @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. $\endgroup$
    – IamKnull
    Commented 2 days ago

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