In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory.
Unconditional quantum typicality Consider a density operator ρ {\displaystyle \rho } with the following spectral decomposition:
ρ = ∑ x p X ( x ) | x ⟩ ⟨ x | . {\displaystyle \rho =\sum _{x}p_{X}(x)\vert x\rangle \langle x\vert .}
The weakly typical subspace is defined as the span of all vectors such that the sample entropy H ¯ ( x n ) {\displaystyle {\overline {H}}(x^{n})} of their classical label is close to the true entropy H ( X ) {\displaystyle H(X)} of the distribution
p X ( x ) {\displaystyle p_{X}(x)} :
T δ X n ≡ span { | x n ⟩ : | H ¯ ( x n ) − H ( X ) | ≤ δ } , {\displaystyle T_{\delta }^{X^{n}}\equiv {\text{span}}\left\{\left\vert x^{n}\right\rangle :\left\vert {\overline {H}}(x^{n})-H(X)\right\vert \leq \delta \right\},}
where
H ¯ ( x n ) ≡ − 1 n log ( p X n ( x n ) ) , {\displaystyle {\overline {H}}(x^{n})\equiv -{\frac {1}{n}}\log(p_{X^{n}}(x^{n})),}
H ( X ) ≡ − ∑ x p X ( x ) log p X ( x ) . {\displaystyle H(X)\equiv -\sum _{x}p_{X}(x)\log p_{X}(x).}
The projector Π ρ , δ n {\displaystyle \Pi _{\rho ,\delta }^{n}} onto the typical subspace of ρ {\displaystyle \rho } is defined as
Π ρ , δ n ≡ ∑ x n ∈ T δ X n | x n ⟩ ⟨ x n | , {\displaystyle \Pi _{\rho ,\delta }^{n}\equiv \sum _{x^{n}\in T_{\delta }^{X^{n}}}\vert x^{n}\rangle \langle x^{n}\vert ,}
where we have "overloaded" the symbol
T δ X n {\displaystyle T_{\delta }^{X^{n}}} to refer also to the set of δ {\displaystyle \delta } -typical sequences:
T δ X n ≡ { x n : | H ¯ ( x n ) − H ( X ) | ≤ δ } . {\displaystyle T_{\delta }^{X^{n}}\equiv \left\{x^{n}:\left\vert {\overline {H}}\left(x^{n}\right)-H(X)\right\vert \leq \delta \right\}.}
The three important properties of the typical projector are as follows:
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