Typical subspace explained

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

\rho with the following spectral decomposition:\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 thatthe sample entropy \overline{H}( x ^{n} ) of their classicallabel is close to the true entropy H( X) of the distributionp _{X} ( x) T ^{X ^{n}} _{\delta} \equiv span \left\{ \left\vert x ^{n} \right\rangle :\left\vert \overline{H}( x ^{n} ) - H( X) \right\vert \leq\delta\right\} , where\overline{H}( x ^{n} ) \equiv - (1)/(n) log ( p _{X ^{n}} ( x ^{n} ) ) , H( X) \equiv - \sum _{x} p _{X} ( x) log p _{X} (x) . The projector \Pi ^{n} _{\rho,\delta} onto the typical subspace of \rho isdefined as\Pi ^{n} _{\rho,\delta} \equiv\sum _{x ^{n} \in T ^{X ^{n}} _{\delta}} \vert x ^{n} \rangle \langle x ^{n} \vert , where we have "overloaded" the symbolT ^{X ^{n}} _{\delta} to refer also to the set of \delta -typical sequences:T ^{X ^{n}} _{\delta} \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:Tr \left\{ \Pi ^{n} _{\rho,\delta} \rho ^{⊗} \right\} \geq 1- \epsilon, Tr \left\{ \Pi ^{n} _{\rho,\delta} \right\} \leq 2 ^{n\left[} , 2 ^{-n\left[} \Pi ^{n} _{\rho,\delta} \leq\Pi ^{n} _{\rho,\delta} \rho ^{⊗} \Pi ^{n} _{\rho,\delta} \leq 2 ^{-n\left[ H(} \Pi ^{n} _{\rho,\delta} , where the first property holds for arbitrary \epsilon,\delta> 0 andsufficiently large n .

Conditional quantum typicality

Consider an ensemble

\left\{ p _{X} ( x) ,\rho _{x} \right\} _{x\inl{X} } of states. Suppose that each state \rho _{x} has thefollowing spectral decomposition:\rho _{x} = \sum _{y} p _{Y|X} ( y|x) \vert y _{x} \rangle \langle y _{x} \vert . Consider a density operator \rho _{x ^{n}} which is conditional on a classicalsequence x ^{n} \equiv x _{1} … x _{n} :\rho _{x ^{n}} \equiv\rho _{x _{1}} ⊗ … ⊗ \rho _{x _{n}} . We define the weak conditionally typical subspace as the span of vectors(conditional on the sequence x ^{n} ) such that the sample conditional entropy\overline{H}( y ^{n} |x ^{n} ) of their classical labels is closeto the true conditional entropy H( Y|X) of the distributionp _{Y|X} ( y|x) p _{X} ( x) T ^{Y ^{n} |x ^{n}} _{\delta} \equiv span \left\{ \left\vert y _{x ^{n}} ^{n} \right\rangle :\left\vert \overline{H}( y ^{n} |x ^{n} ) - H( Y|X) \right\vert \leq\delta\right\} , where\overline{H}( y ^{n} |x ^{n} ) \equiv - (1)/(n) log \left(p _{Y ^{n} |X ^{n}} ( y ^{n} |x ^{n} ) \right) , H( Y|X) \equiv - \sum _{x} p _{X} ( x) \sum _{y} p _{Y|X} ( y|x) log p _{Y|X} ( y|x) . The projector \Pi _{\rho _{x ^{n}} ,\delta} onto the weak conditionally typicalsubspace of \rho _{x ^{n}} is as follows:\Pi _{\rho _{x ^{n}} ,\delta} \equiv\sum _{y ^{n} \in T ^{Y ^{n} |x ^{n}} _{\delta}} \vert y ^{n} _{x ^{n}} \rangle \langle y ^{n} _{x ^{n}} \vert , where we have again overloaded the symbol T ^{Y ^{n} |x ^{n}} _{\delta} to referto the set of weak conditionally typical sequences:T ^{Y ^{n} |x ^{n}} _{\delta} \equiv\left\{ y ^{n} :\left\vert \overline{H}\left(y ^{n} |x ^{n} \right) - H( Y|X) \right\vert \leq\delta\right\} . The three important properties of the weak conditionally typical projector areas follows:E _{X ^{n}} \left\{ Tr \left\{ \Pi _{\rho _{X ^{n}} ,\delta} \rho _{X ^{n}} \right\} \right\} \geq 1- \epsilon, Tr \left\{ \Pi _{\rho _{x ^{n}} ,\delta} \right\} \leq 2 ^{n\left[ H(} , 2 ^{-n\left[} \Pi _{\rho _{x ^{n}} ,\delta} \leq\Pi _{\rho _{x ^{n}} ,\delta} \rho _{x ^{n}} \Pi _{\rho _{x ^{n}} ,\delta} \leq 2 ^{-n\left[} \Pi _{\rho _{x ^{n}} ,\delta} , where the first property holds for arbitrary \epsilon,\delta> 0 andsufficiently large n , and the expectation is with respect to thedistribution p _{X ^{n}} ( x ^{n} ) .

See also

• Classical capacity
• Quantum information theory

References

• Wilde, Mark M., 2017, Quantum Information Theory, Cambridge University Press, Also available at eprint arXiv:1106.1145