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

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Consider a density operator   with the following spectral decomposition:

 

The weakly typical subspace is defined as the span of all vectors such that the sample entropy   of their classical label is close to the true entropy   of the distribution  :

 

where

 
 

The projector   onto the typical subspace of   is defined as

 

where we have "overloaded" the symbol   to refer also to the set of  -typical sequences:

 

The three important properties of the typical projector are as follows:

 
 
 

where the first property holds for arbitrary   and sufficiently large  .

Conditional quantum typicality

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Consider an ensemble   of states. Suppose that each state   has the following spectral decomposition:

 

Consider a density operator   which is conditional on a classical sequence  :

 

We define the weak conditionally typical subspace as the span of vectors (conditional on the sequence  ) such that the sample conditional entropy   of their classical labels is close to the true conditional entropy   of the distribution  :

 

where

 
 

The projector   onto the weak conditionally typical subspace of   is as follows:

 

where we have again overloaded the symbol   to refer to the set of weak conditionally typical sequences:

 

The three important properties of the weak conditionally typical projector are as follows:

 
 
 

where the first property holds for arbitrary   and sufficiently large  , and the expectation is with respect to the distribution  .

See also

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References

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