in the process of learning that bit, Alice still needs to send $m=\Omega
\left(  n\right)  $ qubits for Bob to succeed with high probability.

These results say that the exponential descriptive complexity of quantum states cannot be effectively harnessed for classical data storage, but they do not bound the number of practically meaningful ‘degrees of freedom’ in a quantum state used for purposes other than storing data.  For example, a quantum state could be useful for computation, or it could be a physical system worthy of study in its own right.  The question then becomes, what useful information *can* we give about an $n$-qubit state using a ‘reasonable’ number (say, $\operatorname*{poly}\left(  n\right)  $) of classical bits?

One approach to this question is to identify special subclasses of quantum states for which a faithful approximation can be specified using only $\operatorname*{poly}\left(  n\right)  $ bits.  This has been done, for example, with matrix product states [@vidal] and ‘tree states’ [@aar:mlin].  A second approach is to try to describe an *arbitrary* $n$-qubit state $\rho$ concisely, in such a way that the state $\widetilde{\rho}$ recovered from the description is close to $\rho$ with respect to some natural subclass of *measurements*.  This has been done 