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In physics we attempt to infer the rules governing a system given only the results of imprecise measurements.
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Note here that M M is a POVM element and ℰ † ( M ) \mathcal{E}^{\dagger}(M) represents the result of one of Bob’s yes/no measurements applied to A A . We do not assume that Bob can measure all observables of the form ℰ † ( A ) \mathcal{E}^{\dagger}(A) ; in particular, Bob is not assumed to be able to measure the projections P j P_{j} occurring in the spectral decomposition ℰ † ( B ) = ∑ j a j P j \mathcal{E}^{\dagger}(B)=\sum\displaylimits_{j}a_{j}P_{j} . Rather, by measuring B = ∑ j b j Q j B=\sum\displaylimits_{j}b_{j}Q_{j} on his system, Bob effectively measures the POVM with elements ℰ † ( Q j ) \mathcal{E}^{\dagger}(Q_{j}) on Alice’es system
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This operational interpretation assumes that Bob is actually able to make joint quantum measurements on all his copies, which may be overly optimistic in general. A more appropriate quantity might be based on the task of characterising distinguishability using only LOCC measurements. As will become evident, however, our framework is easily applied to arbitrary information metrics
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We observe that, even though ℰ ( ρ ) \mathcal{E}(\rho) may always be very mixed, the effective state ρ \rho may perfectly well be taken to be pure
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This notion of approximate equivalence does not give us an equivalence relation at this stage because it is neither reflexive nor transitive
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We assume, for simplicity, that ρ \rho has full rank so that all possible features X X are traceless Hermitian operators
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2097
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