Sampling mixed quantum states

Seminar date and time: 
Friday, 11 May, 2018 - 14:00
Contact: 
frederic.dupuis@loria.fr
Affiliation: 
Université de Lorraine
Author: 
Frederic Dupuis
Location: 
GIQ Seminar Room (C5/262)

We investigate sampling procedures that certify that an arbitrary 
quantum state on $n$ subsystems is close to an ideal mixed state 
$\varphi^{\otimes n}$ for a given reference state $\varphi$, up to 
errors on a few positions. This task makes no sense classically: it 
would correspond to certifying that a given bitstring was generated 
according to some desired probability distribution. However, in the 
quantum case, this is possible if one has access to a prover who can 
supply a purification of the mixed state.

In this work, we introduce the concept of mixed-state certification, and 
we show that a natural sampling protocol offers secure certification in 
the presence of a possibly dishonest prover: if the verifier accepts 
then he can be almost certain that the state in question has been 
correctly prepared, up to a small number of errors.

We then apply this result to two-party quantum coin-tossing. Given that 
strong coin tossing is impossible, it is natural to ask ``how close can 
we get". This question has been well studied and is nowadays well 
understood from the perspective of the bias of individual coin tosses. 
We approach and answer this question from a different---and somewhat 
orthogonal---perspective, where we do not look at individual coin tosses 
but at the global entropy instead. We show how two distrusting parties 
can produce a common high-entropy source, where the entropy is an 
arbitrarily small fraction below the maximum (except with negligible 
probability).

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