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A mathematical foundation for self-testing: Lifting common assumptions
In this work we study the phenomenon of self-testing from the first
principles, aiming to place this versatile concept on a rigorous mathematical
footing. Self-testing allows a classical verifier to infer a quantum mechanical
description of untrusted quantum devices that she interacts with in a black-box
manner. Somewhat contrary to the black-box paradigm, existing self-testing
results tend to presuppose conditions that constrain the operation of the
untrusted devices. A common assumption is that these devices perform a
projective measurement of a pure quantum state. Naturally, in the absence of
any prior knowledge it would be appropriate to model these devices as measuring
a mixed state using POVM measurements, since the purifying/dilating spaces
could be held by the environment or an adversary.
We prove a general theorem allowing to remove these assumptions, thereby
promoting most existing self-testing results to their assumption-free variants.
On the other hand, we pin-point situations where assumptions cannot be lifted
without loss of generality. As a key (counter)example we identify a quantum
correlation which is a self-test only if certain assumptions are made.
Remarkably, this is also the first example of a correlation that cannot be
implemented using projective measurements on a bipartite state of full Schmidt
rank. Finally, we compare existing self-testing definitions, establishing many
equivalences as well as identifying subtle differences.Comment: 43 pages, 2 figure