On the hardness of conversion from entangled proof into separable one
arxiv(2024)
摘要
A quantum channel whose image approximates the set of separable states is
called a disentangler, which plays a prominent role in the investigation of
variants of the computational model called Quantum Merlin Arthur games, and has
potential applications in classical and quantum algorithms for the separability
testing and NP-complete problems. So far, two types of a disentangler,
constructed based on ϵ-nets and the quantum de Finetti theorem, have
been known; however, both of them require an exponentially large input system.
Moreover, in 2008, John Watrous conjectured that any disentangler requires an
exponentially large input system, called the disentangler conjecture. In this
paper, we show that both of the two known disentanglers can be regarded as
examples of a strong disentangler, which is a disentangler approximately
breaking entanglement between one output system and the composite system of
another output system and the arbitrarily large environment. Note that the
strong disentangler is essentially an approximately entanglement-breaking
channel while the original disentangler is an approximately
entanglement-annihilating channel, and the set of strong disentanglers is a
subset of disentanglers. As a main result, we show that the disentangler
conjecture is true for this subset, the set of strong disentanglers, for a wide
range of approximation parameters without any computational hardness
assumptions.
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