Impossibility of quantum string commitment
Rahul Jain · arXiv (Cornell University) · 2005
Quantum string commitment (QSC) schemes were introduced in [BCH+ 05]. Let Alice be the committer. Let ρx be the state of Bob’s qubits at the end of the commit phase of an honest run of a QSC protocol when Alice commits x ∈ {0, 1} n. Let ˜px be the maximum probability with which a cheating Alice can reveal x. Let a = log ∑ x ˜px. [BCH+ 05] showed that for single execution of the protocol an (n, a, b)-Ξ-QSC with a+b+5 log(2+4 √ (2))−1 < n is impossible, where b is the One shot Holevo Ξ information of the ensemble E = {1/2n, ρx}. We show that an (n, a, b)-χ-QSC with a + 32b + 58 < n is impossible where b is the Holevo χ information of the ensemble E = {1/2n, ρx}. We also show that if for all ensembles E = {px, ρx} obtained by varying px and fixed ρx, χ(E) ≤ b, then Alice can successfully reveal any x with probability ≥ 2−32b−58. Our results are weaker in terms of constant in front of b and the additive constant but they are stronger in that for any ensemble E, Ξ(E) ≥ χ(E). 1