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We establish two results regarding the query complexity of bounded-error randomized algorithms. * Bounded-error separation theorem. There exists a total function $f : \{0,1\}^n \to \{0,1\}$ whose $\epsilon$-error randomized query complexity…

This paper has been withdrawn by the authors, due a oversimplified decoherence model. It will be substituted by a new work.

This paper has been withdrawn by the authors due to crucial error in the main proof (located in Section 2.4). The authors apologize for any inconveniences.

This paper has been withdrawn by the author due to a crucial sign error in Proposition 3.1.

This paper has been withdrawn by the author(s), due a crucial error on the entanglement of $\Gamma$ registers.

This paper has been withdrawn by the author due to a crucial error in the last lines in the proof of Lemma 3.3.

This paper has been withdrawn due to a error in Theorem 3.1.

This paper has been withdrawn by the authors, due a crucial mistake in Lemma 2

This paper has been withdrawn by the author due to the presented idea is wrong.

This paper has been withdrawn by the authors. We have discovered an error in the evaluation of the diagram, which invalidates our conclusion.

This paper has been withdrawn by the authors due to some fatal errors in the analysis.

The paper has been withdrawn due to numerical error.

This paper has been withdrawn by the author because Lemma 3 is incorrect. This mistake is crucial in this paper.

Lifting theorems are theorems that bound the communication complexity of a composed function $f\circ g^{n}$ in terms of the query complexity of $f$ and the communication complexity of $g$. Such theorems constitute a powerful generalization…

This paper has been withdrawn by the author, due a critical mistake on page 3.

This paper has been withdrawn by the author, due to an error in Proposition 2.2.

This paper has been withdrawn by the author due to a crucial sign error in Theorem 3.4.

This paper has been withdrawn by the authors, due an error involving the weak* convergence argument in section 2

This paper has been withdrawn by the author due to an error estimate in Lemma 3.1.

This paper has been withdrawn by the author because Proposition 1 is not valid.