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==Notations== * <math>HMP_4</math>-states: <math>|\alpha(x)\rangle=\dfrac{1}{2}\sum_{1\leq i\leq4}(-1)^{x_i}|i\rangle</math>, <math>x\in\{0, 1\}^4</math> * for <math>m, a, b \in \{0, 1\}</math>, <math>(x, m, a, b) \in HMP_4 </math> if <math> b = \begin{cases} x_1 \oplus x_{2+m} & \text{if } a = 0 \\ x_{3-m} \oplus x_4 & \text{if } a = 1 \end{cases}</math> * <math>HMP_4</math>-queries: An <math>HMP_4</math>-query is an element <math>m \in \{0, 1\}</math>. A valid answer to the query w.r.t. <math>x \in \{0, 1\}^4</math> is a pair <math>(a, b) \in \{0, 1\} \times \{0, 1\}</math>, such that <math>(x, m, a, b) \in HMP_4</math>. An <math>HMP_4</math> -state can be used to answer an <math>HMP_4</math> -query with certainty: If <math> m = 0 </math>, let <math> v_1 \overset{def}{=}\dfrac{|1\rangle+|2\rangle}{\sqrt{2}} </math> <math> v_2 \overset{def}{=}\dfrac{|1\rangle-|2\rangle}{\sqrt{2}} </math> <math> v_3 \overset{def}{=}\dfrac{|3\rangle+|4\rangle}{\sqrt{2}} </math> <math> v_4 \overset{def}{=}\dfrac{|3\rangle-|4\rangle}{\sqrt{2}} </math> otherwise (m = 1), let <math> v_1 \overset{def}{=}\dfrac{|1\rangle+|3\rangle}{\sqrt{2}} </math> <math> v_2 \overset{def}{=}\dfrac{|1\rangle-|3\rangle}{\sqrt{2}} </math> <math> v_3 \overset{def}{=}\dfrac{|2\rangle+|4\rangle}{\sqrt{2}} </math> <math> v_4 \overset{def}{=}\dfrac{|2\rangle-|4\rangle}{\sqrt{2}} </math> Measure <math>|\alpha(x_i)\rangle</math> in the basis <math>{v_1, v_2, v_3, v_4}</math>, and let <math>(a, b)</math> be <math>(0, 0)</math> if the outcome is <math>v_1</math>; <math>(0, 1)</math> in the case of <math>v_2</math>; <math>(1, 0)</math> in the case of <math>v_3</math>; <math>(1, 1)</math> in the case of <math>v_4</math>. Then <math>(x, m, a, b) \in HMP_4</math> always.
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