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Weak String Erasure: Difference between revisions
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** <math>A_\mathcal{I}</math> denotes the sub-string of <math>A_1^n</math> whose bit are in <math>\mathcal{I}</math>. | ** <math>A_\mathcal{I}</math> denotes the sub-string of <math>A_1^n</math> whose bit are in <math>\mathcal{I}</math>. | ||
* Let us define the following function. | * Let us define the following function. | ||
<math>\gamma(x):= \begin | <math>\gamma(x):= \begin{cases} x, & \text{ if } x>1/2 \\ g^{-1}(x), & \text{ if } x\leq 1/2 \end{cases} </math></br> | ||
where <math>g(x):= h(x)+x-1</math>, and <math>h(x):=-x\log(x)-(1-x)\log(1-x)</math>. </br> | |||
where <math>g(x):= h(x)+x-1</math>, and <math>h(x):=-x\log(x)-(1-x)\log(1-x)</math>.</br> | |||
We will use the shorthand <math>A_1^n</math> to denote the string <math>A_1,\ldots,A_n</math>. We denote <math>[n]</math> for the set <math>\{1,\ldots,n\}</math>. <math>H</math> is the Hadamard gate, and by convention <math>H^0=\mathcal{I}</math> and <math>H^1=H</math>. | We will use the shorthand <math>A_1^n</math> to denote the string <math>A_1,\ldots,A_n</math>. We denote <math>[n]</math> for the set <math>\{1,\ldots,n\}</math>. <math>H</math> is the Hadamard gate, and by convention <math>H^0=\mathcal{I}</math> and <math>H^1=H</math>. | ||