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Weak String Erasure
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==Notation== ** <math>n</math> the total number of rounds. ** <math>A_1^n</math> denotes the string <math>A_1,\ldots,A_n</math>. ** <math>H</math> denotes the Hadamard gate. <math>H^0=I</math> and <math>H^1=H</math> ** <math>X_1^n</math> is the random string Alice sends to Bob by the WSE protocol. ** <math>\Theta_i</math> encodes Alice's choice of basis in round <math>i</math>. ** <math>\hat X_1^n</math> is Bob's outcomes measurement. ** <math>\hat \Theta_i</math> encodes Bob's choice of basis in round <math>i</math>. ** <math>M</math> denotes dishonest Bob quantum memory, that can store a state of dimension at most $d$. ** <math>\Delta t</math> is a duration during which both parties will wait. ** <math>\epsilon>0</math> is a security parameter. ** <math>\mathcal{I}</math> denotes the set of rounds where Alice and Bob have chosen the same basis. ** <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. <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>
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