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Weak String Erasure: Difference between revisions
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==Notations Used== | ==Notations Used== | ||
** <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):= x \text{ if } x>1/2 </math></br> | |||
<math>\quad :=g^{-1}(x) \text{ if } x\leq 1/2</math>, where <math>g(x):= h(x)+x-1$, and $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=\id</math> and <math>H^1=H</math>. | |||
==Hardware Requirements== | ==Hardware Requirements== | ||
* Network Stage: [[:Category: Prepare and Measure Network Stage|Prepare and Measure]]. | * Network Stage: [[:Category: Prepare and Measure Network Stage|Prepare and Measure]]. | ||