Editing Weak String Erasure

Weak String Erasure (WSE) is a two-party functionality, say between Alice and Bob, that allows Alice to send a random bit string to Bob, in such a way that Alice is guarantied that a fraction (ideally half) of the bits are lost during the transmission. However, Alice should not know which bits Bob has received, and which bits have been lost.
'''Tags:''' Universal Cloning, optimal cloning, N to M cloning, symmetric cloning,  [[Asymmetric Universal 1-2 Cloning|Asymmetric Cloning]],  [[Quantum Cloning]], copying quantum states, [[:Category: Quantum Functionality|Quantum Functionality]][[Category: Quantum Functionality]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]], [[Probabilistic Cloning]] 
[[:Category: Two Party Protocols|Two-Party Protocols]] [[Two Party Protocols]], weak string erasure, [[Category: Building Blocks]][[:Category: Building Blocks|Building Blocks]], [[Bounded Storage Model]]

==Assumptions==
*We assume that the adversary has access to a quantum memory that is bounded in size, say the memory is of at most log2(d) qubits. See [[Bounded/Noisy Storage Model]]
* The transmission of qubit is noiseless/lossless.
* The preparation devices (for quantum state) are assumed to be fully characterized and trusted. We assume the same for the measurement devices.

==Outline==
Alice and Bob first agree on a duration <maths>Delta t</math> that should correspond to an estimation of the time needed to make any known quantum memory decoheres. The protocol can be decomposed into three parts. 
*'''Distribution''' This step involves preparation, exchange and measurement of quantum states. Alice chooses a random basis among the X or Z bases. She then chooses at random one of the two states of this basis, prepares this state and sends it over to Bob. Upon receiving the state from Alice, Bob will
choose a random basis among the X or Z bases, and measure the incoming qubit in this basis. They both record the basis they have used, Bob also records his measurement outcome, and Alice record which state she has sent. Both parties repeat this procedure n times. 
*'''Waiting time''' Both parties will wait for time <maths>Delta t</math>. This is to force a malicious party to store part his quantum state in his memory. Of course he will be limited by the size of his quantum memory.
* '''Classical post-processing''' Alice will send to Bob, the bases she has used to prepare her states. Bob will erase all the measurement outcomes of the rounds where he measured in a different basis than Alice has prepared the state.</br> The losses in the transmission happen in the distribution step when Bob measure the incoming qubit in different basis as the one Alice has chosen for the preparation.

==Notations Used==
**<math>|\psi\rangle^{\otimes N}:</math> N initial states
**<math>|\psi^{\perp}\rangle:</math> The state orthogonal to <math>|\psi\rangle</math>
**<math>R_{j}:</math> ancillary and internal states of the QCM
**<math>U_{N,M}:</math> Unitary operation describing the QCM
**<math>F_{N \rightarrow M}:</math> Fidelity of the USQCM showing how close the M output copies are to the N original states
==Properties==
*'''The protocol-'''
**assumes that all of the original states are identical and also all of the output copies will be identical at the end of the protocol (In other words, the final output state belongs to the symmetric subspace of M qubits).
**assumes that the protocol is an approximate deterministic cloning protocol, meaning that in every round it produces approximate copies of the original states.
*'''Fidelity of the <math>N \rightarrow M</math> UQCM:''' <math>F_{N \rightarrow M} = \frac{MN + M + N}{M(N + 2)}</math>
*'''Special case of 1 qubit to 2 qubits:''' <math>F_{1 \rightarrow 2} = \frac{5}{6}</math>

==Pseudo Code==
'''Input:''' j qubits where <math>R_{j}</math> are ancillary and internal states of the QCM.

<u>'''Stage 1'''</u> State preparation
#Prepare N initial states: <math>|\psi\rangle^{\otimes N}</math> and <math>M - N</math> blank states: <math>|0\rangle^{\otimes M - N}</math>
<u>'''Stage 2'''</u> Unitary transformation
# Perform the following unitary transformation on input state <math>|N\psi\rangle = |\psi\rangle^{\otimes N} |0\rangle^{\otimes M - N} |R\rangle</math>
<math>U_{N,M} |N\psi\rangle = \sum_{j=0}^{M-N} \alpha_{j} |(M - j)\psi, j\psi^{\perp}\rangle \otimes R_{j}(\psi)</math></br>
where <math>\alpha_{j} = \sqrt{\frac{N + 1}{M + 1}} \sqrt{\frac{(M - N)!(M - j)!}{(M - N - j)! M!}}</math></br>
<u>'''Stage 3:'''</u> Trace out the QCM state
#Trace out the state of the QCM in <math>R_{j}</math> states.

==Further Information==
<div style='text-align: right;'>''*contributed by Jeremy Ribeiro''</div>