Editing Wiesner Quantum Money

A classical money/banknote has a unique serial number and the bank can provide a verification according to these serial numbers. However, Wiesner suggests that a quantum money has also a number of isolated two-state quantum system and the two-state systems are located in one of four states.

'''Tags:''' [[:Category: Multi Party Protocols|Multi Party Protocols]], non-local games, [[:Category: Quantum Enhanced Classical Functionality|Quantum Enhanced Classical Functionality]], [[:Category: Specific Task|Specific Task]] 
[[Category: Prepare and Measure Network Stage]]
[[Category: Specific Tasks]]
[[Category: Quantum Enhanced Classical Functionality]]
[[Category: Multi Party Protocols]]

==Assumptions==
* The two-state systems must be isolated from the rest of universe, roughly.
* When Wiesner wrote his thesis, there was no device operating in which the phase coherence of a two-state system was preserved for longer than about a second.
==Outline==
Let the money have twenty isolated systems <math>S_i\in\{a, b, \alpha, \beta\}, i=1,...,20</math>. 
* The Mint creates two random binary sequences of twenty digits <math>M_i,N_i\in\{0,1\}</math> where <math>i=1,...,20</math>. Then, two-state systems are placed in one of four states <math>a, b, \alpha, \beta</math>. 
# Bank prepares a pair of orthonormal base states for each state system. Then the two-state system is located in one of four states <math>a, b, \alpha, \beta</math>
# The bank records all polarizations and their serial numbers. On the banknote/quantum money the serial number is plain, while polarizations are kept hidden.
# If the money is returned to the Mint, it checks whether each isolated system is still in its initial state or not. 
Note that since no one except the Mint knows <math>M_i</math> and <math>N_i</math>, even if someone copies the money, he cannot recover the polarization.

==Notation==
*<math>S_i</math>= Isolated system
*<math>M_i,N_i</math>= Random binary sequences
*<math>a,b,\alpha,\beta</math>= States 
==Requirements== 
* Network stage: quantum memory network
==Properties== 
* The scheme requires a central bank for verifying the money
* Pairs of conjugate variables has the same relation with Heisenberg uncertainty principle
* The success probability of the adversary in guessing the state of the target quantum money is <math>(3/4)^N</math>  
==Pseudocode==
'''Input''': Product state of <math>N</math> qubit and a serial number</br>
'''Output''': approved/rejected </br>
'''Stage 1: Preparation'''
# The Mint generate a quantum money composed of two component <math>(|S\rangle,k)</math> where <math>k</math> is the serial number of the banknote and <math>S</math> is a product state of <math>N</math> qubits. Each qubit is randomly chosen from the set <math>\{|0\rangle,|1\rangle,|+\rangle,|-\rangle\}</math> 
# Serial numbers and their states are recorded and kept at the Mint
'''Stage 2: Verification''' 
# The Mint looks for the serial number and the corresponding measurement basis in its database. Thus, each qubit is measured in the right basis,<math>\{|0\rangle,|1\rangle\}</math> or <math>\{|+\rangle,|-\rangle\}</math>.
# The Mint outputs 1 if the result of the measurement corresponds with the data stored in its database, otherwise it returns 0.
==Furthermore Information==
http://users.cms.caltech.edu/~vidick/teaching/120_qcrypto/wiesner.pdf