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: Multi Party Protocols, non-local games, Quantum Enhanced Classical Functionality, Specific Task

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 $S_{i}\in \{a,b,\alpha ,\beta \},i=1,...,20$. The mint creates two random binary sequences of twenty digits $M_{i},N_{i}\in \{0,1\}$ where $i=1,...,20$. Then, two-state systems are placed in one of four states $a,b,\alpha ,\beta$ as shown in Fig.

1. Bank prepares a pair of orthonormal base states for each state system. Then the two-state system is located in one of four states $a,b,\alpha ,\beta$
2. The bank records all polarizations and their serial numbers. On the banknote/quantum money the serial number can be seen, while polarizations are kept secret.
3. 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 $M_{i}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_i} , even if someone copies the money, he cannot recover the polarization.

Notation

• $S_{i}$= Isolated system
• $M_{i},N_{i}$= Random binary sequences
• $a,b,\alpha ,\beta$= 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3/4)^N}

Pseudocode

Input: ​Product state of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} qubit and a serial number
Output: ​approved/rejected
Stage 1: Preparation

1. The Mint generate a quantum money composed of two component $(|S\rangle ,k)$ where $k$ is the serial number of the banknote and $S$ is a product state of $N$ qubits. Each qubit is randomly chosen from the set $\{|0\rangle ,|1\rangle ,|+\rangle ,|-\rangle \}$
2. Serial numbers and their states are recorded and kept at the Mint

Stage 2: Verification

1. The Mint looks for the serial number and the corresponding measurement basis in its database. Thus, each qubit is measured in the right basis,$\{|0\rangle ,|1\rangle \}$ or $\{|+\rangle ,|-\rangle \}$.
2. 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