The classical money scheme involves the Bank distributing notes to untrusted users. Each note has a unique serial number attached to it and this number provides a basis for the verification of the note when the user wants to use it for a transaction. However, in the classical world, nothing prevents a user with sufficient resources to be able to forge the note and create more notes than what he originally had in possession. In the 1980s, Wiesner proposed the idea of quantum money to create unforgeable bank notes. The unforgeability of the note relied on the no-cloning property of quantum mechanics. In this example protocol, the banknotes are several BB84 states prepared by the Bank, who then distributes them to the untrusted users. When the user needs to carry out a transaction with his note, he sends it to the Bank for verification, who then authenticates the validity of the note. Based on the no-cloning property of quantum mechanics, Wiesner showed information-theoretic security against a forger of bank notes.

Conjugate Coding

Let the money have $n$ isolated systems $S_{i} \\\in \\\{a,b,\\\alpha,\\\beta\\\}$, $i = 1,\\\ldots,n$.

• The Mint creates two random binary sequences of $n$ digits $M_{i}, N_{i} \\\in \\\{0,1\\\}$ where $i = 1,\\\ldots,n$. Then, two-state systems are placed in one of four states $a, b, \\\alpha, \\\beta$.

1. 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 is plain, while polarizations are kept hidden.
   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 $N_{i}$, even if someone copies the money, he cannot recover the polarization.

• The quantum money state which is a two-state system 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.

Network stage: quantum memory network.

• 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  $(\\\frac{3}{4})^n$ 

$ \\\textbf{Input:} \\\quad $ Product state of $N$ qubits and a serial number

$ \\\textbf{Output:} \\\quad $ approved/rejected

$ \\\textbf{Stage 1: Preparation} $

1. The Mint generates a quantum money composed of two components $(|S\\\rangle, k)$ 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.

$ \\\textbf{Stage 2: Verification} $

1. The Mint looks up the serial number and the corresponding measurement basis in its database. Each qubit is measured in the correct basis, $\\\{|0\\\rangle, |1\\\rangle\\\}$ or $\\\{|+\\\rangle, |-\\\rangle\\\}$.
2. The Mint outputs $1$ if the measurement results match the data stored in its database; otherwise it outputs $0$.

Conjugate Coding