# Quantum Perspective on Digital Money: Towards a Quantum-Powered Financial System
## 4 Integration of Quantum Money with Quantum Networks
### Quantum Teleportation Mechanism  
Quantum teleportation enables the transfer of a quantum state between two parties, traditionally named [PERSON] (sender) and [PERSON] (receiver), by utilizing a shared entangled pair of qubits and classical communication [12; 31], see a conceptual scheme is shown in Figure 1. Teleportation protocol has been demonstrated experimentally [32] and appears a reliable tool for secure money transfer in the quantum-empowered banking system. When [PERSON] wants to send a quantum state Equation (1) (representing a certain amount of quantum money) to [PERSON], she begins by performing a Bell-state measurement on the currency state and her part of the entangled pair. This measurement collapses [PERSON]'s part of the entangled state, encoding the information about the currency onto [PERSON]'s entangled qubit through quantum correlations.  
Suppose, [PERSON] and [PERSON] share an entangled pair composed of particles \(A\) and \(B\), initially in one of the Bell states given in Equation (9). Then, the total system, consisting of the quantum money qubit encoded on particle \(C\), as defined in Equation (1), and the entangled pair, is given by:  
\[\ket{\psi}_{C}\otimes\ket{\Phi^{+}}_{AB}=(\alpha\ket{0}_{C}+\beta\ket{1}_{C}) \otimes\frac{1}{\sqrt{2}}(\ket{0}_{A}\otimes\ket{0}_{B}+\ket{1}_{A}\otimes \ket{1}_{B}). \tag{12}\]  
Rewriting this state in the Bell basis for [PERSON]'s two qubits (the money qubit and her half of the entangled pair), we obtain:  
\[\begin{split}\ket{\psi}_{C}\otimes\ket{\Phi^{+}}_{AB}=& \frac{1}{2}[\ket{\Phi^{+}}_{AC}(\alpha\ket{0}_{B}+\beta\ket{1}_{B})+ \ket{\Phi^{-}}_{AC}(\alpha\ket{0}_{B}-\beta\ket{1}_{B})\\ &+\ket{\Psi^{+}}_{AC}(\alpha\ket{1}_{B}+\beta\ket{0}_{B})+\ket{ \Psi^{-}}_{CA}(\alpha\ket{1}_{B}-\beta\ket{0}_{B})].\end{split} \tag{13}\]  
[PERSON] then performs a Bell-state measurement on her two qubits associated with particles \(A\) and \(C\). This collapses the state into one of the four Bell states \(\{\ket{\Phi^{+}}_{AC},\ket{\Phi^{-}}_{AC},\ket{\Psi^{+}}_{AC},\ket{\Psi^{-}}_{ AC}\}\), each corresponding to a specific transformation of [PERSON]'s qubit.  
Figure 1: Schematic representation of quantum teleportation utilizing a shared entangled state.  
[PERSON] transmits her measurement outcome (two classical bits) to [PERSON] via a classical communication channel. Based on this outcome, [PERSON] applies one of the following corrective operations to recover \(\ket{\psi}_{C}\):  
* If [PERSON]'s outcome is \(\ket{\Phi^{+}}_{AC}\), [PERSON] does nothing.
* If [PERSON]'s outcome is \(\ket{\Phi^{-}}_{AC}\), [PERSON] applies \(Z\) (Pauli-Z gate).
* If [PERSON]'s outcome is \(\ket{\Psi^{+}}_{AC}\), [PERSON] applies \(X\) (Pauli-X gate).
* If [PERSON]'s outcome is \(\ket{\Psi^{-}}_{AC}\), [PERSON] applies \(XZ\) (Pauli-X followed by Pauli-Z gate).  
Thus, [PERSON] reconstructs the original quantum state \(\ket{\psi}_{C}\), which is a qubit representing the amount of quantum money as in Equation (1).  
Importantly, the original quantum state ceases to exist in [PERSON]'s system after her measurement, ensuring that the no-cloning theorem is preserved. This makes quantum teleportation an inherently secure method for transferring quantum money without risk of duplication.