# Quantum Perspective on Digital Money: Towards a Quantum-Powered Financial System
## 2 Defining Quantum Money and Quantum Wallets
### Protecting Quantum Money from Decoherence  
Quantum states are inherently fragile and can easily lose their distinctive properties due to interactions with their environment--a process known as decoherence [12; 17]. Decoherence poses a significant challenge for quantum money because it can cause the quantum states encoding the currency to degrade, compromising both their authenticity and security. Even minimal interference from surrounding particles or fields can disturb these states, leading to the gradual loss of the \"quantum fingerprint\" that makes each unit of quantum money unique and secure.  
One approach to combating decoherence is to encode quantum money in decoherence-free subspaces (DFS). DFS are specific subspaces within a quantum system where quantum states remain protected from certain types of environmental noise, essentially isolating the information from disruptive influences [18]. These subspaces are typically defined in systems that exhibit symmetries, where the environmental noise affects only certain parts of the quantum system and leaves the states in the DFS unaffected.  
One can think of DFS as \"quiet zones\" or \"safe rooms\" within a noisy environment, where the quantum information is shielded from the disruptive effects around it, allowing the quantum money's delicate states to remain stable and reliable.  
Mathematically, DFS can be understood in terms of the interaction between the system and the environment. For instance, suppose a system of two qubits \(\ket{\psi}=\alpha\ket{00}+\beta\ket{11}\) is subject to some form of noise. If the noise affects the qubits symmetrically, then the states \(\ket{00}\) and \(\ket{11}\) can remain invariant under the action of this noise. The key property of DFS is that they encode quantum information in such a way that noise acting on one qubit does not affect the encoded information, which is preserved across the system.  
In general, the noise model acting on a quantum system is described by a superoperator \(\mathcal{E}\). For a state \(\rho\), the action of noise is:  
\[\rho^{\prime}=\mathcal{E}(\rho) \tag{4}\]  
For DFS to be effective, the evolution of quantum states within the subspace must satisfy  
\[\mathcal{E}(\ket{\psi})=\ket{\psi} \tag{5}\]  
for all states \(\ket{\psi}\) in the DFS, ensuring that these states remain unaffected by the noise.  
For example, a valid DFS under collective bit-flip noise, where both qubits are flipped simultaneously, consists of the Bell states:  
\[\ket{\Psi^{+}}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})\quad\text{ and }\quad \quad\ket{\Psi^{-}}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10}). \tag{6}\]  
These states are eigenstates of the collective bit-flip operator \(X^{\otimes 2}\), meaning they remain unchanged up to a global phase under simultaneous flips:  
\[X^{\otimes 2}\ket{\Psi^{+}}=\ket{\Psi^{+}},\quad X^{\otimes 2}\ket{\Psi^{-}}= \ket{\Psi^{-}}. \tag{7}\]  
Since the states Equation (6) are unaffected by collective bit-flip errors, they can serve as an effective DFS for encoding quantum money. By encoding quantum money in DFS, the coherence of quantum states can be preserved for longer periods, making them more stable during storage and transfer.  
This increased stability is essential for the practical use of quantum money, ensuring that it maintains its secure quantum state across various processes. Furthermore, DFS can be designed to protect against specific types of decoherence, such as phase noise or amplitude damping, by exploiting symmetries in the noise model. This property allows for the robust encoding and verification of quantum money, making it a promising approach for secure quantum financial systems.