# Quantum Perspective on Digital Money: Towards a Quantum-Powered Financial System
## 3 Verification Methods for Quantum Money in Quantum Wallets
### Non-Destructive Quantum Verification  
Non-destructive or weak measurement techniques may allow partial verification of quantum money without entirely collapsing the states. In quantum mechanics, measurements typically cause the collapse of a quantum state, reducing the superposition to one of its basis states. Weak measurements, however, are designed to extract limited information about a quantum state while minimizing this collapse. This is achieved by coupling the system to an ancillary meter system, allowing only partial information to be gathered about the system's state. Mathematically, weak measurements are described within the framework of weak values, as introduced by [PERSON], [PERSON], and [PERSON] [23]. Unlike projective measurements, weak measurements involve a weak interaction between the system and the measurement apparatus, such that the quantum state is only minimally disturbed [24]. A weak measurement of an observable \(\hat{O}\) for a quantum state \(|\psi_{i}\rangle\), followed by post-selection onto a final state \(|\psi_{f}\rangle\), yields the weak value [25]:  
\[O_{w}=\frac{\langle\psi_{f}|\hat{O}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i} \rangle}. \tag{8}\]  
In a typical weak measurement scenario, if the system is initially in a superposition Equation (1), the weak measurement provides information about both \(\alpha\) and \(\beta\) without fully collapsing the state. The expected shift in the measurement pointer is related to the weak value rather than the standard expectation value. Weak measurements provide limited information about the system without fully collapsing its superposition, making them useful for quantum verification tasks like checking the presence of quantum money without fully revealing its state. While this approach does not yield exact information on the monetary amount, it can confirm the presence of quantum states with minimal disturbance, offering partial verification that avoids full state collapse. This feature of weak measurement is particularly advantageous in the context of quantum money, where it is crucial to verify authenticity while preserving the integrity of the encoded quantum state. Building on this concept, methods from quantum computing, such as non-destructive discrimination of Bell states, could also play a critical role in verifying quantum money. Bell states are a class of maximally entangled quantum states that are fundamental to many quantum protocols, including quantum teleportation and quantum key distribution. Techniques used to distinguish Bell states without collapsing their entanglement could be adapted to verify quantum money non-destructively. For instance, if the quantum states representing the money include entangled components or similar quantum properties, protocols for Bell state discrimination could help determine the integrity and amount of quantum money in a wallet. One notable approach is the gate-based protocol for non-destructive Bell state discrimination proposed by [PERSON] [26; 27]. This protocol employs quantum gates, such as Hadamard and CNOT gates, to manipulate and distinguish quantum states without destroying the underlying quantum information. By mapping quantum states onto distinguishable configurations, these operations allow for reliable identification without collapsing the superposition or entanglement. Applying such techniques to quantum money, verification protocols could confirm not only the presence of valid quantum states but also their monetary value, all while preserving the coherence of the quantum wallet. For example, non-destructive discrimination methods could ensure that quantum money remains viable for further transactions or storage, reducing risks associated with state collapse during verification. These capabilities would provide users with an innovative means of validating their quantum currency securely and efficiently, much like checking the balance of a traditional wallet but without risking the integrity of the quantum states.