# Quantum Perspective on Digital Money: Towards a Quantum-Powered Financial System
## 2 Defining Quantum Money and Quantum Wallets  
### Quantum Money  
Quantum money can be defined as symbolic values encoded in quantum states. Each amount of money would be represented by a unique quantum state, \(\ket{\$}\), giving it a \"quantum fingerprint\" that ensures authenticity and prevents forgery. In some quantum money schemes, such as the one proposed in _Quantum Money from Knots_[11], each quantum state is also associated with a classical serial number that can be publicly authenticated, further supporting verification without enabling duplication.  
Qubit (quantum bit), the fundamental unit of quantum information, is a typical example of a quantum state. A qubit is a linear superposition of two orthogonal basis states \(\ket{0}\) and \(\ket{1}\), and is described by the combination [12]:  
\[\ket{\psi}=\alpha\ket{0}+\beta\ket{1} \tag{1}\]  
where \(\alpha\) and \(\beta\) are complex numbers, often called the amplitudes of the states \(\ket{0}\) and \(\ket{1}\), respectively. These amplitudes determine the probability of measuring the qubit in either of the two basis states. For example, the probability of measuring the qubit in state \(\ket{0}\) is \(\ket{\alpha}\), and the probability of measuring the qubit in state \(\ket{1}\) is \(\ket{\beta}\)\({}^{2}\). The definition of a qubit in Equation (1) reflects the principle of quantum superposition.  
To better understand superposition intuitively, one can think of the famous [PERSON]'s cat thought experiment. Imagine a cat in a sealed box that can be simultaneously alive and dead until the box is opened and observed. Similarly, a qubit is not just in a state of 0 or 1 but exists in a combination of both states at the same time. It is only when measured that the qubit 'collapses' to either 0 or 1 with certain probabilities determined by the amplitudes \(\alpha\) and \(\beta\). This unique property enables quantum computers and quantum money to perform tasks and provide security features impossible for classical bits.  
Since quantum states are normalized, we require that:  
\[|\alpha|^{2}+|\beta|^{2}=1, \tag{2}\]  
which ensures that the total probability of measuring the qubit in one of the basis states is 1. The specific combinations of \(\alpha\) and \(\beta\) would correspond to the different values that quantum money can represent, and these quantum states would serve as the \"signature\" of the money. For instance, one could assign different quantum states (represented by different values of \(\alpha\) and \(\beta\)) to represent different amounts of currency, ensuring a secure and unique encoding of the money.  
One of the key properties that makes quantum money particularly secure is its immunity to counterfeiting, which arises from the no-cloning theorem [7]. The no-cloning theorem states that it is impossible to make an exact copy of an arbitrary quantum state. In mathematical terms, suppose we have a quantum state \(|\psi\rangle\) as in Equation (1), and an auxiliary system prepared in an initial state \(|e\rangle\). The no-cloning theorem asserts that there does not exist a unitary operation \(U\) that can clone the state \(|\psi\rangle\) in such a way that:  
\[U(|\psi\rangle\otimes|e\rangle)\longrightarrow|\psi\rangle\otimes|\psi \rangle\,. \tag{3}\]  
The no-cloning theorem means that a quantum state \(|\psi\rangle\) cannot be perfectly duplicated. If we try to clone it, the result will either be an imperfect copy or a detectable error in the process. This is somewhat analogous to trying to perfectly photocopy a delicate sculpture made of sand--any attempt to reproduce it by touching or measuring it disturbs its shape. Thus, quantum money encoded in unique quantum states cannot be replicated without detection, ensuring its authenticity and preventing forgery. Any attempt to copy the quantum money would either fail or be detectable, making it a highly secure form of currency.