In quantum mechanics, non-orthogonal states refer to quantum states that cannot be perfectly distinguished from each other. Unlike orthogonal states, which can be reliably distinguished using an appropriate measurement, non-orthogonal states have a non-zero overlap, making it impossible to determine the exact state with certainty. ### Key Characteristics: 1. **Overlap and Distinguishability**: - Non-orthogonal states have a non-zero inner product, meaning they overlap in Hilbert space. This overlap prevents perfect discrimination between the states. For instance, if |ψ⟩ and |ϕ⟩ are non-orthogonal, ⟨ψ|ϕ⟩ ≠ 0 [[2](https://qubit.guide/4.9-distinguishing-non-orthogonal-states.html)]. 2. **Quantum Cryptography**: - Non-orthogonal states are fundamental in quantum cryptography. They provide security for protocols like quantum key distribution (QKD) because any attempt to distinguish or clone these states introduces errors that can be detected by the communicating parties [[5](https://www.researchgate.net/publication/269646939_Quantum_Public-Key_Cryptosystem_Using_Non-orthogonal_States)]. 3. **No-Cloning Theorem**: - The no-cloning theorem states that it is impossible to create an exact copy of an arbitrary unknown quantum state. This theorem applies particularly to non-orthogonal states, adding another layer of security in quantum cryptographic protocols [[3](https://quantumcomputing.stackexchange.com/questions/21400/using-distinguishability-of-non-orthogonal-states-to-create-a-cloning-device)]. 4. **Measurement Challenges**: - Measuring non-orthogonal states introduces [[uncertainty]]. While orthogonal states can be perfectly distinguished by a suitable measurement, non-orthogonal states can only be distinguished probabilistically, often using quantum measurements that have inherent error rates [[4](https://www.reddit.com/r/QuantumComputing/comments/p9rk7o/are_orthogonal_quantum_states_distinguishable/)]. 5. **Applications**: - Beyond cryptography, non-orthogonal states are used in various quantum information processes and communication protocols, exploiting their properties to enhance security and functionality in quantum systems [[1](https://arxiv.org/abs/quant-ph/9502021#:~:text=All%20existing%20quantum%20cryptosystems%20use,(duplicated)%20by%20an%20eavesdropper)]. Understanding non-orthogonal states is crucial for advancing technologies in quantum computing and communication, where they play a pivotal role in ensuring secure and efficient information transfer.