Quantum Cloning

Functionality

The no-cloning theorem in Quantum Mechanics states that it is impossible to create a perfect copy of arbitrary unknown quantum sates. However, the imperfect cloning is possible in many different ways. The cloning protocols are either approximate cloning, meaning that at all the rounds they produce approximately similar copies, or they are probabilistic protocols which means that probabilistically they produce the exact copies.

Tags: Building block, no-go theorems, Quantum restrictions

Protocols

• Approximate cloning protocols for discrete quantum systems Cloning protocols for discrete quantum systems (DV) have been included in this section. The simple case of copying a qubit is an example of these cloning machines. The more general case of these quantum cloning machines is N $\rightarrow$ M cloner which will produce M identical copies of N initial states. The discrete quantum cloning machines can be divided into two main categories: universal and non-universal.
  • Universal quantum cloning protocols: Universal cloning machines produce copies of any arbitrary states. These protocols produce copies which are approximately close to the original state at every round. Also, a universal cloning machine can be
    • Symmetric or Optimal Cloning
    • Asymmetric Cloning
  • Non-universal quantum cloning protocols It is possible to have a cloning machine which is not universal and these machines have their own functionalities and advantages. In this category, we have
    • State-dependent cloning
    • Phase-covariant cloning.
• Approximate cloninng protocols for continous variables The cloning of several quantum states such as photonic states which are in the regime of continuous variables (CV) have been presented in this section. The N $\rightarrow$ M approximate Gaussian cloning protocol is the most important cloning protocol in CV with a wide variety of application in photonic or quantum oscillator systems.
• Probabilistic cloninng Another way of having an imperfect cloner is to have a probabilistic cloning machine will produce the copies with some probability of success. In these protocols coping task can succeed with probability, but if it is successful, we can always obtain perfect copie. The probabilistic cloning machines will no longer consists of unitary operations only. But these machines are represented by quantum maps instead. This quantum cloning machine is useful, in particular, in studying the B92 quantum key distribution protocol.

Use Case

Signing e-Marksheet, Financial Transactions, Software Distribution, Cryptocurrencies, e-voting

Properties

All QDS protocols are divided into two phases, distribution and messaging. Distribution phase enables sender to generate private keys (kept secret with sender) and public keys (information distributed to recipients) while messaging phase enables exchange of messages using the above keys. For simlicity, most protocols use the case of three parties, one sender (Seller) and two recipients (Buyer and Verifier) exchanging one-bit classical messages signed by Quantum Digital Signatures (QDS).

• A QDS scheme is correct if a message signed by a genuine sender is accepted by a recipient with unit probability.
• A QDS scheme is secure if no one but the sender can sign a message such that it is accepted by a recipient with non-negligible probability.
• Transferability means that at any point a recipient (buyer) can prove it to another recipient (verifier) that the concerned message has been signed by the claimed sender (Seller).
• Unforgeability ensures that a dishonest recipient (buyer) can neither alter a DS nor sign a message with a fake DS (DS that has not come from a genuine sender) and forward it to other recipients (verifier) successfully.
• Non-Repudiation implies that at any point a dishonest sender (seller) cannot deny having signed the message sent to a genuine recipient (Buyer).

Discussion

1. AA (2015) Discusses various classical and quantum digital signature schemes
2. Wallden P. (2018) (In preparation): Discusses the development of Quantum Digital Signatures from the first protocol by Gottesman and Chuang, elaborating advancements in further protocols to turn it into a practical QDS scheme.