# Quantum Bit Commitment

This example protocol achieves the task of bit commitment securely by using a relativistic scheme. In bit commitment, the committer "commits" to a particular bit value. The receiver knows nothing about the committed bit value until the committer chooses to do so (hiding property). The receiver has a guarantee that once committed, the committer cannot change the committed bit value (binding property). Information-theoretic secure bit commitment cannot be done with non-relativistic schemes see this review paper [1].

## Assumptions

• Quantum theory is correct.
• The background space-time is approximately Minkowski.
• The committer can signal at precisely light speed.
• All information processing is instantaneous.

## Outline

Both the receiver and the committer have 2 agents each which are the parties they send their qubits to and receive the committed value from. The agents are light-like separated from the committer.

The receiver securely pre-prepares a set of qubits randomly chosen from the BB84 states and sends them to the committer. To commit to the bit 0, the committer measures the received qubits in the standard basis and in Hadamard basis to commit to 1. The committer then sends the outcomes to their agents over secure classical channels. To unveil the committed bit, the committer's agents reveal the outcomes to the receiver's agents. The receiver's agents then check if the outcomes they have received are the same and consistent with the states sent to the committer. If the check passes, the receiver accepts the commitment.

## Notation

• ${\displaystyle N}$: Number of random qubits used in the commitment.
• ${\displaystyle |\psi _{i}\rangle }$: Random BB84 qubit with index ${\displaystyle i}$.
• ${\displaystyle P}$: Space-time origin point for the Minkowski space which is the position of the committer.
• ${\displaystyle Q_{0}}$: Commiter's first agent.
• ${\displaystyle Q_{1}}$: Commiter's second agent.
• ${\displaystyle Q_{0}^{'}}$: Receiver's first agent.
• ${\displaystyle Q_{1}^{'}}$: Receiver's second agent.

## Requirements

• Secure classical channels between the parties and their agents.
• Basic state preparation abilities for the receiver.
• Instantaneous measurement capabilities for the committer.

## Properties

• There is no need of quantum memory for the parties.
• The protocol is unconditionally secure.

## Protocol Description

The committer and the receiver agree on the space-time origin point P and two light-like separated points where the two agents of each party will be stationed.

### Commitment Phase

1. Prepare a set of ${\displaystyle N}$ qubits ${\displaystyle |\psi _{i}\rangle _{i=1}^{N}}$ chosen independently and randomly from the BB84 states - ${\displaystyle \{|0\rangle ,|1\rangle ,|+\rangle ,|-\rangle \}}$.
2. Send the qubits to the commiter at point P.

Commiter

1. To commit to 0, measure in the ${\displaystyle \{|0\rangle ,|1\rangle \}}$ basis.
2. To commit to 1, measure in the ${\displaystyle \{|+\rangle ,|-\rangle \}}$ basis.
3. Send the measurement outcomes to the agents ${\displaystyle Q_{0}}$ and ${\displaystyle Q_{1}}$ via the secure classical channels.

### Unveiling Phase

Committer

1. The committer's agents reveal the measurement outcomes to the receiver's agents ${\displaystyle Q'_{0}}$ and ${\displaystyle Q'_{1}}$.