How is the fidelity between the shared state (rho_{AB}) and the maximally entangled state (|Phi^+rangle) used to determine the security of the BB84 protocol?

by EITCA Academy / Saturday, 15 June 2024 / Published in Cybersecurity, EITC/IS/QCF Quantum Cryptography Fundamentals, Security of Quantum Key Distribution, Security of BB84, Examination review

The fidelity between the shared state $\rho_{AB}$ and the maximally entangled state $|\Phi^+\rangle$ is a critical metric in determining the security of the BB84 protocol, a cornerstone of quantum key distribution (QKD). To understand this relationship, it is essential to consider the fundamentals of quantum cryptography, the principles underlying the BB84 protocol, and the role of entanglement and fidelity in assessing the security of the generated cryptographic keys.

Quantum key distribution (QKD) allows two parties, commonly referred to as Alice and Bob, to generate a shared, secret random key, which can be used for secure communication. The BB84 protocol, introduced by Charles Bennett and Gilles Brassard in 1984, is the first and one of the most widely studied QKD protocols. It leverages the principles of quantum mechanics to ensure the security of the key distribution process. The security of BB84 is fundamentally rooted in the no-cloning theorem and the disturbance caused by eavesdropping.

In the BB84 protocol, Alice prepares qubits in one of four possible states, using two mutually unbiased bases (MUBs): the computational basis $\{|0\rangle, |1\rangle\}$ and the diagonal basis $\{|\pm\rangle\}$ , where $|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$ and $|-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$ . Alice randomly chooses one of these four states and sends the qubits to Bob through a quantum channel. Bob, upon receiving the qubits, randomly measures each qubit in either the computational or diagonal basis. After the quantum transmission, Alice and Bob publicly announce their chosen bases and discard the results where their bases do not match. The remaining bits form the raw key, which undergoes further post-processing steps such as error correction and privacy amplification to produce the final secret key.

The security of the BB84 protocol hinges on the ability to detect any eavesdropping attempt by an adversary, commonly referred to as Eve. When Eve tries to intercept and measure the qubits, she inevitably introduces errors due to the principles of quantum mechanics. This disturbance can be quantified by comparing the shared state $\rho_{AB}$ between Alice and Bob with an ideal maximally entangled state $|\Phi^+\rangle$ .

The maximally entangled state $|\Phi^+\rangle$ is given by:

$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}.$

This state represents the highest degree of entanglement between two qubits, implying perfect correlations between the measurement outcomes of Alice and Bob.

Fidelity is a measure of the "closeness" or "similarity" between two quantum states. For two density matrices $\rho$ and $\sigma$ , the fidelity $F(\rho, \sigma)$ is defined as:

$F(\rho, \sigma) = \left( \text{Tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right)^2.$

In the context of the BB84 protocol, the fidelity between the shared state $\rho_{AB}$ and the maximally entangled state $|\Phi^+\rangle$ is used to assess the integrity of the quantum channel and the presence of any eavesdropping.

A high fidelity $F(\rho_{AB}, |\Phi^+\rangle)$ close to 1 indicates that the shared state $\rho_{AB}$ is very similar to the maximally entangled state $|\Phi^+\rangle$ , suggesting that the quantum channel is secure and free from significant eavesdropping. Conversely, a low fidelity indicates that the shared state has deviated significantly from the maximally entangled state, implying potential eavesdropping activity.

To illustrate this, consider the following example: Suppose Alice and Bob share a state $\rho_{AB}$ after the quantum transmission. They perform quantum tomography to reconstruct $\rho_{AB}$ and calculate the fidelity with respect to $|\Phi^+\rangle$ . If the fidelity is found to be $F(\rho_{AB}, |\Phi^+\rangle) = 0.98$ , this high value suggests that the shared state is nearly maximally entangled, and the quantum channel is secure. If, however, the fidelity is $F(\rho_{AB}, |\Phi^+\rangle) = 0.7$ , this lower value indicates that the shared state has been significantly disturbed, likely due to eavesdropping.

The relationship between fidelity and security can be further understood through the concept of the quantum bit error rate (QBER). The QBER is the fraction of bits that differ between Alice's and Bob's raw keys. In the absence of eavesdropping, the QBER is primarily due to imperfections in the quantum channel and the measurement devices. However, eavesdropping introduces additional errors, increasing the QBER.

There is a direct connection between the fidelity of the shared state and the QBER. A higher QBER corresponds to a lower fidelity, as the presence of errors indicates a deviation from the ideal maximally entangled state. In the BB84 protocol, if the QBER exceeds a certain threshold (typically around 11%), the protocol is considered insecure, and the key must be discarded. This threshold is derived from the fact that beyond this point, the information gained by Eve through her eavesdropping exceeds the information advantage that Alice and Bob can extract through error correction and privacy amplification.

To quantify the security of the BB84 protocol, one often employs the concept of the secret key rate, which is the rate at which secure key bits can be generated. The secret key rate $R$ is given by:

$R = 1 - 2h(Q),$

where $h(Q)$ is the binary entropy function and $Q$ is the QBER. The binary entropy function $h(Q)$ is defined as:

$h(Q) = -Q \log_2(Q) - (1-Q) \log_2(1-Q).$

As the QBER increases, $h(Q)$ increases, reducing the secret key rate. When the QBER reaches the threshold, the secret key rate drops to zero, indicating that no secure key can be generated.

In practice, to ensure the security of the BB84 protocol, Alice and Bob must perform several steps:

1. Channel Parameter Estimation: Alice and Bob estimate the QBER by comparing a subset of their raw key bits. This estimation provides an indication of the fidelity of the shared state.

2. Error Correction: Alice and Bob apply error correction protocols to reconcile their raw keys. The efficiency of error correction depends on the QBER.

3. Privacy Amplification: To counteract any information Eve may have gained, Alice and Bob apply privacy amplification techniques, reducing the length of the key while increasing its security.

4. Parameter Verification: Alice and Bob verify that the QBER is below the acceptable threshold, ensuring that the fidelity of the shared state is sufficiently high to guarantee security.

The use of fidelity as a metric for the security of the BB84 protocol underscores the importance of entanglement and quantum correlations in quantum cryptography. By ensuring that the shared state $\rho_{AB}$ remains close to the maximally entangled state $|\Phi^+\rangle$ , Alice and Bob can detect and mitigate the effects of eavesdropping, thereby maintaining the integrity and confidentiality of the distributed cryptographic key.

EITCA Academy

How is the fidelity between the shared state (rho_{AB}) and the maximally entangled state (|Phi^+rangle) used to determine the security of the BB84 protocol?

Other recent questions and answers regarding EITC/IS/QCF Quantum Cryptography Fundamentals:

More questions and answers:

EITCA Academy is a part of the European IT Certification framework

EITCA Academy

SIGN IN YOUR ACCOUNT TO HAVE ACCESS TO DIFFERENT FEATURES

FORGOT YOUR DETAILS?

CREATE ACCOUNT

How is the fidelity between the shared state (rho_{AB}) and the maximally entangled state (|Phi^+rangle) used to determine the security of the BB84 protocol?

Other recent questions and answers regarding EITC/IS/QCF Quantum Cryptography Fundamentals:

More questions and answers: