What role does the overlap (C) of measurement operators play in defining the entropic uncertainty relation in the context of QKD?

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

The overlap $C$ of measurement operators plays a important role in defining the entropic uncertainty relation within the context of Quantum Key Distribution (QKD). To understand this role comprehensively, it is necessary to consider the fundamental principles of quantum mechanics, the nature of entropic uncertainty relations, and their application in ensuring the security of QKD protocols.

In quantum mechanics, the Heisenberg Uncertainty Principle is a well-known concept, which states that certain pairs of physical properties, such as position and momentum, cannot be simultaneously measured with arbitrary precision. This principle is mathematically represented by the inequality involving the standard deviations of these observables. However, in the context of QKD, a more relevant form of uncertainty is captured by entropic uncertainty relations, which are expressed in terms of the Shannon entropy or other entropy measures.

Entropic uncertainty relations provide a bound on the sum of the uncertainties (entropies) of the outcomes of two incompatible measurements performed on a quantum system. The overlap $C$ of the measurement operators is a key parameter in these relations, as it quantifies the degree of incompatibility between the measurements. Specifically, the overlap is defined as:

$C = \max_{i,j} |\langle \psi_i | \phi_j \rangle|^2$

where $\{|\psi_i\rangle\}$ and $\{|\phi_j\rangle\}$ are the eigenstates of the two measurement operators. The overlap $C$ ranges between 0 and 1, with $C = 0$ indicating that the measurements are perfectly incompatible (orthogonal bases), and $C = 1$ indicating that the measurements are perfectly compatible (identical bases).

The entropic uncertainty relation involving the overlap $C$ can be expressed as:

$H(X) + H(Z) \geq \log_2 \left(\frac{1}{C}\right)$

where $H(X)$ and $H(Z)$ represent the Shannon entropies of the outcomes of measurements in the $X$ and $Z$ bases, respectively. This inequality implies that the sum of the uncertainties of the two measurements is lower-bounded by a term that depends on the overlap $C$ . The smaller the overlap $C$ , the larger the lower bound, indicating greater uncertainty.

In the context of QKD, the entropic uncertainty relation is employed to ensure the security of the key distribution process. QKD protocols, such as BB84, rely on the principles of quantum mechanics to enable secure communication between two parties, typically referred to as Alice and Bob. The security of these protocols is based on the fact that any eavesdropping attempt by an adversary (Eve) will introduce disturbances in the quantum states being transmitted, which can be detected by Alice and Bob.

The entropic uncertainty relation provides a quantitative measure of the disturbance introduced by Eve. When Alice and Bob perform measurements in different bases (e.g., $X$ and $Z$ bases in the BB84 protocol), the overlap $C$ of these measurement operators determines the lower bound on the sum of the uncertainties of their measurement outcomes. If Eve attempts to gain information about the key by measuring the quantum states, she will inevitably introduce additional uncertainty, thereby increasing the total entropy.

For instance, consider the BB84 protocol, where Alice prepares qubits in one of four possible states: $|0\rangle$ , $|1\rangle$ , $|+\rangle$ , and $|-\rangle$ , where $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$ . These states form two mutually unbiased bases (MUBs): the computational basis $\{|0\rangle, |1\rangle\}$ and the diagonal basis $\{|+\rangle, |-\rangle\}$ . The overlap $C$ between these bases is given by:

$C = \max_{i,j} |\langle \psi_i | \phi_j \rangle|^2 = \frac{1}{2}$

Using the entropic uncertainty relation, we have:

$H(X) + H(Z) \geq \log_2 \left(\frac{1}{C}\right) = \log_2 (2) = 1$

This means that the sum of the uncertainties in the $X$ and $Z$ bases must be at least 1 bit. If Eve tries to measure the qubits in an attempt to gain information about the key, she will introduce errors that increase the uncertainty of the measurement outcomes for Alice and Bob. By comparing their measurement results, Alice and Bob can estimate the amount of information Eve might have gained and decide whether to abort the protocol or proceed with key distillation and error correction.

Furthermore, the overlap $C$ also plays a role in the security proofs of QKD protocols. Security proofs often involve bounding the information that Eve can gain about the key while ensuring that the legitimate parties can still extract a secure key. The entropic uncertainty relation, with the overlap $C$ as a parameter, provides a mathematical framework for deriving these bounds.

For example, in the security analysis of the BB84 protocol, one can use the entropic uncertainty relation to derive a bound on the conditional von Neumann entropy $H(X|E)$ , which quantifies the uncertainty of Alice's key bit $X$ given Eve's information $E$ . The relation can be expressed as:

$H(X|E) + H(X|B) \geq \log_2 \left(\frac{1}{C}\right)$

where $H(X|B)$ is the conditional entropy of Alice's key bit given Bob's measurement outcome. This bound ensures that even if Eve possesses some information about the key, the remaining uncertainty is sufficient to guarantee the security of the key after error correction and privacy amplification.

In practical implementations of QKD, the overlap $C$ can be experimentally determined by characterizing the measurement devices and the quantum states used in the protocol. Accurate knowledge of $C$ is essential for assessing the security of the QKD system and for optimizing the key generation rate. For instance, in the presence of device imperfections or noise, the actual overlap $C$ may differ from the ideal value, and this must be taken into account in the security analysis.

To illustrate the importance of the overlap $C$ with an example, consider a scenario where Alice and Bob use slightly misaligned measurement bases due to experimental imperfections. Suppose the overlap $C$ is found to be 0.55 instead of the ideal value of 0.5. The entropic uncertainty relation then becomes:

$H(X) + H(Z) \geq \log_2 \left(\frac{1}{0.55}\right) \approx 0.87$

This indicates that the sum of the uncertainties is lower than in the ideal case, which could potentially affect the security of the protocol. Alice and Bob would need to account for this reduced bound in their error correction and privacy amplification procedures to ensure that the final key remains secure.

The overlap $C$ of measurement operators is a fundamental parameter in defining the entropic uncertainty relation, which in turn plays a critical role in the security analysis of QKD protocols. By quantifying the degree of incompatibility between measurement bases, the overlap $C$ determines the lower bound on the sum of the uncertainties of measurement outcomes. This bound is essential for detecting eavesdropping attempts and for deriving security proofs that guarantee the confidentiality of the distributed key. Accurate characterization of $C$ and careful consideration of its implications are vital for the practical implementation and security assurance of QKD systems.

EITCA Academy

What role does the overlap (C) of measurement operators play in defining the entropic uncertainty relation in the context of QKD?

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

What role does the overlap (C) of measurement operators play in defining the entropic uncertainty relation in the context of QKD?

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

More questions and answers: