In this blog post, I examine the error probability introduced by an eavesdropper, Eve, in the BB84 protocol when she employs a measurement basis at a generic angle. The standard 50% error rate arises from the specific 45 degree angle between BB84 bases. I will show how the error probability varies with Eve's chosen angle, demonstrating that the 45 degree configuration maximizes the error, enhancing the security of the quantum key distribution against interception attempts. Understanding this angle dependence is key to appreciating the robustness of BB84.
In this blog post, I will explain how the BB84 protocol ensures secure key distribution even in the presence of an eavesdropper, Eve. After Bob measures the received photons and announces his bases, Alice reveals the correct bases. They discard mismatches and reconcile the rest. To detect eavesdropping, they sacrifice a subset of their key. If Eve intercepts photons and guesses the wrong basis, her measurements introduce errors. By comparing the sacrificed bits, Alice and Bob can statistically detect Eve's presence through an increased error rate, ensuring the security of their quantum key exchange.
In this blog post, I explore the BB84 protocol, a Quantum Key Distribution (QKD) method. I will explain how this method leverages the principles of quantum mechanics, specifically photon polarization, to establish secure communication channels. I will detail the quantum states involved, the measurement bases, and the fundamental concept of non-commuting observables that underpin the security of this protocol. By examining the probabilities of measurement outcomes, I aim to clarify how BB84 allows for the secure exchange of cryptographic keys.
In this blog post, I explore the concept of perfect secrecy in cryptography, focusing on the one-time pad. I explain how this encryption method, when used correctly, achieves unbreakable security, as mathematically proven by Claude Shannon. My discussion covers the conditions for perfect secrecy, including key randomness, length, and non-reuse. I present Shannon's theorem and its proof to demonstrate why the one-time pad is considered perfectly secure.
In this blog post, I explore the concept of quantum cloning and the fundamental no-cloning theorem. I will demonstrate why, unlike classical information, perfect copies of arbitrary quantum states cannot be created. This principle, rooted in quantum mechanics, has implications for quantum cryptography and our understanding of quantum information. I will explain the theorem using polarized photons as a concrete example, highlighting its importance in securing quantum communication and preventing faster-than-light communication.
In this blog post, I explore how to statistically determine the polarization of photons through repeated measurements. While a single photon measurement is inherently limited by quantum principles, examining multiple, identically polarized photons allows us to infer their initial quantum state. I discuss how measuring photons with polarizers in different orientations, like along the x-axis and potentially another angle, helps resolve ambiguities and provides a clearer picture of the photon's polarization, whether linear, circular, or elliptical.
In this blog post, I introduce qubits, the basic block of quantum information science, highlighting their unique ability to exist in superposition—simultaneously representing 0 and 1. I will explain how this quantum property differs fundamentally from classical bits and unlocks new computational paradigms. From photon polarization to superconducting circuits, I will survey the diverse physical systems capable of embodying qubits, emphasizing the coherence time as a key metric for practical applications.
In this blog post, I extend the analysis of polarizing beam-splitters to consider measurements at an arbitrary angle. I derive the quantum observable that describes polarization measurement along an arbitrary direction. I show the step-by-step derivation of the matrix representation of this observable in the default basis. I then verify that the eigenvalues are +1 and -1 as expected for a polarization measurement. Finally, I explicitly calculate the eigenstates corresponding to these eigenvalues, confirming they align with the expected polarization states.
In this blog post, I explore the function of polarizing beam-splitters in measuring photon polarization. I detail how these devices separate photons based on their polarization states, leading to distinct measurement outcomes. I calculate the probabilities of transmission and reflection for a photon with arbitrary polarization. I discuss why a single measurement provides limited information, requiring multiple measurements for accurate polarization estimation. Finally, I address the no-cloning theorem and its implications for photon measurement, highlighting the fundamental quantum limits involved.
In this blog post, I explore a single-photon polarization, bridging the gap between classical electromagnetism and quantum mechanics. I will show how the polarization of a single photon can be understood as a quantum state within a two-dimensional space. This quantum approach mirrors the classical description of polarization vectors, offering a clear picture of how a single light particle can exhibit polarization properties.
In this blog post, I conclude my investigation into the Mach-Zehnder interferometer with squeezed vacuum by calculating the variance of the balanced output signal. Through detailed quantum mechanical derivations, I obtain an analytical expression for this variance and subsequently determine the signal-to-noise ratio (SNR) for the interferometer. My results demonstrate that the use of a squeezed vacuum input significantly enhances the sensitivity of the interferometer, surpassing the standard quantum limit. This final analysis underscores the practical advantages of employing squeezed states of light in precision measurement and quantum sensing applications.
In this blog post, I extend my analysis of the Mach-Zehnder interferometer with a squeezed vacuum input to compute the expectation value of the squared balanced signal. Building upon the previous derivation of the linear signal difference, I tackle the more complex task of squaring the output signal difference operator. This involves expanding and rearranging intricate expressions containing creation and annihilation operators for both squeezed vacuum and coherent states. My step-by-step approach aims to provide the detailed calculations required to understand the quantum statistical properties of the balanced signal in this interferometer configuration.
In this blog post, I explore the quantum mechanics of a Mach-Zehnder interferometer where a squeezed vacuum state enters one input and a coherent state the other. I derive the expectation value for the difference in output signals, considering the impact of squeezed vacuum on the interferometer's behavior. My analysis highlights how quantum states at the input influence the measurement outcomes, revealing interesting aspects of quantum interferometry and signal detection. This investigation provides insights into the fundamental principles governing these quantum optical systems.
In this blog post, I explore the squeezed vacuum state and its potential to revolutionize quantum noise reduction. Squeezed vacuum, unlike conventional vacuum, exhibits unique properties. While maintaining a zero average field and satisfying the fundamental Heisenberg uncertainty principle, it displays a non-zero average photon number. By manipulating vacuum fluctuations, in particular the P quadrature fluctuations using negative R, I show how it's possible to achieve noise levels below the standard quantum limit in phase measurements.
In this blog post, I explore the measurement of a quantum state's quadrature using a balanced beam splitter and a quasi-classical state. By analyzing the expectation value and variance of the difference in photon numbers at the output ports, I show how, in the specific case of a balanced Mach-Zehnder interferometer with a strong quasi-classical state in one input channel, the measurement effectively extracts the P quadrature of the quantum state in the other input channel. This method provides a way to characterize the statistical properties of the input quantum state, including vacuum fluctuations, through repeated measurements of the balanced signal.