The objective of this module is to introduce students to the fundamental principles
of quantum cryptography and demonstrate how quantum physics concepts can be applied
to ensure secure information processing, data transmission and communication.
Students will acquire the necessary quantum foundations and understand how quantum phenomena
such as superposition, measurement, and entanglement provide new approaches to cryptographic security.
At the end of this module, students will be able to describe and analyze a quantum cryptographic protocols
(for example BB84), explaining its operation and security principles.
Cryptography is the study of methods and techniques for developing and using coded algorithms to protect information and ensure secure communication, so that it can only be accessed by authorized parties possessing the appropriate decryption key or mechanism. In other words, cryptography protects communication by preventing unauthorized access to information.
Image generated with AI assistance
Today, cryptography plays a vital role in many aspects of everyday life, including electronic commerce, secure web communication, protection of confidential information, banking systems, healthcare technologies, and national security applications.
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From Classical to Quantum Cryptography
Classical cryptography relied primarily on substitution and transposition techniques to protect information.
Well-known examples include the Caesar cipher and the Vigenère cipher.
Beginning in the 1970s, modern cryptography evolved into a discipline based on mathematics and computer science. Modern cryptographic systems rely on publicly known algorithms and secret cryptographic keys, following Kerckhoffs’s Principle and focuses on two closely related objectives: secure key exchange and secure data encryption. Based on these objectives, modern cryptographic systems are generally classified into symmetric and asymmetric cryptography.
Beginning in the 1970s, modern cryptography evolved into a discipline based on mathematics and computer science. Modern cryptographic systems rely on publicly known algorithms and secret cryptographic keys, following Kerckhoffs’s Principle and focuses on two closely related objectives: secure key exchange and secure data encryption. Based on these objectives, modern cryptographic systems are generally classified into symmetric and asymmetric cryptography.
| Feature | Symmetric Cryptography | Asymmetric Cryptography |
|---|---|---|
| Principle | The same secret key is used for encryption and decryption |
A public key is used for encryption and a private key for decryption |
| Examples |
AES (Advanced Encryption Standard) DES (Data Encryption Standard) OTP (One-Time Pad) |
RSA (Rivest–Shamir–Adleman) ECC (Elliptic Curve Cryptography) DH (Diffie–Hellman Key Exchange) |
| Illustration |
Image generated with AI assistance
|
Image generated with AI assistance
|
| Advantages | Fast and efficient for large amounts of data | Enables secure key exchange and digital signatures |
| Limitations | Requires the secure distribution of a shared secret key over an insecure channel - the key distribution problem |
Computationally more expensive than symmetric methods; Quantum computers may break widely used cryptosystems; for example, Shor's algorithm can efficiently factor large integers, threatening RSA and related cryptosystems. |
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Exercise: One-Time Pad
Objective: Learn how the One-Time Pad (OTP) encryption method allows two parties, Alice and Bob, to communicate securely using a shared secret key.
The OTP is an encryption method that can provide perfect secrecy when used correctly. Its security relies on three fundamental rules:
- The key must be truly random.
- The key length must be equal to the message length.
- The key must be used only once for a single message.
The One-Time Pad principle can be implemented using different representations of characters. For example, characters may be represented by their ASCII codes or by binary values, in which case encryption is performed using the XOR operation.
The table below illustrates the OTP principle using ASCII codes. Each character of the message and the key is first converted into its numerical ASCII representation. Encryption is then performed using modular arithmetic on these numerical values.
| Position | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Message | H | E | L | O | O |
| ASCII Code of Message | 72 | 69 | 76 | 79 | 79 |
| Key | k | E | y | 1 | 9 |
| ASCII Code of Key | 107 | 69 | 121 | 49 | 57 |
| Ciphertext ASCII Code: \((m[i] + k[i]) \bmod 256 \) | 179 | 138 | 197 | 128 | 136 |
| Decryption | |||||
| Ciphertext Character | ³ | Š | Å | € | ˆ |
| Recovered Plaintext: \((c[i] - k[i]) \bmod 256 \) | H | E | L | O | O |
In this exercise, the OTP is implemented using a custom alphabet of length \(N\), leading to the following general formulas:
Encryption:
\( c[i] = (m[i] + k[i]) \bmod N \)
Decryption:
\( m[i] = (c[i] - k[i]) \bmod N \)
where:
- \(m[i]\) is the index of the message character at position \(i\),
- \(k[i]\) is the index of the key character at position \(i\),
- \(c[i]\) is the index of the ciphertext character at position \(i\),
- \(N\) is the size of the alphabet.
Choose Your Role:
Alice: Encrypt a message
1. Enter a message.
2. a secret key.
3. Encrypt the message using the One-Time Pad method.
4. Share the ciphertext and the secret key with Bob.
2. a secret key.
3. Encrypt the message using the One-Time Pad method.
4. Share the ciphertext and the secret key with Bob.
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Exercise: Asymmetric (Public-Key) Cryptography
RSA was introduced in 1978 by Ronald Rivest, Adi Shamir, and Leonard Adleman. It uses a public key for encryption and a private key for decryption. Its security relies on the difficulty of factoring the large number \(n = pq\).
Encryption: \( C \equiv M^{e} \pmod n \)
Decryption: \( M \equiv C^{d} \pmod n \)
where \(M\) is the plaintext, \(C\) the ciphertext, \((e,n)\) the public key, and \((d,n)\) the private key. The exponents \(e\) and \(d\) are chosen such that \[ e\,d \equiv 1 \pmod{\varphi(n)}. \] This relationship ensures that a message encrypted with the public key exponent \(e\) can be recovered using the private key exponent \(d\).
Generate RSA Key Pair:
Encryption: enter the message to be encrypted and the RSA public key in the format \(e,n\)
Decryption: enter the ciphertext as comma-separated integers and the RSA private key in the format \(d,n\)
In modern cryptographic systems, security is based on computational difficulty of solving certain mathematical problems. Quantum computers could solve some of these problems efficiently, posing a threat to cryptographic systems.
Security is not absolute. Depends on technological limitations
Quantum cryptography relies on the laws of quantum mechanics. Any attempt to intercept quantum information disturbs the quantum state, making eavesdropping detectable.
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Foundations of Quantum Information and Computing
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Postulates of Quantum Mechanics for Information Processing
The postulates of quantum mechanics are fundamental principles that define how information can be represented,
manipulated, and measured within the quantum computing model.
Postulate 1: State Space
Associated with any isolated quantum system is a complex vector space with an inner product (a Hilbert space),
called the state space of the system. The system is completely described by its state vector \(|\psi \rangle \),
which is a unit vector in this space.
- Represents qubits and their superpositions
- Describes initialization of quantum registers
Postulate 2: Time Evolution
The evolution of a closed quantum system from time \(t_0\) to \(t_1\) is described by the unitary transformation:
$$
|\psi(t_1)\rangle = U |\psi(t_0)\rangle, \quad U^\dagger U = I
$$
- Quantum gates correspond to unitary operators
- The state \(|\psi \rangle \) changes via quantum gates
- All quantum algorithms are sequences of unitary operations (quantum circuits)
- Ensures reversibility of quantum computation
Postulate 3: Measurement
Quantum measurements are described by a collection of
measurement operators \(\{ M_m \}\), acting on the state space of the system. Each operator corresponds to a possible measurement outcome \(m\).
If the system is in the state \(|\psi \rangle \) immediately before the measurement, the probability of obtaining outcome \(m\) is
\(
p(m) = \langle \psi | M_m^\dagger M_m | \psi \rangle.
\)
The state of the system after measurement becomes
The state of the system after measurement becomes
$$
| \psi \rangle \xrightarrow{\text{Measurement}} \frac{M_m |\psi \rangle}{\sqrt{{\langle\psi|} M_m^\dagger M_m |\psi \rangle}}.
$$
The measurement operators satisfy the completeness relation
\(
\sum_m M_m^\dagger M_m = I,
\)
which ensures that the probabilities of all possible outcomes sum to one:
$$
\sum_m p(m) = \sum_m \langle \psi | M_m^\dagger M_m | \psi \rangle = 1 .
$$
- Explains probabilistic outcomes and collapse of superpositions
- Converts qubit superpositions into classical bits for readout
- Determines the output of quantum algorithms
- Implements projective measurements in quantum circuits
Postulate 4: Composition
The state space of a composite physical system is the tensor product of the state spaces of its subsystems.
If subsystem \(i\) is prepared in the state \(|\psi_i\rangle\), the joint state of the total system is
\( |\psi_1\rangle \otimes | \psi_2 \rangle \otimes ... \otimes |\psi_n\rangle\).
- Describes multi-qubit systems and entanglement
- Allows operations on subsystems or individual qubits
- Fundamental for quantum circuits, quantum teleportation, and entanglement-based algorithms
Qubits (quantum bits) are the basic units of quantum information, analogous to bits in classical computing.
While a classical bit can exist in only one of two states, \(0\) or \(1\), a qubit can exist in a superposition of the basis states \(|0 \rangle\) or \(|1 \rangle\):
\begin{aligned}
\hspace{5em}
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle \hspace{5em} \hfill (1)
\end{aligned}
where \(\alpha, \beta \in \mathbb{C}\) are complex amplitudes. The quantities \(|\alpha|^2\) and \(|\beta|^2\) represent the probabilities of measuring the states \(|0 \rangle\) and \(|1 \rangle\), respectively. The normalization condition ensures that the total probability is 1:
\(
|\alpha|^2 + |\beta|^2 = 1.
\)
- In Dirac (bra–ket) notation, commonly used in quantum theory and quantum algorithms (see Eq.(1))
- In column-vector (matrix) representation, which is useful for computations and simulations, the same physical state can be written as:
$$ |\psi \rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}, \quad |0 \rangle \equiv \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad |1 \rangle \equiv \begin{bmatrix} 0 \\ 1 \end{bmatrix} $$
-
For geometric visualization, the Bloch sphere representation is commonly used:
\begin{aligned} |\psi \rangle = \cos \frac{\theta}{2} |0 \rangle + e^{i \phi} \sin \frac{\theta}{2} |1 \rangle \hspace{1em} \hfill (2) \\ \end{aligned} $$ \alpha = \cos \frac{\theta}{2}, \quad \beta = e^{i \phi} \sin \frac{\theta}{2}, $$where \(\theta, \phi \in \mathbb{R}\) are the polar and azimuthal angles. For a single qubit, the global phase factor has no physically observable effect and can be omitted.
Qubits' key properties:
- Superposition: A qubit can exist in a combination of \(|0 \rangle\) and \(|1 \rangle \) simultaneously.
- Measurement: When measured, a qubit collapses to either \(0\) or \(1 \) with probabilities determined by its state.
- Entanglement: Qubits can be correlated with each other in ways that have no classical counterpart.
- Interference: Quantum states can reinforce or cancel each other, enabling powerful computational effects.
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Superposition
The principle of superposition allows a qubit to exist in a combination of its basis states, \(|0 \rangle\) and \(|1 \rangle \) (Eq.(1)).
For an \(n\)-qubit system, the state can be a superposition of all \(2^n\) basis states:
Each \(|i\rangle\) represents a binary combination of the \(n\) qubits' basic states and can be expressed as a tensor product of the individual qubits:
For a two-qubit system (\(n = 2\)), there are four computational basis states: \(|00\rangle\), \(|01\rangle\), \(|10\rangle\), and \(|11\rangle\), where \(|01\rangle = |0\rangle \otimes |1\rangle,\) meaning the first qubit is \(|0\rangle\) and the second is \(|1\rangle\). The general state of a two-qubit system is a superposition of all four states:
In column-vector form:
For an \(n\)-qubit system, the state can be a superposition of all \(2^n\) basis states:
\begin{aligned}
|\psi \rangle = \sum_{i=0}^{2^n-1} \alpha_i |i \rangle, \hspace{5em} \hfill (3)
\end{aligned}
where \(\alpha _i \in \mathbb{C}\) and \(\sum_{i=0}^{2^n-1} |\alpha _i |^2 = 1\).
Each \(|i\rangle\) represents a binary combination of the \(n\) qubits' basic states and can be expressed as a tensor product of the individual qubits:
$$
|q_1 q_2 \dots q_n \rangle = |q_1 \rangle \otimes |q_2 \rangle \otimes \dots \otimes|q_n \rangle ,
$$
where \(q_1, q_2, \dots, q_n\) denote the states of qubits 1 through \(n\), ordered left to right.
For a two-qubit system (\(n = 2\)), there are four computational basis states: \(|00\rangle\), \(|01\rangle\), \(|10\rangle\), and \(|11\rangle\), where \(|01\rangle = |0\rangle \otimes |1\rangle,\) meaning the first qubit is \(|0\rangle\) and the second is \(|1\rangle\). The general state of a two-qubit system is a superposition of all four states:
$$
|\psi\rangle = \alpha _0 |00\rangle + \alpha _1 |01\rangle + \alpha _2 |10\rangle + \alpha _3 |11\rangle,
$$
with \(\sum_{i=0}^{3} |\alpha _i |^2 = 1 \).
In column-vector form:
$$
|\psi\rangle
= \alpha_0
\begin{bmatrix}1\\0\\0\\0\end{bmatrix}
+ \alpha_1
\begin{bmatrix}0\\1\\0\\0\end{bmatrix}
+ \alpha_2
\begin{bmatrix}0\\0\\1\\0\end{bmatrix}
+ \alpha_3
\begin{bmatrix}0\\0\\0\\1\end{bmatrix}
=
\begin{bmatrix}
\alpha_0 \\
\alpha_1 \\
\alpha_2 \\
\alpha_3
\end{bmatrix}.
$$
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Mesurements
In quantum mechanics concept of measurement plays a crucial role in determining the outcome of a quantum experiment .
If the measurement operator \(M_m\) is projector onto the eigenstate \(|\phi_m \rangle\):
$$
M_m = | \phi_m \rangle \langle \phi_m |,
$$
the probability of observing that outcome is given by the Born rule:
\begin{aligned}
p(m) = \langle \psi | M_m |\psi \rangle = \langle \psi | \phi_m \rangle \langle \phi_m |\psi \rangle = |\langle \phi_m |\psi \rangle|^2 . \hspace{1em} \hfill (4)
\end{aligned}
After the measurement, the quantum state collapses to the observed eigenstate \(|\phi_m \rangle\):
$$
| \psi \rangle \xrightarrow{\text{measure m}} \frac{M_m | \psi \rangle}{\sqrt{p(m)}} = |\phi_m \rangle.
$$
Measurement in Quantum Computing
- Probabilistic nature: Measurement is inherently probabilistic. The probabilities of different outcomes are determined by the amplitudes of the quantum state according to the Born rule.
- State collapse: During measurement, the system transitions from a superposition to a definite state, turning the qubit into a classical bit. This phenomenon, known as wavefunction collapse, is a fundamental concept in quantum mechanics.
- Irreversibility: Measurement is irreversible — performing a measurement destroys information about the original superposition, which cannot be recovered.
- Partial measurements: Measuring one qubit can influence correlated qubits due to entanglement (Postulate 4, composition).
- Multiple runs: Because outcomes are probabilistic, quantum computers typically run the same circuit multiple times (called shots) to estimate the distribution of results. The final result is a statistical distribution of outcomes, not a single deterministic value.
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Entanglement
Quantum entanglement is a fundamental property of quantum systems in which two or more qubits become correlated
in such a way that the state of one qubit cannot be described independently of the state of the others,
even when separated by large distances. This is a uniquely quantum-mechanical phenomenon that has no classical equivalent.
For two arbitrary quantum systems \(A\) and \(B\), with respective Hilbert spaces \(H_A\) and \(H_B\), a pure state is entangled if:
The canonical examples of maximally entangled two-qubit states, known as the Bell states, are:
Quantum entanglement is a key resource in quantum cryptography and quantum communication. It enables entanglement-based Quantum Key Distribution (QKD) protocols, such as E91 and BBM92, as well as quantum teleportation and future quantum networking technologies, and forms the foundation of future quantum networking technologies.
For two arbitrary quantum systems \(A\) and \(B\), with respective Hilbert spaces \(H_A\) and \(H_B\), a pure state is entangled if:
-
It cannot be written as a tensor product of individual qubit states:
$$ |\psi \rangle \neq |\psi_A\rangle \otimes |\psi_B\rangle $$
-
Equivalently, the reduced density matrix of at least one subsystem is mixed:
$$ \text{Tr}(\rho_A^2) < 1 , \quad \text{or} \quad \text{Tr}(\rho_B^2) < 1 $$
The canonical examples of maximally entangled two-qubit states, known as the Bell states, are:
$$ |\Phi^\pm \rangle = \frac{|00\rangle \pm |11\rangle}{\sqrt{2}}, $$
$$ |\Psi^\pm \rangle = \frac{|01\rangle \pm |10\rangle}{\sqrt{2}}, $$
For the \(|\Psi^+\rangle\) Bell state, the joint measurement yields either \(|01\rangle\) or \(|10\rangle\) with equal probability,
but if the first qubit is found in state \(|0\rangle\), the second qubit is always found in \(|1\rangle\), and vice versa.
Thus, measurement causes the collapse of the joint quantum state.
Quantum entanglement is a key resource in quantum cryptography and quantum communication. It enables entanglement-based Quantum Key Distribution (QKD) protocols, such as E91 and BBM92, as well as quantum teleportation and future quantum networking technologies, and forms the foundation of future quantum networking technologies.
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Quantum gates
Quantum gates are reversible, unitary, and linear operations
that manipulate qubits and serve as the fundamental building blocks of quantum circuits,
analogous to how classical logic gates manipulate classical bits in digital circuits.
The properties of quantum gates are defined by the postulates of quantum mechanics (Postulate 2).
The output state can be obtained by applying a unitary matrix \(U\) to the initial state: $$ |\psi'\rangle = U |\psi\rangle $$ where \(U^\dagger U = U U^\dagger = I\), and \(U^\dagger\) denotes the conjugate transpose of \(U\). This ensures reversibility and conservation of probability.
The properties of quantum gates are defined by the postulates of quantum mechanics (Postulate 2).
The output state can be obtained by applying a unitary matrix \(U\) to the initial state: $$ |\psi'\rangle = U |\psi\rangle $$ where \(U^\dagger U = U U^\dagger = I\), and \(U^\dagger\) denotes the conjugate transpose of \(U\). This ensures reversibility and conservation of probability.
Choose a quantum gate to see its matrix representation and learn how it affects qubits. Both single-qubit and multi-qubit (marked with *) gates are presented.
×
The identity gate is the identity matrix - square matrix with ones on the main diagonal and zeros elsewhere.
Leaves the qubit unchanged:
\(I|0\rangle = |0\rangle\)
\(I|1\rangle = |1\rangle \)
\(I|0\rangle = |0\rangle\)
\(I|1\rangle = |1\rangle \)
$$ I = \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix} $$
×
The Pauli gates \((X,Y,Z)\) are the three Pauli matrices \((\sigma _{x},\sigma _{y},\sigma _{z})\) that act on a single qubit.
The Pauli X, Y and Z equate, to a rotation around the \(x\), \(y\) and \(z\) axes of the Bloch sphere by \(\pi\) radians, respectively.
The Pauli X, Y and Z equate, to a rotation around the \(x\), \(y\) and \(z\) axes of the Bloch sphere by \(\pi\) radians, respectively.
Pauli-X (NOT) similar to the classical NOT gate,
Flips computational basis states:
\( X|{0}\rangle = |{1}\rangle \)
\( X|{1}\rangle = |{0}\rangle \)
Flips computational basis states:
\( X|{0}\rangle = |{1}\rangle \)
\( X|{1}\rangle = |{0}\rangle \)
$$X = \sigma_{x} = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix} $$
Pauli-Y flips the computational basis states,
Adds a relative phase of \( \pm i \):
\( Y|{0}\rangle = i|{1}\rangle \)
\( Y|{1}\rangle = -i|{0}\rangle\)
Adds a relative phase of \( \pm i \):
\( Y|{0}\rangle = i|{1}\rangle \)
\( Y|{1}\rangle = -i|{0}\rangle\)
$$Y = \sigma_{y} = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix} $$
Pauli-Z leaves \(|0\rangle\) unchanged,
Adds a relative phase of \(\varphi = \pi\) to \(|1\rangle\):
\( Z|{0}\rangle = |{0}\rangle \)
\( Z|{1}\rangle = -|{1}\rangle \)
Adds a relative phase of \(\varphi = \pi\) to \(|1\rangle\):
\( Z|{0}\rangle = |{0}\rangle \)
\( Z|{1}\rangle = -|{1}\rangle \)
$$Z = \sigma_{z} = \begin{pmatrix}1 & 0 \\0 & -1\end{pmatrix} $$
×
The Hadamard gate is one of the most important single-qubit gates
used to create superpositions and enable parallelism in quantum computations.
When applied to a qubit initialized in a classical state \(|0\rangle\) or \(|1\rangle\),
the Hadamard gate transforms it into an equal superposition of both states,
making it a fundamental tool in quantum circuits.
It converts the computational (Z) basis \({|0\rangle, |1\rangle} \) into the X-basis \({|+\rangle, |-\rangle} \). Up to a global phase, this corresponds to a \(\pi\)-rotation around the \( (\hat{x} + \hat{z})/\sqrt{2}\) axis on the Bloch sphere.
It converts the computational (Z) basis \({|0\rangle, |1\rangle} \) into the X-basis \({|+\rangle, |-\rangle} \). Up to a global phase, this corresponds to a \(\pi\)-rotation around the \( (\hat{x} + \hat{z})/\sqrt{2}\) axis on the Bloch sphere.
\( H|{0}\rangle = \frac{|{0}\rangle + |{1}\rangle}{\sqrt{2}} = |+\rangle \)
\(H|{1}\rangle = \frac{|{0}\rangle - |{1}\rangle}{\sqrt{2}} = |-\rangle\)
\(H|{1}\rangle = \frac{|{0}\rangle - |{1}\rangle}{\sqrt{2}} = |-\rangle\)
$$ H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix} $$
×
The phase shift is a family of single-qubit gates that map the basis states
\( |0\rangle \mapsto |0\rangle \) and \(|1\rangle \mapsto e^{i\varphi }|1\rangle \).
Applying the gate modifies the phase of the quantum state. This is equivalent to a rotation about
the \(z\)-axis on the Bloch sphere by \(\varphi\) radians.
The phase shift gates are represented by the matrix (\(P^{\dagger }(\varphi )=P(-\varphi )\)):
$$ P(\varphi )={\begin{pmatrix}1&0\\0&e^{i\varphi }\end{pmatrix}}$$
where \(\varphi\) is the phase shift with the \(2\pi\) period.
Leaves \(|0\rangle\) unchanged and adds a relative phase of \(\varphi = \pi/2\) to \(|1\rangle\):
Leaves \(|0\rangle\) unchanged and adds a relative phase of \(\varphi = \pi/2\) to \(|1\rangle\):
\( S|0\rangle = |0\rangle \)
\( S|1\rangle = i|1\rangle \)
\( S|1\rangle = i|1\rangle \)
$$ S = \begin{pmatrix}1 & 0 \\0 & i\end{pmatrix} $$
×
The phase shift is a family of single-qubit gates that map the basis states
\( |0\rangle \mapsto |0\rangle \) and \(|1\rangle \mapsto e^{i\varphi }|1\rangle \).
Applying the gate modifies the phase of the quantum state. This is equivalent to a rotation about
the \(z\)-axis on the Bloch sphere by \(\varphi\) radians.
The phase shift gates are represented by the matrix (\(P^{\dagger }(\varphi )=P(-\varphi )\)):
$$ P(\varphi )={\begin{pmatrix}1&0\\0&e^{i\varphi }\end{pmatrix}}$$
where \(\varphi\) is the phase shift with the \(2\pi\) period.
Leaves \(|0\rangle\) unchanged and adds a relative phase of \(\varphi = \pi/4\) to \(|1\rangle\):
Leaves \(|0\rangle\) unchanged and adds a relative phase of \(\varphi = \pi/4\) to \(|1\rangle\):
\( T|0\rangle = |0\rangle \)
\( T|1\rangle = e^{i\pi/4}|1\rangle \)
\( T|1\rangle = e^{i\pi/4}|1\rangle \)
$$ T = \begin{pmatrix}1 & 0 \\0 & e^{i\pi/4}\end{pmatrix} $$
×
Controlled-NOT (CNOT or controlled Pauli-X) is a two-qubit gate. The second (target) qubit is flipped only
if the first (control) qubit is \(|1\rangle\).
$$
|00\rangle \rightarrow |00\rangle,\quad
|01\rangle \rightarrow |01\rangle,\quad
|10\rangle \rightarrow |11\rangle,\quad
|11\rangle \rightarrow |10\rangle
$$
$$ \text{CNOT} =\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix} $$
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Single-Qubit State Evolution on the Bloch Sphere
For visualizing the evolution of a single-qubit state, select a quantum gate to apply to the current state.
The gate transforms the state vector, represented as a point on the Bloch sphere.
The movement of this point shows how the amplitudes and relative phase change under single-qubit operations.
\(\theta\):
\(\phi\):
$$
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle: \alpha = 1, \beta = 0
$$
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Quantum Cryptography: Applications
Quantum Key Distribution (QDP) is based on the fact that measuring an unknown quantum state inevitably disturbs it
Main Categories of Quantum Key Distribution Protocols
| Category | Protocols |
|---|---|
| Prepare-and-Measure | BB84, B92, DPS-QKD, SARG04, COW |
| Entanglement-Based | E91, BBM92 |
| Enhanced Practical Security | MDI-QKD, Twin-Field QKD |
Recent research has focused on improving the security and practical deployment of QKD through protocols such as Measurement-Device-Independent QKD (MDI-QKD), Twin-Field QKD (TF-QKD), Device-Independent QKD (DI-QKD), and Continuous-Variable QKD (CV-QKD). Among these protocols, BB84 is the first and most widely studied QKD protocol and serves as the starting point for understanding quantum cryptography.
Among the quantum key distribution protocols introduced above, we will use BB84 as an example to illustrate the fundamental principles of quantum cryptography. First proposed by Charles Bennett and Gilles Brassard in 1984, the BB84 protocol enables two parties to establish a shared secret key whose security relies on the fundamental laws of quantum mechanics.
|
Goal Establish a shared secret key while detecting eavesdropping Quantum Resource Polarization states of single photons encoded in two measurement bases Principle Measuring an unknown quantum state disturbs it, introducing detectable errors Advantages • Detects eavesdropping • Security based on quantum mechanics • Solves the key distribution problem Limitations • Requires specialized quantum hardware • Limited communication distance • Generates secret keys rather than encrypting messages directly |
Image generated with AI assistance
|
After the BB84 protocol has established a shared secret key, the key can be used with a symmetric encryption algorithm such as AES or the One-Time Pad to protect confidential information.
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Exercise: BB84 Quantum Key Distribution Protocol
Objective: Learn how the BB84 quantum key distribution protocol enables Alice and Bob to generate a shared secret key and determine
whether it can be used securely or should be discarded due to possible eavesdropping.
The maine stages of the BB84 protocol (see figure above):
- Alice prepares photons: Alice generates random bits and randomly chooses a basis (+ or x) for each bit. The resulting quantum states are encoded into photons and transmitted to Bob.
- Bob measures photons: For each received photon, Bob randomly chooses a measurement basis (+ or x) and records the result.
- Basis comparison: Alice and Bob publicly reveal only the bases they used. The bit values remain secret.
- Sifting: Alice and Bob keep only the positions where their bases match. These bits form the sifted key.
- Error estimation: Alice and Bob publicly compare a small random subset of the sifted key to estimate the error rate.
- Decision: If the error rate is sufficiently low, the remaining bits can be used to generate a shared secret key. Otherwise, the key is discarded.
- Secure key generation: The final shared key can then be used with a symmetric encryption method, such as AES or the One-Time Pad.
Choose the number of photons used for the BB84 protocol:
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Quantum Gate Challenge
To demonstrate how the principles of quantum computing can be applied to encryption and information protection,
we propose the Quantum Gate Challenge as an illustrative and pedagogical example.
The goal of this challenge is to find the secret key used to encrypt a phrase. The key consists of a sequence of quantum gates forming a quantum circuit that serves as the encryption mechanism. By applying the key (the chosen encryption quantum circuit) in reverse order, the original phrase can be recovered.
The goal of this challenge is to find the secret key used to encrypt a phrase. The key consists of a sequence of quantum gates forming a quantum circuit that serves as the encryption mechanism. By applying the key (the chosen encryption quantum circuit) in reverse order, the original phrase can be recovered.
We start by preparing the quantum register according to the classical message, encoding classical bits into qubits:
- If the bit is 0, keep the qubit in the state |0⟩.
- If the bit is 1, apply an X gate (bit-flip gate) to transform |0⟩ into |1⟩.
2.
The phrase is encrypted using a key 
composed of two randomly selected gates:
composed of two randomly selected gates:
3. Enter the letters corresponding to Gate 1 and Gate 2 to find the key.
For decryption, the gates are applied in the inverse order:
4. Measurement collapses the quantum state into classical bits;
the result reveals the original message if the correct key was used.
Warning: According to the no-cloning theorem, each new attempt
to find the key for the chosen phrase requires repeating stages 1–3.
The No-Cloning Theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This guarantees security because any attempt to measure or clone a quantum state disturbs it, revealing the presence of an eavesdropper.
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