Lecture Notes on Quantum Information: Super Dense Coding, Teleportation, Error Correction, and Entropy

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February 5, 2025

Introduction

This lecture covers several key topics in quantum information theory, building upon previous discussions and setting the stage for future topics on quantum computer architecture. We will explore super dense coding, a protocol to transmit two classical bits using a single qubit, quantum teleportation, enabling the transfer of quantum states, quantum error correction, addressing the challenge of noise in quantum systems, and finally, an introduction to quantum information theory and entropy, contrasting it with classical Shannon entropy. These topics are fundamental to understanding quantum communication and computation.

Super Dense Coding

Overview of Super Dense Coding

Super dense coding is a quantum communication protocol that enables Alice to transmit two classical bits to Bob by sending only a single qubit. This remarkable feat is achieved by exploiting pre-shared entanglement between Alice and Bob. The cornerstone of this protocol is that Alice and Bob initially possess a maximally entangled state, specifically a Bell pair.

Protocol Setup: Shared Entanglement

Prior to initiating communication, Alice and Bob establish a shared entangled pair of qubits. Let us consider they share the Bell state $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$. Alice is given the first qubit, and Bob receives the second qubit of this entangled pair. This entangled pair serves as the quantum resource for super dense coding.

Encoding by Alice

To transmit two classical bits, Alice applies a specific unitary operation to her qubit. The choice of operation depends on the two bits she intends to send:

To send the classical bits “00”: Alice applies the Identity operator ($I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$).
To send the classical bits “01”: Alice applies the Pauli-Z operator ($Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$).
To send the classical bits “10”: Alice applies the Pauli-X operator ($X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$).
To send the classical bits “11”: Alice applies the $iY$ operator ($iY = i\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$).

After applying the appropriate unitary operation, Alice transmits her qubit to Bob through a quantum channel.

Decoding by Bob and Measurement

Upon receiving the qubit from Alice, Bob now holds both qubits. The combined state is now one of the four Bell states, depending on Alice’s operation. To decode the message, Bob performs a measurement that distinguishes between these four Bell states.

For instance, if Alice applies the Pauli-Z gate to send “01”, the initial Bell pair $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$ is transformed into: \[(Z \otimes I) \left|{\beta_{00}}\right\rangle = (Z \otimes I) \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle) = \frac{1}{\sqrt{2}}(Z\left|{0}\right\rangle \otimes \left|{0}\right\rangle + Z\left|{1}\right\rangle \otimes \left|{1}\right\rangle) = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle - \left|{11}\right\rangle) = \left|{\beta_{01}}\right\rangle.\] Similarly, applying $X$ and $iY$ gates results in $\left|{\beta_{10}}\right\rangle$ and $\left|{\beta_{11}}\right\rangle$ respectively. The complete set of Bell states is: \[\begin{aligned}\left|{\beta_{00}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle) \\\left|{\beta_{01}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{00}\right\rangle - \left|{11}\right\rangle) \\\left|{\beta_{10}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{01}\right\rangle + \left|{10}\right\rangle) \\\left|{\beta_{11}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{01}\right\rangle - \left|{10}\right\rangle)\end{aligned}\] To differentiate these Bell states, Bob can perform a measurement in the Bell basis. Alternatively, Bob can apply a quantum circuit consisting of a CNOT gate followed by Hadamard gates on both qubits, and then measure in the computational basis $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$.

Let’s examine the decoding circuit for the case where Alice sent “00” (Identity gate), so the state is $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$. Bob applies a CNOT gate with the first qubit as control and the second as target: \[CNOT_{12} \left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(CNOT_{12}\left|{00}\right\rangle + CNOT_{12}\left|{11}\right\rangle) = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{10}\right\rangle).\] Next, Bob applies a Hadamard gate to the first qubit: \[(H \otimes I) CNOT_{12} \left|{\beta_{00}}\right\rangle = (H \otimes I) \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{10}\right\rangle) = \frac{1}{\sqrt{2}}(H\left|{0}\right\rangle\left|{0}\right\rangle + H\left|{1}\right\rangle\left|{1}\right\rangle) = \left|{00}\right\rangle.\] By performing similar calculations for the other Bell states, it can be shown that measuring in the computational basis after applying the CNOT and Hadamard gates allows Bob to uniquely determine which of the four Bell states he has, and thus decode the two classical bits Alice intended to send. The measurement outcomes directly correspond to Alice’s input bits.

Efficiency and Resource Considerations

Super dense coding achieves a transmission of two classical bits per qubit, doubling the classical capacity of a qubit in terms of classical information conveyed. However, this efficiency relies critically on the pre-shared entanglement. The initial distribution of the Bell pair is a necessary resource. Without pre-shared entanglement, this protocol would not be possible.

Security Implications

An eavesdropper, Eve, intercepting the qubit transmitted by Alice gains access to only one qubit of the entangled system. Without possessing Bob’s qubit, Eve cannot fully decode the two classical bits encoded by Alice. Furthermore, any attempt by Eve to measure or interact with the transmitted qubit will likely disturb the entanglement between Alice and Bob’s qubits. Alice and Bob can implement security checks by verifying the entanglement fidelity of their shared pairs. Deviations from expected correlations, due to Eve’s intervention, can alert Alice and Bob to potential eavesdropping, similar to security measures in quantum key distribution protocols like BB84. Eve would need to measure or interact with the qubit in a way that is detectable to Alice and Bob to gain information, thus compromising her stealth.

Entanglement is Key: Relies on pre-shared Bell pairs between sender and receiver.
Increased Capacity: Transmits two classical bits using a single qubit.
Decoding via Bell Measurement: Receiver needs to distinguish Bell states to decode the message.
Security: Offers inherent security against eavesdropping due to entanglement sensitivity.
Resource Trade-off: Exchanges pre-shared entanglement for increased communication efficiency.

Quantum Teleportation

Introduction to Quantum Teleportation

Quantum teleportation is a protocol that enables the transfer of an unknown quantum state from Alice to Bob without physically transmitting the quantum system itself. This process relies on pre-shared entanglement and classical communication. Suppose Alice possesses a qubit in an unknown state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ that she wishes to teleport to Bob.

Steps of the Quantum Teleportation Protocol

The quantum teleportation protocol consists of the following steps:

Preparation of Entangled Pair: Alice and Bob pre-share a maximally entangled Bell pair. A common choice is $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$. Alice retains the first qubit (qubit 1), and Bob receives the second qubit (qubit 2).
Bell State Measurement by Alice: Alice interacts her qubit in the unknown state $\left|{\psi}\right\rangle$ (qubit 3) with her half of the Bell pair (qubit 1). She applies a CNOT gate with qubit 3 as the control and qubit 1 as the target, followed by a Hadamard gate on qubit 3.
Measurement in Computational Basis: Alice measures qubits 3 and 1 in the computational basis $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$. This measurement yields two classical bits, say $c_1$ and $c_2$.
Classical Communication to Bob: Alice communicates the measurement outcomes $c_1$ and $c_2$ to Bob via a classical communication channel.
Conditional Quantum Operations by Bob: Based on the classical bits $c_1$ and $c_2$ received from Alice, Bob applies specific Pauli gates to his qubit (qubit 2) to recover the original state $\left|{\psi}\right\rangle$. The operations are conditional:
- If $c_1 = 0, c_2 = 0$: Bob applies the Identity gate ($I$).
- If $c_1 = 0, c_2 = 1$: Bob applies the Pauli-X gate ($X$).
- If $c_1 = 1, c_2 = 0$: Bob applies the Pauli-Z gate ($Z$).
- If $c_1 = 1, c_2 = 1$: Bob applies the Pauli-ZX gate ($ZX$).
After these operations, Bob’s qubit (qubit 2) is in the state $\left|{\psi}\right\rangle$.

Quantum Circuit for Teleportation

The quantum circuit representing the teleportation protocol is depicted below:

Quantum Teleportation Circuit. Qubit 3 is the unknown state (|{}) to be teleported. Qubits 1 and 2 are the entangled Bell pair. (M) denotes measurement in the computational basis, and (X^{c_2}) and (Z^{c_1}) are Pauli gates applied conditionally based on classical measurement outcomes (c_2) and (c_1).

In this circuit, $X^{c_2}$ denotes applying the Pauli-X gate if the classical bit $c_2 = 1$ and doing nothing if $c_2 = 0$. Similarly, $Z^{c_1}$ denotes applying the Pauli-Z gate if $c_1 = 1$ and doing nothing if $c_1 = 0$. The output state $\left|{\psi'}\right\rangle$ on qubit 2 is the teleported state, ideally identical to the input state $\left|{\psi}\right\rangle$.

Entanglement as a Quantum Resource

The pre-shared Bell pair is the essential quantum resource that enables teleportation. Entanglement creates a quantum correlation between Alice’s and Bob’s qubits, allowing for the non-local transfer of quantum information. Initially, the combined state of the three qubits is: \[\left|{\Psi_0}\right\rangle = \left|{\psi}\right\rangle \otimes \left|{\beta_{00}}\right\rangle = (\alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle) \otimes \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle) = \frac{1}{\sqrt{2}} (\alpha\left|{000}\right\rangle + \alpha\left|{011}\right\rangle + \beta\left|{100}\right\rangle + \beta\left|{111}\right\rangle).\] After Alice applies the CNOT and Hadamard gates and performs measurements, the entanglement is utilized to project Bob’s qubit into a state that, after classical corrections, becomes the original state $\left|{\psi}\right\rangle$.

Classical Communication and State Reconstruction

Alice’s measurements are crucial as they project the quantum state into classical information (bits $c_1$ and $c_2$). This classical information is then communicated to Bob. Crucially, Bob cannot obtain the state $\left|{\psi}\right\rangle$ until he receives and applies the corrections based on Alice’s classical measurement outcomes. The classical communication step ensures that no information is transmitted faster than light, preserving causality. The measurement outcomes from Alice act as instructions for Bob to reconstruct the teleported state from his half of the entangled pair.

Causality is Preserved

Quantum teleportation does not violate the principle of causality. Although the quantum state is transferred from Alice to Bob, no information travels faster than the speed of light. Alice’s measurement outcomes are random, and these outcomes are necessary for Bob to reconstruct the original quantum state. The transmission of classical information from Alice to Bob is limited by the speed of light. Therefore, teleportation is not a method for faster-than-light communication; it is a protocol for quantum state transfer utilizing pre-existing entanglement and classical communication.

State Transfer, Not Copying: Teleportation moves a quantum state, it does not create a copy at the sender’s location, consistent with the no-cloning theorem.
Requires Entanglement and Classical Communication: Relies on pre-shared entanglement and classical communication channel.
Bell Basis Measurement: Alice performs a Bell state measurement to project the state and obtain classical information.
Conditional Correction by Bob: Bob applies corrections based on classical information from Alice to retrieve the original quantum state.
Causality Compliant: Does not violate causality as classical communication is necessary for state reconstruction.

Quantum Error Correction

Classical Error Correction: Repetition Code

In classical communication, noise in the channel can cause bit-flip errors. A fundamental technique to mitigate these errors is error correction coding. One of the simplest examples is the repetition code. To transmit a logical bit, say ‘0’, we encode it as ‘000’, and ‘1’ is encoded as ‘111’. At the receiver end, if errors occur, we can decode the intended bit by majority voting. For instance, if ‘000’ is sent and ‘010’ is received, majority voting correctly decodes it back to ‘0’.

To analyze the effectiveness, consider a binary symmetric channel with a bit-flip probability $p$. For the repetition code, an error in decoding occurs if two or more bits in the triplet are flipped. The probability of this happening is the sum of probabilities of exactly two flips and exactly three flips: \[P(\text{error}) = \binom{3}{2} p^2 (1-p)^1 + \binom{3}{3} p^3 (1-p)^0 = 3p^2(1-p) + p^3 = 3p^2 - 3p^3 + p^3 = 3p^2 - 2p^3 = p^2(3 - 2p).\] For $p < 1/2$, $3 - 2p > 2$, so $3p - 2p^2 > p$, and for sufficiently small $p$, $3p^2 - 2p^3 < p$. Specifically, for $p < 1/2$, we can show that $3p^2 - 2p^3 < p$ is equivalent to $3p - 2p^2 < 1$, or $2p^2 - 3p + 1 > 0$, which factors as $(2p - 1)(p - 1) > 0$. This holds for $p < 1/2$ or $p > 1$. Since we consider $p < 1/2$, the probability of error is reduced from $p$ to $3p^2 - 2p^3$.

Quantum Error Correction: Three-Qubit Bit-Flip Code

Quantum systems are even more susceptible to errors than classical systems, suffering from decoherence and various types of quantum noise, including bit-flip and phase-flip errors. To protect quantum information, we need quantum error correction codes. The simplest quantum error correction code is the three-qubit bit-flip code, analogous to the classical repetition code.

Definition 1 (Three-Qubit Bit-Flip Code). To encode a qubit state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ against bit-flip errors, we map the logical qubits as follows: \[\left|{0}\right\rangle_L \equiv \left|{\overline{0}}\right\rangle = \left|{000}\right\rangle, \quad \left|{1}\right\rangle_L \equiv \left|{\overline{1}}\right\rangle = \left|{111}\right\rangle.\] Thus, the encoded state for $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ is: \[\left|{\psi}\right\rangle_L = \alpha\left|{\overline{0}}\right\rangle + \beta\left|{\overline{1}}\right\rangle = \alpha\left|{000}\right\rangle + \beta\left|{111}\right\rangle.\]

This encoding can be implemented using quantum circuits. One such circuit, as suggested in the transcript, uses CNOT gates. Starting with $\left|{\psi}\right\rangle\left|{00}\right\rangle$, we can apply CNOT gates to achieve the encoding: \[\begin{aligned}(\text{CNOT}_{12} \otimes I) (\text{CNOT}_{13} \otimes I) (\left|{\psi}\right\rangle\left|{00}\right\rangle) &= (\text{CNOT}_{12} \otimes I) (\text{CNOT}_{13} \otimes I) (\alpha\left|{0}\right\rangle\left|{00}\right\rangle + \beta\left|{1}\right\rangle\left|{00}\right\rangle) \\&= (\text{CNOT}_{12} \otimes I) (\alpha\left|{0}\right\rangle\left|{00}\right\rangle + \beta\left|{1}\right\rangle\left|{10}\right\rangle) \\&= \alpha\left|{0}\right\rangle\left|{00}\right\rangle + \beta\left|{1}\right\rangle\left|{11}\right\rangle = \alpha\left|{000}\right\rangle + \beta\left|{111}\right\rangle = \left|{\psi}\right\rangle_L.\end{aligned}\] Here, $\text{CNOT}_{ij}$ denotes a CNOT gate with qubit $i$ as control and qubit $j$ as target.

Syndrome Measurement for Error Detection

To detect bit-flip errors without directly measuring the encoded qubit in the computational basis (which would collapse the superposition), we perform syndrome measurements. For the three-qubit bit-flip code, we can measure the parity of pairs of qubits using the operators $Z_1Z_2$ and $Z_2Z_3$. These operators commute with the encoded subspace and can detect bit-flip errors.

Consider the action of $Z_1Z_2$ and $Z_2Z_3$ on the encoded states $\left|{\overline{0}}\right\rangle = \left|{000}\right\rangle$ and $\left|{\overline{1}}\right\rangle = \left|{111}\right\rangle$: \[\begin{aligned}Z_1Z_2 \left|{000}\right\rangle &= \left|{000}\right\rangle, \quad Z_1Z_2 \left|{111}\right\rangle = \left|{111}\right\rangle \\Z_2Z_3 \left|{000}\right\rangle &= \left|{000}\right\rangle, \quad Z_2Z_3 \left|{111}\right\rangle = \left|{111}\right\rangle\end{aligned}\] If a bit flip occurs, for example, on the first qubit, transforming $\left|{000}\right\rangle \rightarrow \left|{100}\right\rangle$, the measurements change: \[\begin{aligned}Z_1Z_2 \left|{100}\right\rangle &= (Z_1 \left|{1}\right\rangle) \otimes (Z_2 \left|{0}\right\rangle) \otimes \left|{0}\right\rangle = (-\left|{1}\right\rangle) \otimes \left|{0}\right\rangle \otimes \left|{0}\right\rangle = -\left|{100}\right\rangle \\Z_2Z_3 \left|{100}\right\rangle &= \left|{100}\right\rangle\end{aligned}\] By measuring the eigenvalues of $Z_1Z_2$ and $Z_2Z_3$, we obtain error syndromes that indicate which qubit, if any, has flipped. The possible syndromes and corresponding errors are:

Syndrome ( $+1, +1$ ): No error.
Syndrome ( $-1, +1$ ): Error on qubit 1.
Syndrome ( $-1, -1$ ): Error on qubit 2.
Syndrome ( $+1, -1$ ): Error on qubit 3.

Here, the syndrome is represented as (eigenvalue of $Z_1Z_2$, eigenvalue of $Z_2Z_3$). These measurements do not distinguish between $\left|{\overline{0}}\right\rangle$ and $\left|{\overline{1}}\right\rangle$ within the code space, only detect errors.

Error Correction

Based on the error syndrome, we can apply a corrective operation to revert the bit-flip error. If the syndrome indicates an error on qubit $i$, we apply a Pauli-X gate $X_i$ to qubit $i$. For example, if the syndrome is $(-1, +1)$, indicating an error on qubit 1, and the state is $\left|{100}\right\rangle$, applying $X_1$ corrects the error: $X_1\left|{100}\right\rangle = \left|{000}\right\rangle = \left|{\overline{0}}\right\rangle$. If the syndrome is $(+1, +1)$, no correction is needed (Identity operation).

No-Cloning Theorem and Error Correction

Quantum error correction is consistent with the no-cloning theorem. We are not creating copies of an arbitrary unknown quantum state. Instead, we are encoding one logical qubit into a redundant physical system (three qubits in this case) to protect the quantum information from noise. The redundancy allows us to detect and correct errors, but it does not enable the cloning of the encoded quantum state. The no-cloning theorem prohibits perfect copying of unknown quantum states, and quantum error correction operates within these constraints by focusing on protecting quantum information rather than replicating it.

Redundancy for Protection: Encodes one logical qubit into multiple physical qubits to protect against errors.
Error Syndrome Measurement: Uses parity check measurements (e.g., $Z_1Z_2, Z_2Z_3$) to detect errors without collapsing the encoded information.
Error Correction based on Syndrome: Applies corrective operations (e.g., Pauli-X gates) based on the measured error syndrome to revert errors.
Bit-Flip Correction: The three-qubit code specifically corrects bit-flip errors. Other codes are needed for phase-flip and general errors.
No Cloning Issue: Quantum error correction does not violate the no-cloning theorem; it protects information, not copies states.

Quantum Information Theory and Entropy

Introduction to Quantum Information Theory and Entropy

This section introduces the fundamental concepts of quantum information theory, focusing on entropy as a measure of information and uncertainty in quantum systems. We will contrast quantum entropy with its classical counterpart, Shannon entropy, and discuss the role of density operators in defining quantum entropy. A useful resource for further reading is the book "Quantum Thinking for Computer Scientists," particularly the chapters starting around page 288, which provide a concise introduction to these topics.

Definition 2 (Classical Shannon Entropy). In classical information theory, Shannon entropy quantifies the average amount of information gained when observing the outcome of a random variable. For a discrete random variable $X$ with possible outcomes $\{x_1, x_2, ..., x_n\}$ and corresponding probabilities $\{p_1, p_2, ..., p_n\}$, the Shannon entropy $H(X)$ is defined as: \[H(X) = - \sum_{i=1}^{n} p_i \log_2 p_i.\] Shannon entropy measures the average uncertainty associated with the random variable $X$. It represents the average number of bits needed to describe the outcome of $X$.

Definition 3 (Density Operators). Density operators are essential for describing quantum states, especially when dealing with mixed states, which are probabilistic ensembles of pure quantum states, or when the quantum state is not precisely known. For an ensemble of pure states $\{\left|{\psi_i}\right\rangle\}$ with corresponding probabilities $\{p_i\}$, the density operator $\rho$ is given by: \[\rho = \sum_{i} p_i \left|{\psi_i}\right\rangle\left\langle{\psi_i}\right|.\] Density operators provide a comprehensive description of quantum states, encompassing both pure and mixed states, and are fundamental in quantum statistical mechanics and quantum information theory.

Definition 4 (Von Neumann Entropy). Von Neumann entropy is the quantum analogue of Shannon entropy, extending the concept of entropy to quantum states described by density operators. For a density operator $\rho$, the Von Neumann entropy $S(\rho)$ is defined as: \[S(\rho) = - \text{Tr}(\rho \log_2 \rho) = - \sum_{i} \lambda_i \log_2 \lambda_i,\] where $\{\lambda_i\}$ are the eigenvalues of the density operator $\rho$. The trace operation ensures that the entropy is independent of the choice of basis. Von Neumann entropy quantifies the quantum uncertainty or mixedness of a quantum state. For a pure state $\left|{\psi}\right\rangle$, the density operator is $\rho = \left|{\psi}\right\rangle\left\langle{\psi}\right|$, which has one eigenvalue equal to 1 and all others 0, resulting in $S(\rho) = 0$, indicating zero entropy for pure states.

Relationship between Shannon and Von Neumann Entropy

Von Neumann entropy generalizes Shannon entropy. In cases where the density operator is diagonal in a particular basis, and measurements are performed in the same basis, Von Neumann entropy reduces to Shannon entropy. Consider the spectral decomposition of the density operator $\rho = \sum_i \lambda_i\lambda_i \left|{i}\right\rangle\left\langle{i}\right|$, where $\{\left|{i}\right\rangle\}$ are the eigenvectors and $\{\lambda_i\}$ are the eigenvalues. If we consider measurements in the basis $\{\left|{i}\right\rangle\}$, the probabilities of obtaining each outcome $\left|{i}\right\rangle$ are given by the eigenvalues $\lambda_i$. In this scenario, the Von Neumann entropy $S(\rho) = - \sum_i \lambda_i \log_2 \lambda_i$ is formally equivalent to the Shannon entropy of the probability distribution $\{\lambda_i\}$.

The example in the lecture transcript highlights that Shannon entropy can be basis-dependent if one considers probabilities of measurement outcomes in a basis different from the eigenbasis of the density operator. However, Von Neumann entropy, defined through eigenvalues, provides a basis-independent measure of quantum entropy. It represents the minimum possible Shannon entropy achievable by measuring the quantum state in any basis, which is obtained when measuring in the eigenbasis of the density operator.

Quantum Channel Capacity and Quantum Shannon Theorems

Quantum Shannon theorems extend the concepts of classical channel capacity and Shannon’s coding theorems to the quantum domain. These theorems define the fundamental limits on the rate at which quantum information can be reliably transmitted over quantum channels. Analogous to classical information theory, there are quantum Shannon theorems for both noiseless and noisy quantum channels. These theorems establish the capacity of quantum channels to transmit qubits, considering the unique properties of quantum information and noise.

Shannon Entropy: Measures uncertainty of classical random variables, basis-dependent in quantum context if measurement basis is not specified relative to the state.
Von Neumann Entropy: Measures quantum uncertainty of quantum states (density operators), basis-independent, generalizes Shannon entropy.
Pure States: Von Neumann entropy is zero for pure states, reflecting no quantum uncertainty.
Mixed States: Von Neumann entropy is greater than zero for mixed states, quantifying the degree of mixedness.
Connection: When a quantum state is measured in the eigenbasis of its density operator, Von Neumann entropy reduces to the Shannon entropy of the resulting probability distribution of measurement outcomes.

Conclusion

This lecture has explored fundamental concepts in quantum information, highlighting the power and intricacies of quantum communication and computation. We began with super dense coding, demonstrating how entanglement can double the classical information capacity of a qubit, achieving more efficient communication. Quantum teleportation was then examined, illustrating the transfer of unknown quantum states through entanglement and classical communication, crucially without violating causality. We addressed the challenge of noise in quantum systems with an introduction to quantum error correction, specifically the three-qubit bit-flip code, showcasing the principles of encoding, syndrome measurement, and error correction. Finally, we introduced quantum information theory and Von Neumann entropy, contrasting it with classical Shannon entropy and establishing it as the fundamental measure of uncertainty in quantum states.

These topics—super dense coding, teleportation, error correction, and quantum entropy—collectively form a cornerstone for understanding advanced quantum technologies. Future lectures will delve deeper into related areas, including quantum channel capacities, more sophisticated quantum error correction techniques, and the architectural principles of quantum computers, building upon the foundational knowledge established here.

--- title: "Lecture Notes on Quantum Information: Super Dense Coding, Teleportation, Error Correction, and Entropy" author: "Your Name" date: "2025-02-05" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This lecture covers several key topics in quantum information theory, building upon previous discussions and setting the stage for future topics on quantum computer architecture. We will explore super dense coding, a protocol to transmit two classical bits using a single qubit, quantum teleportation, enabling the transfer of quantum states, quantum error correction, addressing the challenge of noise in quantum systems, and finally, an introduction to quantum information theory and entropy, contrasting it with classical Shannon entropy. These topics are fundamental to understanding quantum communication and computation. # Super Dense Coding ## Overview of Super Dense Coding Super dense coding is a quantum communication protocol that enables Alice to transmit two classical bits to Bob by sending only a single qubit. This remarkable feat is achieved by exploiting pre-shared entanglement between Alice and Bob. The cornerstone of this protocol is that Alice and Bob initially possess a maximally entangled state, specifically a Bell pair. ## Protocol Setup: Shared Entanglement Prior to initiating communication, Alice and Bob establish a shared entangled pair of qubits. Let us consider they share the Bell state $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$. Alice is given the first qubit, and Bob receives the second qubit of this entangled pair. This entangled pair serves as the quantum resource for super dense coding. ## Encoding by Alice To transmit two classical bits, Alice applies a specific unitary operation to her qubit. The choice of operation depends on the two bits she intends to send: - To send the classical bits "00": Alice applies the Identity operator ($I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$). - To send the classical bits "01": Alice applies the Pauli-Z operator ($Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$). - To send the classical bits "10": Alice applies the Pauli-X operator ($X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$). - To send the classical bits "11": Alice applies the $iY$ operator ($iY = i\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$). After applying the appropriate unitary operation, Alice transmits her qubit to Bob through a quantum channel. ## Decoding by Bob and Measurement Upon receiving the qubit from Alice, Bob now holds both qubits. The combined state is now one of the four Bell states, depending on Alice's operation. To decode the message, Bob performs a measurement that distinguishes between these four Bell states. For instance, if Alice applies the Pauli-Z gate to send "01", the initial Bell pair $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$ is transformed into: $$(Z \otimes I) \left|{\beta_{00}}\right\rangle = (Z \otimes I) \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle) = \frac{1}{\sqrt{2}}(Z\left|{0}\right\rangle \otimes \left|{0}\right\rangle + Z\left|{1}\right\rangle \otimes \left|{1}\right\rangle) = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle - \left|{11}\right\rangle) = \left|{\beta_{01}}\right\rangle.$$ Similarly, applying $X$ and $iY$ gates results in $\left|{\beta_{10}}\right\rangle$ and $\left|{\beta_{11}}\right\rangle$ respectively. The complete set of Bell states is: $$\begin{aligned}\left|{\beta_{00}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle) \\\left|{\beta_{01}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{00}\right\rangle - \left|{11}\right\rangle) \\\left|{\beta_{10}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{01}\right\rangle + \left|{10}\right\rangle) \\\left|{\beta_{11}}\right\rangle &= \frac{1}{\sqrt{2}}(\left|{01}\right\rangle - \left|{10}\right\rangle)\end{aligned}$$ To differentiate these Bell states, Bob can perform a measurement in the Bell basis. Alternatively, Bob can apply a quantum circuit consisting of a CNOT gate followed by Hadamard gates on both qubits, and then measure in the computational basis $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$. Let's examine the decoding circuit for the case where Alice sent "00" (Identity gate), so the state is $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$. Bob applies a CNOT gate with the first qubit as control and the second as target: $$CNOT_{12} \left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(CNOT_{12}\left|{00}\right\rangle + CNOT_{12}\left|{11}\right\rangle) = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{10}\right\rangle).$$ Next, Bob applies a Hadamard gate to the first qubit: $$(H \otimes I) CNOT_{12} \left|{\beta_{00}}\right\rangle = (H \otimes I) \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{10}\right\rangle) = \frac{1}{\sqrt{2}}(H\left|{0}\right\rangle\left|{0}\right\rangle + H\left|{1}\right\rangle\left|{1}\right\rangle) = \left|{00}\right\rangle.$$ By performing similar calculations for the other Bell states, it can be shown that measuring in the computational basis after applying the CNOT and Hadamard gates allows Bob to uniquely determine which of the four Bell states he has, and thus decode the two classical bits Alice intended to send. The measurement outcomes directly correspond to Alice's input bits. ## Efficiency and Resource Considerations Super dense coding achieves a transmission of two classical bits per qubit, doubling the classical capacity of a qubit in terms of classical information conveyed. However, this efficiency relies critically on the pre-shared entanglement. The initial distribution of the Bell pair is a necessary resource. Without pre-shared entanglement, this protocol would not be possible. ## Security Implications An eavesdropper, Eve, intercepting the qubit transmitted by Alice gains access to only one qubit of the entangled system. Without possessing Bob's qubit, Eve cannot fully decode the two classical bits encoded by Alice. Furthermore, any attempt by Eve to measure or interact with the transmitted qubit will likely disturb the entanglement between Alice and Bob's qubits. Alice and Bob can implement security checks by verifying the entanglement fidelity of their shared pairs. Deviations from expected correlations, due to Eve's intervention, can alert Alice and Bob to potential eavesdropping, similar to security measures in quantum key distribution protocols like BB84. Eve would need to measure or interact with the qubit in a way that is detectable to Alice and Bob to gain information, thus compromising her stealth. ::: tcolorbox - **Entanglement is Key**: Relies on pre-shared Bell pairs between sender and receiver. - **Increased Capacity**: Transmits two classical bits using a single qubit. - **Decoding via Bell Measurement**: Receiver needs to distinguish Bell states to decode the message. - **Security**: Offers inherent security against eavesdropping due to entanglement sensitivity. - **Resource Trade-off**: Exchanges pre-shared entanglement for increased communication efficiency. ::: # Quantum Teleportation ## Introduction to Quantum Teleportation Quantum teleportation is a protocol that enables the transfer of an unknown quantum state from Alice to Bob without physically transmitting the quantum system itself. This process relies on pre-shared entanglement and classical communication. Suppose Alice possesses a qubit in an unknown state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ that she wishes to teleport to Bob. ## Steps of the Quantum Teleportation Protocol The quantum teleportation protocol consists of the following steps: 1. **Preparation of Entangled Pair**: Alice and Bob pre-share a maximally entangled Bell pair. A common choice is $\left|{\beta_{00}}\right\rangle = \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle)$. Alice retains the first qubit (qubit 1), and Bob receives the second qubit (qubit 2). 2. **Bell State Measurement by Alice**: Alice interacts her qubit in the unknown state $\left|{\psi}\right\rangle$ (qubit 3) with her half of the Bell pair (qubit 1). She applies a CNOT gate with qubit 3 as the control and qubit 1 as the target, followed by a Hadamard gate on qubit 3. 3. **Measurement in Computational Basis**: Alice measures qubits 3 and 1 in the computational basis $\{\left|{0}\right\rangle, \left|{1}\right\rangle\}$. This measurement yields two classical bits, say $c_1$ and $c_2$. 4. **Classical Communication to Bob**: Alice communicates the measurement outcomes $c_1$ and $c_2$ to Bob via a classical communication channel. 5. **Conditional Quantum Operations by Bob**: Based on the classical bits $c_1$ and $c_2$ received from Alice, Bob applies specific Pauli gates to his qubit (qubit 2) to recover the original state $\left|{\psi}\right\rangle$. The operations are conditional: - If $c_1 = 0, c_2 = 0$: Bob applies the Identity gate ($I$). - If $c_1 = 0, c_2 = 1$: Bob applies the Pauli-X gate ($X$). - If $c_1 = 1, c_2 = 0$: Bob applies the Pauli-Z gate ($Z$). - If $c_1 = 1, c_2 = 1$: Bob applies the Pauli-ZX gate ($ZX$). After these operations, Bob's qubit (qubit 2) is in the state $\left|{\psi}\right\rangle$. ## Quantum Circuit for Teleportation The quantum circuit representing the teleportation protocol is depicted below: <figure id="fig:teleportation_circuit"> <figcaption>Quantum Teleportation Circuit. Qubit 3 is the unknown state $\left|{\psi}\right\rangle$ to be teleported. Qubits 1 and 2 are the entangled Bell pair. $M$ denotes measurement in the computational basis, and $X^{c_2}$ and $Z^{c_1}$ are Pauli gates applied conditionally based on classical measurement outcomes $c_2$ and $c_1$.</figcaption> </figure> In this circuit, $X^{c_2}$ denotes applying the Pauli-X gate if the classical bit $c_2 = 1$ and doing nothing if $c_2 = 0$. Similarly, $Z^{c_1}$ denotes applying the Pauli-Z gate if $c_1 = 1$ and doing nothing if $c_1 = 0$. The output state $\left|{\psi'}\right\rangle$ on qubit 2 is the teleported state, ideally identical to the input state $\left|{\psi}\right\rangle$. ## Entanglement as a Quantum Resource The pre-shared Bell pair is the essential quantum resource that enables teleportation. Entanglement creates a quantum correlation between Alice's and Bob's qubits, allowing for the non-local transfer of quantum information. Initially, the combined state of the three qubits is: $$\left|{\Psi_0}\right\rangle = \left|{\psi}\right\rangle \otimes \left|{\beta_{00}}\right\rangle = (\alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle) \otimes \frac{1}{\sqrt{2}}(\left|{00}\right\rangle + \left|{11}\right\rangle) = \frac{1}{\sqrt{2}} (\alpha\left|{000}\right\rangle + \alpha\left|{011}\right\rangle + \beta\left|{100}\right\rangle + \beta\left|{111}\right\rangle).$$ After Alice applies the CNOT and Hadamard gates and performs measurements, the entanglement is utilized to project Bob's qubit into a state that, after classical corrections, becomes the original state $\left|{\psi}\right\rangle$. ## Classical Communication and State Reconstruction Alice's measurements are crucial as they project the quantum state into classical information (bits $c_1$ and $c_2$). This classical information is then communicated to Bob. Crucially, Bob cannot obtain the state $\left|{\psi}\right\rangle$ until he receives and applies the corrections based on Alice's classical measurement outcomes. The classical communication step ensures that no information is transmitted faster than light, preserving causality. The measurement outcomes from Alice act as instructions for Bob to reconstruct the teleported state from his half of the entangled pair. ## Causality is Preserved Quantum teleportation does not violate the principle of causality. Although the quantum state is transferred from Alice to Bob, no information travels faster than the speed of light. Alice's measurement outcomes are random, and these outcomes are necessary for Bob to reconstruct the original quantum state. The transmission of classical information from Alice to Bob is limited by the speed of light. Therefore, teleportation is not a method for faster-than-light communication; it is a protocol for quantum state transfer utilizing pre-existing entanglement and classical communication. ::: tcolorbox - **State Transfer, Not Copying**: Teleportation moves a quantum state, it does not create a copy at the sender's location, consistent with the no-cloning theorem. - **Requires Entanglement and Classical Communication**: Relies on pre-shared entanglement and classical communication channel. - **Bell Basis Measurement**: Alice performs a Bell state measurement to project the state and obtain classical information. - **Conditional Correction by Bob**: Bob applies corrections based on classical information from Alice to retrieve the original quantum state. - **Causality Compliant**: Does not violate causality as classical communication is necessary for state reconstruction. ::: # Quantum Error Correction ## Classical Error Correction: Repetition Code In classical communication, noise in the channel can cause bit-flip errors. A fundamental technique to mitigate these errors is error correction coding. One of the simplest examples is the repetition code. To transmit a logical bit, say '0', we encode it as '000', and '1' is encoded as '111'. At the receiver end, if errors occur, we can decode the intended bit by majority voting. For instance, if '000' is sent and '010' is received, majority voting correctly decodes it back to '0'. To analyze the effectiveness, consider a binary symmetric channel with a bit-flip probability $p$. For the repetition code, an error in decoding occurs if two or more bits in the triplet are flipped. The probability of this happening is the sum of probabilities of exactly two flips and exactly three flips: $$P(\text{error}) = \binom{3}{2} p^2 (1-p)^1 + \binom{3}{3} p^3 (1-p)^0 = 3p^2(1-p) + p^3 = 3p^2 - 3p^3 + p^3 = 3p^2 - 2p^3 = p^2(3 - 2p).$$ For $p < 1/2$, $3 - 2p > 2$, so $3p - 2p^2 > p$, and for sufficiently small $p$, $3p^2 - 2p^3 < p$. Specifically, for $p < 1/2$, we can show that $3p^2 - 2p^3 < p$ is equivalent to $3p - 2p^2 < 1$, or $2p^2 - 3p + 1 > 0$, which factors as $(2p - 1)(p - 1) > 0$. This holds for $p < 1/2$ or $p > 1$. Since we consider $p < 1/2$, the probability of error is reduced from $p$ to $3p^2 - 2p^3$. ## Quantum Error Correction: Three-Qubit Bit-Flip Code Quantum systems are even more susceptible to errors than classical systems, suffering from decoherence and various types of quantum noise, including bit-flip and phase-flip errors. To protect quantum information, we need quantum error correction codes. The simplest quantum error correction code is the three-qubit bit-flip code, analogous to the classical repetition code. ::: definition **Definition 1** (Three-Qubit Bit-Flip Code). *To encode a qubit state $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ against bit-flip errors, we map the logical qubits as follows: $$\left|{0}\right\rangle_L \equiv \left|{\overline{0}}\right\rangle = \left|{000}\right\rangle, \quad \left|{1}\right\rangle_L \equiv \left|{\overline{1}}\right\rangle = \left|{111}\right\rangle.$$ Thus, the encoded state for $\left|{\psi}\right\rangle = \alpha\left|{0}\right\rangle + \beta\left|{1}\right\rangle$ is: $$\left|{\psi}\right\rangle_L = \alpha\left|{\overline{0}}\right\rangle + \beta\left|{\overline{1}}\right\rangle = \alpha\left|{000}\right\rangle + \beta\left|{111}\right\rangle.$$* ::: This encoding can be implemented using quantum circuits. One such circuit, as suggested in the transcript, uses CNOT gates. Starting with $\left|{\psi}\right\rangle\left|{00}\right\rangle$, we can apply CNOT gates to achieve the encoding: $$\begin{aligned}(\text{CNOT}_{12} \otimes I) (\text{CNOT}_{13} \otimes I) (\left|{\psi}\right\rangle\left|{00}\right\rangle) &= (\text{CNOT}_{12} \otimes I) (\text{CNOT}_{13} \otimes I) (\alpha\left|{0}\right\rangle\left|{00}\right\rangle + \beta\left|{1}\right\rangle\left|{00}\right\rangle) \\&= (\text{CNOT}_{12} \otimes I) (\alpha\left|{0}\right\rangle\left|{00}\right\rangle + \beta\left|{1}\right\rangle\left|{10}\right\rangle) \\&= \alpha\left|{0}\right\rangle\left|{00}\right\rangle + \beta\left|{1}\right\rangle\left|{11}\right\rangle = \alpha\left|{000}\right\rangle + \beta\left|{111}\right\rangle = \left|{\psi}\right\rangle_L.\end{aligned}$$ Here, $\text{CNOT}_{ij}$ denotes a CNOT gate with qubit $i$ as control and qubit $j$ as target. ## Syndrome Measurement for Error Detection To detect bit-flip errors without directly measuring the encoded qubit in the computational basis (which would collapse the superposition), we perform syndrome measurements. For the three-qubit bit-flip code, we can measure the parity of pairs of qubits using the operators $Z_1Z_2$ and $Z_2Z_3$. These operators commute with the encoded subspace and can detect bit-flip errors. Consider the action of $Z_1Z_2$ and $Z_2Z_3$ on the encoded states $\left|{\overline{0}}\right\rangle = \left|{000}\right\rangle$ and $\left|{\overline{1}}\right\rangle = \left|{111}\right\rangle$: $$\begin{aligned}Z_1Z_2 \left|{000}\right\rangle &= \left|{000}\right\rangle, \quad Z_1Z_2 \left|{111}\right\rangle = \left|{111}\right\rangle \\Z_2Z_3 \left|{000}\right\rangle &= \left|{000}\right\rangle, \quad Z_2Z_3 \left|{111}\right\rangle = \left|{111}\right\rangle\end{aligned}$$ If a bit flip occurs, for example, on the first qubit, transforming $\left|{000}\right\rangle \rightarrow \left|{100}\right\rangle$, the measurements change: $$\begin{aligned}Z_1Z_2 \left|{100}\right\rangle &= (Z_1 \left|{1}\right\rangle) \otimes (Z_2 \left|{0}\right\rangle) \otimes \left|{0}\right\rangle = (-\left|{1}\right\rangle) \otimes \left|{0}\right\rangle \otimes \left|{0}\right\rangle = -\left|{100}\right\rangle \\Z_2Z_3 \left|{100}\right\rangle &= \left|{100}\right\rangle\end{aligned}$$ By measuring the eigenvalues of $Z_1Z_2$ and $Z_2Z_3$, we obtain error syndromes that indicate which qubit, if any, has flipped. The possible syndromes and corresponding errors are: - Syndrome ( $+1, +1$ ): No error. - Syndrome ( $-1, +1$ ): Error on qubit 1. - Syndrome ( $-1, -1$ ): Error on qubit 2. - Syndrome ( $+1, -1$ ): Error on qubit 3. Here, the syndrome is represented as (eigenvalue of $Z_1Z_2$, eigenvalue of $Z_2Z_3$). These measurements do not distinguish between $\left|{\overline{0}}\right\rangle$ and $\left|{\overline{1}}\right\rangle$ within the code space, only detect errors. ## Error Correction Based on the error syndrome, we can apply a corrective operation to revert the bit-flip error. If the syndrome indicates an error on qubit $i$, we apply a Pauli-X gate $X_i$ to qubit $i$. For example, if the syndrome is $(-1, +1)$, indicating an error on qubit 1, and the state is $\left|{100}\right\rangle$, applying $X_1$ corrects the error: $X_1\left|{100}\right\rangle = \left|{000}\right\rangle = \left|{\overline{0}}\right\rangle$. If the syndrome is $(+1, +1)$, no correction is needed (Identity operation). ## No-Cloning Theorem and Error Correction Quantum error correction is consistent with the no-cloning theorem. We are not creating copies of an arbitrary unknown quantum state. Instead, we are encoding one logical qubit into a redundant physical system (three qubits in this case) to protect the quantum information from noise. The redundancy allows us to detect and correct errors, but it does not enable the cloning of the encoded quantum state. The no-cloning theorem prohibits perfect copying of unknown quantum states, and quantum error correction operates within these constraints by focusing on protecting quantum information rather than replicating it. ::: tcolorbox - **Redundancy for Protection**: Encodes one logical qubit into multiple physical qubits to protect against errors. - **Error Syndrome Measurement**: Uses parity check measurements (e.g., $Z_1Z_2, Z_2Z_3$) to detect errors without collapsing the encoded information. - **Error Correction based on Syndrome**: Applies corrective operations (e.g., Pauli-X gates) based on the measured error syndrome to revert errors. - **Bit-Flip Correction**: The three-qubit code specifically corrects bit-flip errors. Other codes are needed for phase-flip and general errors. - **No Cloning Issue**: Quantum error correction does not violate the no-cloning theorem; it protects information, not copies states. ::: # Quantum Information Theory and Entropy ## Introduction to Quantum Information Theory and Entropy This section introduces the fundamental concepts of quantum information theory, focusing on entropy as a measure of information and uncertainty in quantum systems. We will contrast quantum entropy with its classical counterpart, Shannon entropy, and discuss the role of density operators in defining quantum entropy. A useful resource for further reading is the book \"Quantum Thinking for Computer Scientists,\" particularly the chapters starting around page 288, which provide a concise introduction to these topics. ::: definition **Definition 2** (Classical Shannon Entropy). *In classical information theory, Shannon entropy quantifies the average amount of information gained when observing the outcome of a random variable. For a discrete random variable $X$ with possible outcomes $\{x_1, x_2, ..., x_n\}$ and corresponding probabilities $\{p_1, p_2, ..., p_n\}$, the Shannon entropy $H(X)$ is defined as: $$H(X) = - \sum_{i=1}^{n} p_i \log_2 p_i.$$ Shannon entropy measures the average uncertainty associated with the random variable $X$. It represents the average number of bits needed to describe the outcome of $X$.* ::: ::: definition **Definition 3** (Density Operators). *Density operators are essential for describing quantum states, especially when dealing with mixed states, which are probabilistic ensembles of pure quantum states, or when the quantum state is not precisely known. For an ensemble of pure states $\{\left|{\psi_i}\right\rangle\}$ with corresponding probabilities $\{p_i\}$, the density operator $\rho$ is given by: $$\rho = \sum_{i} p_i \left|{\psi_i}\right\rangle\left\langle{\psi_i}\right|.$$ Density operators provide a comprehensive description of quantum states, encompassing both pure and mixed states, and are fundamental in quantum statistical mechanics and quantum information theory.* ::: ::: definition **Definition 4** (Von Neumann Entropy). *Von Neumann entropy is the quantum analogue of Shannon entropy, extending the concept of entropy to quantum states described by density operators. For a density operator $\rho$, the Von Neumann entropy $S(\rho)$ is defined as: $$S(\rho) = - \text{Tr}(\rho \log_2 \rho) = - \sum_{i} \lambda_i \log_2 \lambda_i,$$ where $\{\lambda_i\}$ are the eigenvalues of the density operator $\rho$. The trace operation ensures that the entropy is independent of the choice of basis. Von Neumann entropy quantifies the quantum uncertainty or mixedness of a quantum state. For a pure state $\left|{\psi}\right\rangle$, the density operator is $\rho = \left|{\psi}\right\rangle\left\langle{\psi}\right|$, which has one eigenvalue equal to 1 and all others 0, resulting in $S(\rho) = 0$, indicating zero entropy for pure states.* ::: ## Relationship between Shannon and Von Neumann Entropy Von Neumann entropy generalizes Shannon entropy. In cases where the density operator is diagonal in a particular basis, and measurements are performed in the same basis, Von Neumann entropy reduces to Shannon entropy. Consider the spectral decomposition of the density operator $\rho = \sum_i \lambda_i\lambda_i \left|{i}\right\rangle\left\langle{i}\right|$, where $\{\left|{i}\right\rangle\}$ are the eigenvectors and $\{\lambda_i\}$ are the eigenvalues. If we consider measurements in the basis $\{\left|{i}\right\rangle\}$, the probabilities of obtaining each outcome $\left|{i}\right\rangle$ are given by the eigenvalues $\lambda_i$. In this scenario, the Von Neumann entropy $S(\rho) = - \sum_i \lambda_i \log_2 \lambda_i$ is formally equivalent to the Shannon entropy of the probability distribution $\{\lambda_i\}$. The example in the lecture transcript highlights that Shannon entropy can be basis-dependent if one considers probabilities of measurement outcomes in a basis different from the eigenbasis of the density operator. However, Von Neumann entropy, defined through eigenvalues, provides a basis-independent measure of quantum entropy. It represents the minimum possible Shannon entropy achievable by measuring the quantum state in any basis, which is obtained when measuring in the eigenbasis of the density operator. ## Quantum Channel Capacity and Quantum Shannon Theorems Quantum Shannon theorems extend the concepts of classical channel capacity and Shannon's coding theorems to the quantum domain. These theorems define the fundamental limits on the rate at which quantum information can be reliably transmitted over quantum channels. Analogous to classical information theory, there are quantum Shannon theorems for both noiseless and noisy quantum channels. These theorems establish the capacity of quantum channels to transmit qubits, considering the unique properties of quantum information and noise. ::: tcolorbox - **Shannon Entropy**: Measures uncertainty of classical random variables, basis-dependent in quantum context if measurement basis is not specified relative to the state. - **Von Neumann Entropy**: Measures quantum uncertainty of quantum states (density operators), basis-independent, generalizes Shannon entropy. - **Pure States**: Von Neumann entropy is zero for pure states, reflecting no quantum uncertainty. - **Mixed States**: Von Neumann entropy is greater than zero for mixed states, quantifying the degree of mixedness. - **Connection**: When a quantum state is measured in the eigenbasis of its density operator, Von Neumann entropy reduces to the Shannon entropy of the resulting probability distribution of measurement outcomes. ::: # Conclusion This lecture has explored fundamental concepts in quantum information, highlighting the power and intricacies of quantum communication and computation. We began with super dense coding, demonstrating how entanglement can double the classical information capacity of a qubit, achieving more efficient communication. Quantum teleportation was then examined, illustrating the transfer of unknown quantum states through entanglement and classical communication, crucially without violating causality. We addressed the challenge of noise in quantum systems with an introduction to quantum error correction, specifically the three-qubit bit-flip code, showcasing the principles of encoding, syndrome measurement, and error correction. Finally, we introduced quantum information theory and Von Neumann entropy, contrasting it with classical Shannon entropy and establishing it as the fundamental measure of uncertainty in quantum states. These topics---super dense coding, teleportation, error correction, and quantum entropy---collectively form a cornerstone for understanding advanced quantum technologies. Future lectures will delve deeper into related areas, including quantum channel capacities, more sophisticated quantum error correction techniques, and the architectural principles of quantum computers, building upon the foundational knowledge established here.