mod02lec11 - Quantum Computing Concepts: Entanglement and Interference​ - Part 1

Introduction to Quantum Computing Concepts

Overview of Today's Lecture

• The lecture focuses on programming quantum computers, building upon concepts learned in the previous week.

• Key topics include qubit entanglement, Bell states, Bell measurement, quantum teleportation, and interference.

Recap of Previous Concepts

• Review of single qubits and their representation in Dirac notation as kets.

• Measurement principles: a single qubit collapses to either state 0 or state 1 with probabilities summing to one.

• Introduction to single qubit gates as linear transformations represented by matrices.

Understanding Multi-Qubit States

Multipartite States and Gates

• A multipartite state is formed through the tensor product of multiple single qubits.

• Multiple qubit gates apply transformations simultaneously across several qubits while maintaining reversibility.

Hadamard Transformations

• Applying Hadamard transforms increases the state space exponentially (2^n for n qubits).

• The Hadamard gate can create superposition states from basis states, allowing for rapid computation compared to classical methods.

Exploring Entanglement

Product vs. Entangled States

• Multi-qubit states are classified into product states (tensor products of defined states) and entangled states (cannot be expressed as such).

Bell States Definition

• Four maximally entangled two-qubit Bell states are introduced:

• Psi_00: Probability distribution includes only |00⟩ and |11⟩ with equal probability.

• Psi_01: Includes |01⟩ and |10⟩ with equal probability but zero for |00⟩ and |11⟩.

Implications of Entanglement

Measurement Outcomes in Entangled States

• Measuring one qubit in an entangled pair determines the outcome of the other due to inherent correlations.

Understanding Quantum State Collapse

• When measuring a two-qubit state like Psi_00 or Psi_11, outcomes will always yield correlated results (either both zeros or both ones).

Understanding Multipartite States and Bell States

Correlation in Multipartite States

• There is a high correlation between individual parts of a multipartite state, indicating that even when qubits are distanced, entangled states remain connected.

• This phenomenon is crucial for quantum teleportation and important computations.

Producing Bell States

• The process to produce maximally entangled Bell states involves specific input qubit configurations (i, j) leading to corresponding output states.

• A Hadamard gate is applied to the first qubit followed by a CNOT gate to generate the desired output state from the input state.

Example of State Transformation

• For an input state of (0, 0), applying the Hadamard transform results in a superposition represented as 1/sqrt2(0 + 1) . The subsequent application of CNOT maintains the second qubit's state if the control qubit is zero.

• When testing with an input of (1, 0), applying Hadamard transforms leads to different outcomes based on whether the control qubit flips or preserves its state during CNOT operations.

Understanding Bell Measurement

• Bell measurement reverses the process of producing a Bell state; given an entangled state psi_ij , it reveals i and j through measurement. The same circuit used for production can be reversed for measurement purposes.

• The circuit requires first applying CNOT followed by Hadamard gates before measurements can be made on both qubits, ensuring accurate retrieval of original states i and j.

Practical Application: Measuring Input States

• Using psi_00 as an example demonstrates how measurements reveal original inputs after processing through CNOT and Hadamard gates, confirming that these operations effectively retrieve initial values from entangled states.

• Breaking down two-qubit states into simpler forms aids in understanding transformations during quantum operations like Hadamard applications on each component separately.

Quantum State Transformations and Teleportation

Quantum State Transformations

• The Hadamard transform is applied to the first qubit, changing its state from |0⟩ to (|0⟩ + |1⟩)/√2, while the second qubit remains unchanged.

• After distributing the zero state inside, a common factor of 1/2 emerges, leading to states |00⟩, |10⟩, and |-10⟩.

• The simplification results in the final state being |00⟩ with a 100% probability of measuring this outcome when starting from input state |ψ₀₀⟩.

• This measurement certainty simplifies quantum operations since both qubits will always yield zero upon measurement.

• A table illustrates that any input other than |ψ₀₀⟩ will produce specific output states with corresponding probabilities.

Introduction to Quantum Teleportation

• Quantum teleportation is introduced as a practical concept derived from quantum computing and entanglement rather than science fiction.

• Alice and Bob create an entangled Bell state (|ψᵢⱼ⟩), where Alice retains one part and Bob retains another despite physical distance separating them.

• Each participant holds one half of an entangled pair; Alice possesses her part of the Bell state while Bob has his own.

• Alice aims to communicate a different single qubit state (|φ⟩ = α|0⟩ + β|1⟩) to Bob without traveling physically.

• The communication relies on their shared entangled quantum state (Bell state), which facilitates information transfer.

Mechanism of Quantum Teleportation

• A high-level overview shows Alice sending two classical bits after performing measurements on her qubits using a classical communication channel.

• EPR pairs are referenced as foundational for understanding entanglement; they were named after Einstein, Podolsky, and Rosen who studied these phenomena.

• Alice's classical communication allows Bob to infer the original quantum state based on his knowledge of their shared EPR pair and the received bits from Alice.

• The teleportation protocol involves Bell measurements performed by Alice on her known qubits before communicating results to Bob for further processing.

• In this three-qubit system setup, two qubits are known by Alice while Bob knows his part of the EPR pair; this forms the basis for subsequent transformations.

Quantum State Transformation and Measurement

Overview of Quantum States

• Alice communicates her measurement results to Bob, enabling him to apply transformations to recover the qubit φ that she is trying to center.

• The state χ₁ is introduced as a tensor product representation consisting of three qubits: one being φ and the other two forming the state ψ₀₀.

Tensor Product Expansion

• The expansion of χ₁ involves distributing terms from φ (α|0⟩ + β|1⟩) with ψ₀₀, leading to a combination of states including the Bell state.

• After applying a controlled-NOT (CNOT) gate, the new three-qubit state χ₂ is formed, where the first qubit influences whether the second qubit's state flips or remains unchanged.

Effects of CNOT Gate

• The effect of CNOT on χ₂ shows how it preserves or flips states based on the first qubit's value; for instance, if it's 0, the second remains unchanged.

• This transformation leads to specific outcomes in χ₂ depending on whether the first qubit is 0 or 1.

Hadamard Transform Application

• A Hadamard gate is applied to the first qubit in χ₂, resulting in a new state denoted as χ₃. This alters only the first qubit while keeping others intact.

• The Hadamard transform changes |0⟩ into a superposition state (|+⟩), while |1⟩ transforms into another superposition (|-⟩).

Simplification and State Representation

• After simplification, we obtain a three-qubit system that can represent all possible combinations of states from |000⟩ to |111⟩.

• Emphasis is placed on separating out Alice's knowledge from Bob’s part by focusing on just two qubits relevant for measurement.

Probability Distribution and Final State

• By extracting common terms from different states, probabilities for each possible outcome are calculated based on Alice's measurements.

• Bob’s final quantum state (φ') reflects all transformations before receiving Alice’s measurement results. This setup aids in determining φ effectively.

Quantum Measurement and State Inference

Alice's Measurements and Bob's States

• When Alice performs measurements, she can obtain one of four possible outcomes: 00, 01, 10, or 11. The claim is that if Alice measures the state as 00, then Bob's state (denoted as φ') must be α|0⟩ + β|1⟩.

• If Alice measures the state as 10, then Bob's state φ' must change to α|0⟩ - β|1⟩. This relationship is derived from a specific expression that outlines how these states correlate.

• The table of states is constructed based on these expressions. For instance, if Alice’s measurement results in the three-qubit state being |01⟩, then Bob’s corresponding state must be α|1⟩ + β|0⟩. Each measurement leads to a distinct outcome for Bob.

• The original quantum state was defined as α|0⟩ + β|1⟩. Ultimately, the goal is for Bob to infer this original state based on Alice's measurements and their implications on his own quantum state.