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Chapter 5: Algorithms

Quantum Computing - Expanded Lecture Summaries

Sharat Batra

March 15, 2025

Lecture 3: Bloch Sphere, Eigenstates, Eigenvectors, Projection Operator

Key Algorithms

Superdense Coding with Bell States

Key Insight Superdense coding demonstrates how entanglement + quantum operations + classical communication can double the classical communication capacity of a quantum channel. It is a powerful example of quantum advantage and plays a central role in quantum communication and information theory.

In superdense coding, Alice and Bob share an entangled Bell state initially:

Great! Here's the revised Markdown with LaTeX equations for your lecture section on superdense coding, using standard Dirac (ket) notation for all 4 Bell states, and fully compatible with VS Code PDF rendering:

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Superdense Coding

In superdense coding, Alice and Bob share an entangled qubit pair in the Bell state:

\[ \left| \Phi^+ \right\rangle = \frac{1}{\sqrt{2}} \left( \left|00\right\rangle + \left|11\right\rangle \right) \]

Alice can send two classical bits of information by applying one of four unitary operations to her qubit and then sending it to Bob. The shared entangled state allows Bob to recover two bits from just one qubit transmission.

Bell States

The four Bell states used in superdense coding are:

\[ \begin{aligned} \left| \Phi^+ \right\rangle &= \frac{1}{\sqrt{2}} \left( \left|00\right\rangle + \left|11\right\rangle \right) \\ \left| \Phi^- \right\rangle &= \frac{1}{\sqrt{2}} \left( \left|00\right\rangle - \left|11\right\rangle \right) \\ \left| \Psi^+ \right\rangle &= \frac{1}{\sqrt{2}} \left( \left|01\right\rangle + \left|10\right\rangle \right) \\ \left| \Psi^- \right\rangle &= \frac{1}{\sqrt{2}} \left( \left|01\right\rangle - \left|10\right\rangle \right) \end{aligned} \]

Encoding Scheme

Alice applies one of four operations to her qubit, depending on the 2-bit classical message she wants to send:

Classical Bits Operation (on Alice's qubit) Resulting State
00 \(I\) (Identity) $\left
01 \(X\) (Bit flip) $\left
10 \(Z\) (Phase flip) $\left
11 \(iY\) (Bit and phase flip) $\left

The resulting state is then sent to Bob.

Alice's Encoding Operations (Step-by-Step)

  • Message 00: $$ (I \otimes I) \left| \Phi^+ \right\rangle = \left| \Phi^+ \right\rangle $$

  • Message 01: $$ (X \otimes I) \left| \Phi^+ \right\rangle = \left| \Psi^+ \right\rangle $$

  • Message 10: $$ (Z \otimes I) \left| \Phi^+ \right\rangle = \left| \Phi^- \right\rangle $$

  • Message 11: $$ (Y \otimes I) \left| \Phi^+ \right\rangle = i \left| \Psi^- \right\rangle $$

Note: The global phase \(i\) does not affect measurement outcomes.

Bob’s Decoding Procedure

After receiving the qubit from Alice, Bob performs the reverse entanglement operations to recover the original 2 classical bits:

  1. Apply CNOT gate with Alice's qubit as control and Bob's as target.
  2. Apply Hadamard gate to Alice’s qubit.

This transforms each Bell state to a unique computational basis state:

\[ \begin{aligned} \left| \Phi^+ \right\rangle &\rightarrow \left|00\right\rangle \\ \left| \Psi^+ \right\rangle &\rightarrow \left|01\right\rangle \\ \left| \Phi^- \right\rangle &\rightarrow \left|10\right\rangle \\ \left| \Psi^- \right\rangle &\rightarrow \left|11\right\rangle \end{aligned} \]

Bob then measures the two qubits in the computational basis and perfectly retrieves Alice’s 2-bit message.