[Paper Review] Controlled Quantum Dense Coding in a Four-particle Non-maximally Entangled State via Local Measurements

Chang-Bao Fu, Yan Xia+4 more|ArXiv.org|Jan 21, 2006

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper proposes a controlled quantum dense coding scheme using a four-particle non-maximally entangled state, where two controllers (parties 1 and 4) regulate the classical information capacity via local measurements on their qubits. By adjusting measurement angles and leveraging state collapse, the scheme enables Alice and Bob to transmit up to two bits of classical information, with the capacity dependent on both the measurement angles and the entanglement coefficients of the channel.

ABSTRACT

A controlled quantum dense coding scheme is investigated with a four-particle non-maximal quantum channel. The amount of classical information is shown to be capable of being controlled by the controllers through adjustments of the local measurement angles and to depend on the coefficients of the quantum channel; in addition, the four particles are distributed in two inverse ways in such an quantum channel. A restricted condition for distributing the particles to realize quantum dense coding in an arbitrary ($N+2$)-particle quantum channel is proposed.

Motivation & Objective

• To investigate how classical information capacity in quantum dense coding can be controlled via local measurements in a four-particle non-maximally entangled state.
• To analyze the dependence of transmitted information on the coefficients of the quantum channel and the measurement angles of the controllers.
• To determine the valid particle distribution configurations that enable successful quantum dense coding in a four-particle system.
• To derive a general restricted condition for particle distribution in arbitrary (N+2)-particle quantum channels to realize controlled dense coding.
• To extend controlled dense coding beyond maximally entangled states to physically more accessible non-maximally entangled states.

Proposed method

• The scheme uses a four-particle entangled state |ψ⟩ = a|0000⟩ + b|1001⟩ + c|0110⟩ + d|1111⟩, with real coefficients satisfying |a|² + |b|² + |c|² + |d|² = 1.
• Party 4 performs a local measurement on qubit 4 using basis states |+⟩₄ = cosθ₁|0⟩₄ + sinθ₁|1⟩₄ and |−⟩₄ = sinθ₁|0⟩₄ − cosθ₁|1⟩₄, collapsing the state into a three-particle state depending on the outcome.
• Party 1 performs a similar local measurement on qubit 1 using basis states |+⟩₁ = cosθ₂|0⟩₁ + sinθ₂|1⟩₁ and |−⟩₁ = sinθ₂|0⟩₁ − cosθ₂|1⟩₁, further collapsing the state to a two-qubit state between Alice and Bob.
• After successful measurement outcomes (|+⟩₄ and |+⟩₁), the state of qubits 2 and 3 collapses to a two-particle state |φ⟩₂₃ = (a cosθ₁ cosθ₂ + b sinθ₁ sinθ₂)|00⟩₂₃ + (c cosθ₁ cosθ₂ + d sinθ₁ sinθ₂)|11⟩₂₃.
• The classical information capacity is derived as C = 1 + 2|sinγ|² = 1 + 2[1 + cot²γ]⁻¹, where γ is a function of the coefficients and measurement angles.
• A restricted condition is proposed for (N+2)-particle channels: after N controllers perform local measurements, the remaining two particles must form a non-maximally entangled state that is a linear combination of |00⟩ and |11⟩ states, preserving the ability to encode two classical bits.

Experimental results

Research questions

• RQ1 How does the classical information capacity in controlled quantum dense coding depend on the local measurement angles of the controllers?
• RQ2 What is the role of the entanglement coefficients (a, b, c, d) in determining the amount of transmissible classical information?
• RQ3 What are the valid particle distribution configurations that allow successful quantum dense coding in a four-particle non-maximally entangled state?
• RQ4 Can a general condition be derived for distributing particles in an arbitrary (N+2)-particle quantum channel to enable controlled dense coding?
• RQ5 How does the use of a non-maximally entangled state compare to a maximally entangled state in terms of practical feasibility and information capacity?

Key findings

• The classical information capacity C reaches a maximum of two bits when the measurement angles are θ₁ = θ₂ = π/4 and the coefficients are equal (|a| = |b| = |c| = |d| = 1/2), corresponding to a maximally entangled state.
• When |tanγ| < 1, the information capacity is less than two bits, indicating that the channel's entanglement quality and measurement settings jointly limit transmission efficiency.
• Only two particle distribution configurations allow successful dense coding: (1) parties 1 and 4 act as controllers (quantum erasure), with parties 2 and 3 as sender and receiver; (2) the roles of parties 1 and 4 are reversed, with the same outcome.
• Distributions where parties 1 and 4 are not used as controllers fail to produce a usable two-particle entangled state for encoding.
• The scheme achieves controlled dense coding without requiring a maximally entangled channel, making it more feasible for physical implementations where such states are difficult to generate.
• A general restricted condition is established: in an (N+2)-particle channel, after N controllers perform local measurements, the remaining two particles must form a non-maximally entangled state that is a linear combination of |00⟩ and |11⟩ states to preserve two-bit information transmission.