2.3 Quantum superposition and mixed states

Quantum superposition and mixed states are key concepts in understanding the nature of quantum systems. They reveal how particles can exist in multiple states simultaneously and how quantum systems can be described as probabilistic mixtures of different states.

These ideas are crucial for grasping the unique properties of quantum light. They explain phenomena like entanglement and interference, which are essential for quantum technologies like computing and cryptography. Understanding these concepts is vital for exploring the quantum world.

Quantum superposition and its significance

Fundamentals of quantum superposition

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• Quantum superposition is a fundamental principle of quantum mechanics where a quantum system can exist in multiple states simultaneously until it is measured or observed
• Superposition is mathematically represented by a linear combination of the basis states, with coefficients determining the probability amplitudes of each state
  • For example, a qubit can be in a superposition of the states |0⟩ and |1⟩, written as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex coefficients
• The act of measurement or observation collapses the superposition, causing the quantum system to randomly assume one of the possible states according to the probability amplitudes
  • In the qubit example, measuring the superposition state |ψ⟩ will result in either the state |0⟩ or |1⟩, with probabilities |α|² and |β|², respectively

Significance in quantum optics

• In quantum optics, superposition allows for the creation of quantum states of light that exhibit properties not possible with classical light, such as entanglement and quantum interference
  • Entangled photon pairs, such as those produced by spontaneous parametric down-conversion (SPDC), are in a superposition of multiple polarization states
• Quantum superposition enables the development of quantum technologies, such as quantum computing, quantum cryptography, and quantum sensing
  • Quantum key distribution (QKD) protocols, like BB84, rely on the superposition of photon polarization states to ensure secure communication
• Superposition is a crucial ingredient for implementing quantum algorithms and quantum gates, which exploit the parallelism of quantum states to perform certain computations more efficiently than classical computers

Pure vs mixed states

Characteristics of pure states

• Pure states are quantum states that can be described by a single state vector or wave function, representing a definite quantum state with no classical uncertainty
• Pure states exhibit perfect coherence and can be represented by a single point on the Bloch sphere
  • For example, the pure states |0⟩ and |1⟩ are represented by the north and south poles of the Bloch sphere, respectively
• The density matrix of a pure state has a single non-zero eigenvalue equal to 1, while all other eigenvalues are zero
  • For a pure state |ψ⟩, the density matrix is given by ρ = |ψ⟩⟨ψ|, which is a projection operator onto the state vector

Characteristics of mixed states

• Mixed states are statistical ensembles of pure states, where the system is in a combination of different pure states with corresponding probabilities
• Mixed states have reduced coherence and are represented by a point inside the Bloch sphere
  • The maximally mixed state, represented by the center of the Bloch sphere, is an equal mixture of all possible pure states
• The density matrix of a mixed state has multiple non-zero eigenvalues summing to 1
  • For a mixed state, the density matrix is a weighted sum of the density matrices of the constituent pure states: ρ = ∑ᵢ pᵢ |ψᵢ⟩⟨ψᵢ|, where pᵢ is the probability of the system being in the pure state |ψᵢ⟩

Density matrices for quantum states

Constructing density matrices for pure states

• For a pure state |ψ⟩, the density matrix is given by ρ = |ψ⟩⟨ψ|, which is the outer product of the state vector with its conjugate transpose
  • For example, the density matrix of the pure state |ψ⟩ = α|0⟩ + β|1⟩ is given by: ρ = |ψ⟩⟨ψ| = (α|0⟩ + β|1⟩)(α*⟨0| + β*⟨1|) = |α|²|0⟩⟨0| + αβ*|0⟩⟨1| + α*β|1⟩⟨0| + |β|²|1⟩⟨1|
• The density matrix of a pure state is a projection operator, satisfying ρ² = ρ and Tr(ρ²) = 1

Constructing density matrices for mixed states

• For a mixed state, the density matrix is a weighted sum of the density matrices of the constituent pure states: ρ = ∑ᵢ pᵢ |ψᵢ⟩⟨ψᵢ|, where pᵢ is the probability of the system being in the pure state |ψᵢ⟩
  • For example, a mixed state consisting of a 60% probability of being in the state |0⟩ and a 40% probability of being in the state |1⟩ has the density matrix: ρ = 0.6|0⟩⟨0| + 0.4|1⟩⟨1|
• The density matrix is a Hermitian, positive semidefinite matrix with unit trace (Tr(ρ) = 1), ensuring that the probabilities sum to 1

Properties of quantum states using density matrices

Calculating expectation values and probabilities

• The density matrix formalism allows for the calculation of expectation values of observables without explicit knowledge of the state vector, using the trace operation: ⟨A⟩ = Tr(ρA)
  • For example, the expectation value of the Pauli-Z operator (σ_z) for a qubit state ρ is given by ⟨σ_z⟩ = Tr(ρσ_z)
• The probability of measuring a quantum state in a particular basis state |i⟩ is given by the diagonal elements of the density matrix: P(i) = ⟨i|ρ|i⟩
  • For a qubit, the probabilities of measuring the states |0⟩ and |1⟩ are given by P(0) = ρ₀₀ and P(1) = ρ₁₁, respectively

Quantifying purity and entanglement

• The purity of a quantum state can be determined by the trace of the squared density matrix: Tr(ρ²)
  • A pure state has Tr(ρ²) = 1, while a mixed state has Tr(ρ²) < 1
  • The linear entropy, defined as S_L(ρ) = 1 - Tr(ρ²), quantifies the degree of mixedness, with 0 for pure states and 1 for maximally mixed states
• The von Neumann entropy, S(ρ) = -Tr(ρ log ρ), quantifies the amount of uncertainty or mixedness in a quantum state
  • Pure states have zero entropy, while mixed states have non-zero entropy
  • The maximum entropy for a d-dimensional quantum system is log(d), corresponding to the maximally mixed state
• The reduced density matrix, obtained by performing a partial trace over a subsystem, allows for the description of entanglement and correlations between subsystems in a composite quantum system
  • For a bipartite system ρ_AB, the reduced density matrix of subsystem A is given by ρ_A = Tr_B(ρ_AB)
  • Entanglement between subsystems can be quantified using measures such as entanglement entropy, concurrence, or negativity, which are calculated from the reduced density matrices