Reduced Density MatrixEdit
The reduced density matrix is a foundational concept in quantum theory that describes the state of a subsystem when the rest of the system is ignored or not accessible. It provides a precise way to talk about local properties of a part of a larger, possibly entangled, quantum system. Formally, if a composite system AB is described by a density operator ρAB on the joint Hilbert space HA ⊗ HB, the state of subsystem A is given by the reduced density matrix ρA = TrB(ρAB), obtained by tracing out the degrees of freedom of B. This construction is central to the study of open quantum systems, quantum information, and many-body physics, because it lets researchers compute expectation values of observables that act only on A, without requiring full knowledge of the environment.
In simple terms, ρA contains all the statistical information accessible to measurements performed solely on subsystem A. It generalizes the classical idea of a probability distribution over states to the quantum setting, capturing both classical uncertainty and quantum coherence that remains accessible to A. The trace rule for expectation values, ⟨OA⟩ = TrA(OA ρA), holds for any observable OA that acts only on A, making ρA the practical object of study for local physics.
Mathematical formalism
Let ρAB be the density operator describing the joint state of subsystems A and B. The reduced density matrix for A is defined by ρA = TrB(ρAB). This operation satisfies key properties: - ρA is positive semidefinite and has unit trace: TrA(ρA) = 1. - The eigenvalues of ρA lie in [0, 1], and they encode the degree of mixedness (or purity) of the local state.
A particularly important case occurs when the global state is pure, ρAB = |Ψ⟩⟨Ψ|. In that situation, ρA is generally mixed unless |Ψ⟩ is a product state |ψA⟩ ⊗ |ψB⟩. The amount of mixing of ρA is a direct indicator of entanglement between A and B.
Schmidt decomposition provides a clean link between ρA and entanglement. Any pure state |Ψ⟩ ∈ HA ⊗ HB can be written as |Ψ⟩ = ∑i √pi |iA⟩ |iB⟩, with pi ≥ 0 and ∑i pi = 1. The reduced density matrix ρA then has eigenvalues {pi}, and its von Neumann entropy S(ρA) = −TrA(ρA log ρA) measures the entanglement entropy between A and B. When the pi are uniform, as in a maximally entangled state, ρA is maximally mixed and S(ρA) attains its maximum value given the dimensions of the subsystems.
A common intuitive special case is a two-qubit Bell state, such as |Φ+⟩ = (|00⟩ + |11⟩)/√2. The joint state is pure, but the reduced state of either qubit is ρA = ½(|0⟩⟨0| + |1⟩⟨1|), which is maximally mixed. In this scenario, measurements on A alone reveal no definite outcome until a measurement on A is paired with a measurement on B, reflecting the entanglement between the subsystems.
The partial trace operation, TrB, is the formal mechanism behind ρA. Conceptually, it sums over all possible states of the environment B, weighting each by its probability, and leaves a description of A that is compatible with those possibilities. This construction is central to the study of open quantum systems and to techniques like density-matrix renormalization group methods in many-body physics.
Examples
Bell-state example: As noted above, for |Φ+⟩, ρA = ½I, a maximally mixed state on the two-dimensional Hilbert space of A. The same holds for ρB, and S(ρA) = ln 2.
General bipartite pure state: If |Ψ⟩ has Schmidt coefficients √pi, then ρA has eigenvalues {pi} and S(ρA) = −∑i pi log pi. This makes the entanglement entropy directly computable from the local spectrum.
Mixed global state: If ρAB is not pure, ρA may still be mixed or even pure depending on correlations. For any ρAB, ρA = TrB(ρAB) fully determines all local expectations ⟨OA⟩ for OA that act only on A, but it does not in general capture correlations between A and B that would be revealed by measurements on both subsystems.
Applications
Quantum information processing: Subsystems are described by reduced density matrices to analyze local operations, channel capacities, and tomography. Local states determine the outcomes of measurements on individual qubits or registers and form the basis for protocols like quantum key distribution and distributed quantum computation. See Quantum information and Density matrix for broader context, and Partial trace for the operations used to obtain ρA.
Open quantum systems and decoherence: Tracing out the environment models how a system loses coherence due to interactions with its surroundings. The resulting ρA captures decoherence effects and helps explain the emergence of classical-looking statistics in measurements performed on the subsystem. See Decoherence (physics) for related phenomena and interpretations.
Many-body physics and condensed matter: Local reduced density matrices describe subsystems of strongly correlated states and underpin numerical methods such as density-matrix renormalization group (DMRG). They quantify local order, correlations, and thermodynamic behavior within larger lattices. See Density matrix, Schmidt decomposition, and Quantum many-body physics for connected topics.
Quantum thermodynamics and emergence of thermality: Tracing over a large environment can yield a reduced state that resembles a thermal state at an effective temperature, illustrating how local subsystems can exhibit thermodynamic behavior even when the global state is pure or out of equilibrium. See Thermodynamics and Canonical ensemble for related ideas.
Connections and interpretation
Relation to entanglement: The reduced density matrix is the primary tool for quantifying and characterizing entanglement between subsystems. A mixed ρA arising from tracing out B signals nonclassical correlations between A and B when the global state is entangled, as opposed to mere classical statistical correlation.
Local versus global descriptions: ρA encapsulates all locally accessible information about A, but it does not in general reveal all cross-subsystem correlations. For that reason, analyses often consider both ρA and ρAB together, particularly when assessing how local measurements inform global properties.
Interpretational debates: In the foundation of quantum mechanics, there are discussions about how to interpret mixed states arising from tracing out parts of a system. Some view ρA as reflecting genuine quantum uncertainty about A due to entanglement with B, while others emphasize epistemic viewpoints where the mixture reflects incomplete knowledge about a larger, pure state. These discussions touch on broader questions about the nature of quantum states and information, but the operational use of reduced density matrices remains well defined and widely applied in practice.