ts-quantum/dist/quantum/src/operators/algebra.d.ts

TypeScript library for quantum mechanics calculations and utilities

/**
 * Operator algebra extensions for quantum mechanics
 *
 * Provides fundamental operator algebra operations including commutators,
 * anti-commutators, Lie algebraic structures, and more.
 */
import { Complex, IOperator, IStateVector } from '../core/types';
/**
 * Adds two operators
 *
 * @param a First operator
 * @param b Second operator
 * @returns The sum operator
 */
export declare function addOperators(a: IOperator, b: IOperator): IOperator;
/**
 * Subtracts one operator from another
 *
 * @param a First operator
 * @param b Second operator to subtract
 * @returns The difference operator
 */
export declare function subtractOperators(a: IOperator, b: IOperator): IOperator;
/**
 * Calculates the commutator [A,B] = AB - BA between two operators
 *
 * Used for determining whether operators commute, which is essential
 * for determining if observables can be measured simultaneously.
 *
 * @param A First operator
 * @param B Second operator
 * @returns The commutator operator [A,B]
 */
export declare function commutator(A: IOperator, B: IOperator): IOperator;
/**
 * Calculates the anti-commutator {A,B} = AB + BA between two operators
 *
 * Important for fermion systems and in supersymmetry.
 *
 * @param A First operator
 * @param B Second operator
 * @returns The anti-commutator operator {A,B}
 */
export declare function antiCommutator(A: IOperator, B: IOperator): IOperator;
/**
 * Calculates a nested commutator [A, [B, C]] and more complex structures
 *
 * Useful for higher-order perturbation theory and quantum field calculations.
 *
 * @param ops Array of operators to use in nested commutator
 * @param indices Array of pairs of indices specifying the commutator structure.
 *        Each pair [a, b] creates a commutator between operators.
 *        Pairs are processed from last to first (innermost to outermost).
 *        IMPORTANT: This function only supports strictly nested commutators of the form [A,[B,[C,D]]], not branched structures like [[A,B],[C,D]].
 *
 * @example
 * // For a structure [A,[B,C]] with ops = [A, B, C]
 * // indices = [[0, 1], [1, 2]]
 * // 1. First computes [B,C] using indices [1,2] (last pair)
 * // 2. Then computes [A,[B,C]] using [0,1] (first pair)
 *
 * @example
 * // For the Jacobi identity [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]]
 * // term1 = nestedCommutator([X, Y, Z], [[0, 1], [1, 2]])
 * // term2 = nestedCommutator([Y, Z, X], [[0, 1], [1, 2]])
 * // term3 = nestedCommutator([Z, X, Y], [[0, 1], [1, 2]])
 *
 * @note For the LAST (innermost) pair: both indices refer to operators in the ops array
 * @note For ALL OTHER pairs: the first index refers to an operator in the ops array,
 *       while the second operand is always the result of the previous calculation
 *
 * @returns The resulting operator from the nested commutator structure
 *
 * @see createNestedCommutator - For a more intuitive way to create nested commutators
 */
export declare function nestedCommutator(ops: IOperator[], indices: number[][]): IOperator;
/**
 * Calculates the Lie derivative L_A(B) = [A, B]
 *
 * Important in the theory of Lie algebras and quantum mechanics.
 *
 * @param A First operator (generator)
 * @param B Second operator
 * @returns The Lie derivative operator
 */
export declare function lieDerivative(A: IOperator, B: IOperator): IOperator;
/**
 * Implements the Baker-Campbell-Hausdorff formula to calculate exp(A)exp(B)
 *
 * Formula: exp(A)exp(B) = exp(A + B + 1/2[A,B] + 1/12[A,[A,B]] - 1/12[B,[A,B]] + ...)
 * Essential for quantum mechanics when dealing with non-commuting operators.
 *
 * @param A First operator in exponential
 * @param B Second operator in exponential
 * @param order Maximum order of nested commutators to include
 * @returns Approximation of exp(A+B) based on BCH formula
 */
export declare function BCHFormula(A: IOperator, B: IOperator, order?: number): IOperator;
/**
 * Checks if two operators commute (within numerical tolerance)
 *
 * @param A First operator
 * @param B Second operator
 * @param tolerance Numerical tolerance for zero check
 * @returns True if operators commute
 */
export declare function operatorsCommute(A: IOperator, B: IOperator, tolerance?: number): boolean;
/**
 * Calculates the expectation value of a commutator [A,B] for a state
 *
 * Important for uncertainty relations in quantum mechanics.
 *
 * @param state Quantum state
 * @param A First operator
 * @param B Second operator
 * @returns Complex expectation value
 */
export declare function commutatorExpectation(state: IStateVector, A: IOperator, B: IOperator): Complex;
/**
 * Calculates the uncertainty product ΔA·ΔB for a state
 *
 * For the uncertainty principle: ΔA·ΔB ≥ |⟨[A,B]⟩|/2
 *
 * @param state Quantum state
 * @param A First operator
 * @param B Second operator
 * @returns Real number representing uncertainty product
 */
export declare function uncertaintyProduct(state: IStateVector, A: IOperator, B: IOperator): number;
/**
 * Checks if an operator is normal (AA† = A†A)
 *
 * Normal operators have special spectral properties.
 *
 * @param A Operator to check
 * @param tolerance Numerical tolerance
 * @returns True if operator is normal
 */
export declare function isNormalOperator(A: IOperator, tolerance?: number): boolean;
/**
 * Creates an operator from its generating function: exp(iG)
 *
 * Common in quantum mechanics where G is the generator (often Hermitian)
 *
 * @param generator Generator operator G
 * @returns Resulting operator exp(iG)
 */
export declare function operatorFromGenerator(generator: IOperator): IOperator;
/**
 * Creates a nested commutator in a more intuitive way
 *
 * This function provides a simpler interface for creating nested commutators
 * compared to the more complex indexing scheme used by nestedCommutator.
 *
 * @param ops Array of operators in the exact order they should appear in the commutator
 * @returns The resulting operator from the nested commutator structure
 *
 * @example
 * // To compute [X, [Y, Z]]:
 * createNestedCommutator([X, Y, Z]);
 *
 * // To compute [A, [B, [C, D]]]:
 * createNestedCommutator([A, B, C, D]);
 */
export declare function createNestedCommutator(ops: IOperator[]): IOperator;
/**
 * Creates projection operator |ψ⟩⟨ψ| from a state
 *
 * @param state Quantum state to project onto
 * @returns Projection operator
 */
export declare function projectionOperator(state: IStateVector): IOperator;