When is a pure state of three qubits determined by its single-particle reduced density matrices?

Abstract

Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the, so called, Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecisions, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the Greenberger--Horne--Zeilinger (GHZ) state by stochastic local operations and classical communication (SLOCC), and discuss generalizations of our results to an arbitary number of qubits.

19 pages

References in corpus (7)

• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
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• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
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• Multiparticle entanglement as an emergent phenomenon
• Reconstructing quantum states from single-party information
• Almost all four-particle pure states are determined by their two-body marginals
• How many invariant polynomials are needed to decide local unitary equivalence of qubit states?
• Mode-entanglement of Gaussian fermionic states
• Proving genuine multiparticle entanglement from separable nearest-neighbor marginals
• A brief introduction to multipartite entanglement
• Multi-partite $W-$type state is determined by its single particle reduced density matrices
• Bipartite entanglement, spherical actions and geometry of local unitary orbits
• Multipartite quantum correlations: symplectic and algebraic geometry approach
• Introduction to quantum entanglement in many-body systems
• Implications of pinned occupation numbers for natural orbital expansions. II: Rigorous derivation and extension to non-fermionic systems
• Tripartite Entanglement and Quantum Correlation
• A geometric formulation to measure global and genuine entanglement in three-qubit systems
• Universal and distorsion-free entanglement concentration of multiqubit quantum states in the W class
• Quantum tomography of three-qubit generalized Werner states
• On the Symplectic Reduced Space of Three-Qubit Pure States
• Non-Locality$\ne$Quantum Entanglement
• A variation principle for ground spaces
• Enhancing Small Dataset Classification Using Projected Quantum Kernels with Convolutional Neural Networks
• Geometry of two-body correlations in three-qubit states
• Investigating Pure State Uniqueness in Tomography via Optimization
• Solving one-body ensemble N-representability problems with spin