QUANTUM MARGINAL PROBLEM AND REPRESENTATIONS OF THE SYMMETRIC GROUP

By Alexander A. Klyachko. 2004

184 citations recorded by the source corpus.

We discuss existence of mixed stateAB of two (or multy-) com- ponent system HAB = HA HB with reduced density matricesA, �B and given spectraAB,�A,�B. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a5122032389c3d8d8b73c549541f2aaa2fe52936

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Rosa C. Orellana
2. Murat Altunbulak
3. Emmanuel Briand
4. Mercedes H. Rosas
5. Graeme J. Mitchison
6. Matthias Christandl
7. Aram Wettroth Harrow
8. Alexander A. Klyachko

In Mathematical Physics

1. Murat Altunbulak
2. Rosa C. Orellana
3. Mercedes H. Rosas
4. Emmanuel Briand
5. Graeme J. Mitchison
6. Matthias Christandl
7. Aram Wettroth Harrow
8. Vijay Balasubramanian

In Statistics and Probability

1. Murat Altunbulak
2. Rosa C. Orellana
3. Mercedes H. Rosas
4. Graeme J. Mitchison
5. Emmanuel Briand
6. Matthias Christandl
7. Aram Wettroth Harrow

Citations in the corpus

• The Pauli Principle Revisited (2008, methodology): “We also benefit from Convex package by Franz [13], who apply a similar approach to the quantum marginal problem for three qutrits [12,17].”
• Nonzero Kronecker Coefficients and What They Tell us about Spectra (2007, methodology): “ ρA,Spec ρB,Spec ρAB) = (rA,rB,rAB). How can we characterise such triples? It turns out that the admissible spectral triples correspond to Young diagrams (µ,ν,λ) with nonzero Kronecker coefficient gµνλ [4, 13]. This means that the irreducible representation Vλ is contained in the tensor product of Vµ ”
• Bulk curves from boundary data in holography (2013, background): “rix exists in case one is given reduced density matrices for various subsystems. For example, for a spin chain with L = 1 wherein one only knows the reduced density matrices for the individual spins, [58] gave explicit necessary and sucient conditions in terms of the eigenvalues of the reduced densi”
• The stability of the Kronecker product of Schur functions (2009, methodology): “We use the Jacobi-Trudi determinant to extend the definition of sµ to the case where µ is any finite sequence of n integers : sµ = det ( hµj+i−j ) 1≤i,j≤n , (1) where hk is the complete homogeneous symmetric function of degree k.”