Reduction criterion implies majorization?

Cite this problem as Problem 9.


The density matrix of any separable state is majorized by its reductions (the density matrix reduced to one subsystem, e. g. \rho_A = tr_B(\rho_{AB}). This is in fact the strongest separability criterion based on the spectra of a state and one of its reductions. However, it is not known how it is related to other separability criteria like PPT, undistillability or the reduction criterion. The problem is to find out how majorization enters into the known implication chain of separability criteria.


One of the remarkable properties of entangled states is that they can exhibit locally more disorder than globally. The simplest example is the maximally entangled state, which is pure as a whole but it has maximally chaotic reductions. A powerful tool comparing the order/disorder of two systems is majorization and in fact it is a more stringent notion of order/disorder than entropy.

It was proven in [1] that the density matrix of a separable state is majorized by both of its reductions. Hence, majorization yields a separability criterion, which is merely based on the spectra of a state and its reductions.

There are many important separability/entanglement criteria or properties and in most cases the relations between them are well known: Separability ⇒ positivity of the partial transpose [2] ⇒ undistillability [3] ⇒ reduction criterion [4].

The intuition may be, that all these criteria a strictly stronger than majorization, however the matter is not decided yet.

Partial Solutions

Apart from inconclusive numerical search for counterexamples for the implication: reduction criterion \to majorization, the only partial result is derived in [5], where it was shown, that the reduction criterion implies positivity for conditional Renyi entropies for every value of the entropic parameter. Although conditional entropies also measure the proportion between global and local disorder, this result cannot be extended directly to majorization.


The answer to the question is contained in [6], stating that the reduction criterion does imply majorization.

The key idea of the proof is that \rho_A\otimes \mathbb{I}_B\geq \rho_{AB} implies \rho_{AB}^{1/2}\geq \rho_A^{1/2}\otimes \mathbb{I}_B R with \|R\|\leq 1, where  \rho_{AB}  is a bipartite density matrix, and \|\|  is the operator norm. By virtue of this, we can derive the existence of the substochastic matrix S such that \lambda (\rho_{AB}) = S \lambda(\rho_A) , where \lambda(\rho_{AB}) (\lambda(\rho_A) ) is the eigenvalue (column) vector of \rho_{AB} (\rho_{A}). This last equation is equivalent to the weak submajorization relation \lambda(\rho_{AB})\prec w\lambda(\rho_{A}) which is none other than \lambda(\rho_{AB})\prec \lambda(\rho_{A}) in this problem.


[1] M. A. Nielsen and J. Kempe, Separable States Are More Disordered Globally than Locally, Phys. Rev. Lett. 86, 5184 (2001) and quant-ph/0011117 (2000).

[2] A. Peres, Separability Criterion for Density Matrices, Phys. Rev. Lett. 77, 1413

(1996) and quant-ph/9604005 (1996).

[3] M. Horodecki, P. Horodecki, and R. Horodecki, Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature?, Phys. Rev. Lett. 80,

5239 (1998) and quant-ph/9801069 (1998).

[4] M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59, 4206 (1999) and quant-ph/9708015

(1997); J. Cerf, C. Adami, and R. M. Gingrich, Reduction criterion for separability, Phys. Rev. A 60, 898 (1999) and quant-ph/9710001 (1997).

[5] K. G. H. Vollbrecht and M. M. Wolf, Conditional entropies and their relation to entanglement criteria, quant-ph/0202058 (2002).

[6] T. Hiroshima, Majorization criterion for distillability of a bipartite quantum state, quant-ph/0303057 (2003).