Inequalities for the Ranks of Quantum States

Abstract

We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (α-Rényi entropy for α=1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other α-Rényi entropies for $\alpha \in (0,1)\cup (1,\infty )$ satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for $\alpha \in (0,1)$ is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of α=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., α=1) and 0-Rényi entropy are exceptionally interesting measures of entanglement in the multipartite setting.

References in corpus (7)

• On the quantum, classical and total amount of correlations in a quantum state
• Is Entanglement Monogamous?
• The structure of multidimensional entanglement in multipartite systems
• Universal quantum computation with little entanglement
• The entropy vector formalism and the structure of multidimensional entanglement in multipartite systems
• Non-Shannon Information Inequalities in Four Random Variables
• A new inequality for the von Neumann entropy

Cited by in corpus (24)

• Multi-photon entanglement in high dimensions
• Entanglement Certification $-$ From Theory to Experiment
• Automated Search for new Quantum Experiments
• Multipartite entanglement measures
• Witnessing genuine multipartite entanglement with positive maps
• Should Entanglement Measures be Monogamous or Faithful?
• Quantifying high dimensional entanglement with two mutually unbiased bases
• Characterizing multipartite entanglement without shared reference frames
• Quantum Experiments and Graphs III: High-Dimensional and Multi-Particle Entanglement
• Dimensionally sharp inequalities for the linear entropy
• Monogamy of correlations and entropy inequalities in the Bloch picture
• Quantum Experiments and Hypergraphs: Multi-Photon Sources for Quantum Interference, Quantum Computation and Quantum Entanglement
• Digital Discovery of 100 diverse Quantum Experiments with PyTheus
• Nonlocal and controlled unitary operators of Schmidt rank three
• Entanglement cost and entangling power of bipartite unitary and permutation operators
• An algorithm to explore entanglement in small systems
• Thermodynamically optimal creation of correlations
• A complete picture of the four-party linear inequalities in terms of the 0-entropy
• Constraints on Multipartite Quantum Entropies
• Refuting spectral compatibility of quantum marginals
• On two maximally entangled couples
• Thermodynamic Constraints on Information Transmission in Quantum Ensembles
• Graph states of prime-power dimension from generalized CNOT quantum circuit
• Entanglement distillation in terms of a conjectured matrix inequality