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arXiv:quant-ph/0009109 · Physical Review A

Schmidt-number witnesses and bound entanglement

Anna Sanpera, Dagmar Bruß, Maciej Lewenstein

Published Apr 9, 2001• 233 citations
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Abstract

The Schmidt number of a mixed state characterizes the minimum Schmidt rank of the pure states needed to construct it. We investigate the Schmidt number of an arbitrary mixed state by studying Schmidt-number witnesses that detect it. We present a canonical form of such witnesses and provide constructive methods for their optimization. Finally, we present strong evidence that all bound entangled states with positive partial transpose in ${\mathcal{C}}^{3}\ensuremath{\bigotimes}{\mathcal{C}}^{3}$ have Schmidt number 2.

DOI
10.1103/physreva.63.050301

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•Sep 6, 2006•0 citations
The Schmidt number of a mixed state characterizes the minimum Schmidt rank of the pure states needed to construct it. We investigate the Schmidt number of an arbitrary mixed state by studying Schmidt-number witnesses that detect it. We present a canonical form of such witnesses and provide constructive methods for their optimization. Finally, we present strong evidence that all bound entangled states with positive partial transpose in C3\ensuremath⨂C3{\mathcal{C}}^{3}\ensuremath{\bigotimes}{\mathcal{C}}^{3}C3\ensuremath⨂C3 have Schmidt number 2.
A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
Families of kkk-positive maps and Schmidt number witnesses from generalized equiangular measurements
Quantum entanglement is an important resource in many modern technologies, like quantum computation or quantum communication and information processing. Therefore, most interest is given to detect and quantify entangled states. Entanglement degree of bipartite mixed quantum states can be measured using the Schmidt number. Witnesses of the Schmidt number are closely related to kkk-positive linear maps, for which there is no general construction. Here, we use the generalized equiangular measurements to define a family of kkk-positive linear maps and the corresponding Schmidt number witnesses. We present examples of witnesses that detect two-parameter entangled states with Schmidt number k+1k+1k+1 in any dimension. Our approach allows for a more efficient entanglement quantification than other known Schmidt number witnesses constructed from symmetric measurement operators.
Entanglement plays a crucial role in quantum information science and many-body physics, yet quantifying it in mixed quantum many-body systems has remained a notoriously difficult problem. Here, we introduce families of quantitative entanglement witnesses, constructed from partial transpose and realignment moments, which provide rigorous bounds on entanglement monotones as well as entanglement dimensionality. Our witnesses can be efficiently measured using SWAP tests or variants of Bell measurements, thus making them directly implementable on current hardware. Leveraging our witnesses, we present several novel results on entanglement properties of mixed states, both in quantum information and many-body physics. We develop efficient algorithms to test whether mixed states with bounded entropy have low or high entanglement, which previously was only possible for pure states. We also provide an efficient algorithm to test the Schmidt rank using only two-copy measurements, and the operator Schmidt rank using four-copy measurements. Further, our witnesses robustly certify the quantum circuit depth in the presence of noise, as well as the Schmidt rank of mixed states. Finally, we show that the entanglement phase diagram of Haar random states, quantified by the partial transpose negativity, can be fully established solely by computing our witness, a result that also applies to any state 4 -design. Our witnesses can also be efficiently computed for matrix product states, thus enabling the characterization of entanglement in extensive many-body systems. Finally, we make progress on the entanglement required for quantum cryptography, establishing rigorous limits on pseudoentanglement and pseudorandom density matrices with bounded entropy. Our work opens new avenues for quantifying entanglement in large and noisy quantum systems.
We study the structure of k-Schmidt witnesses in infinite dimensions. We show that for any states with Schmidt number greater than , there exists a -Schmidt witnesses of the form to detect it, where and is a finite rank self-adjoint operator. We provide a method to construct -Schmidt witnesses with these forms. An elementary operator criterion to determine the lower bound of the Schmidt number of a state is obtained. We also construct a class of 2-Schmidt witnesses, from which we can detect the entanglement of states by entries of their matrix representations.
We prove the existence of bound entangled states with negative partial transpose (NPT) in any d × d(d ≥ 3) Hilbert space with a simple assumption on Schmidt rank two states. Obviously they belong to the class of conjectured to be bound entangled states by Divincenzo et.al [Phys. Rev. A, 61, 062312(2000)] and by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. PACS number(s): 03.67.Hk, 03.65.Ud. The basic issue on the classification of mixed state entanglement at least on the level of bipartite systems solely depends upon whether there exist bound entangled states or not. The existence of PPT-bound (PPT means positive partial transpose) entangled states [1] and also the existence of NPT N−copy undistillable states [2, 3] for every positive integer N naturally indicates there may exist NPT-bound entangled states. In this work we are able to show the existence of NPT-bound entangled states with a simple assumption on Schmidt rank two states. We first briefly describe the issue and the importance of the problem. In recent years it is found that quantum entanglement is an useful resource in performing several tasks in quantum information theory and quantum communication [4]. Maximally entangled states shared between two parties are essential ingredients in this respect [5]. Now due to the interaction with environment, states are in practice found to be mixed. However there is a process called distillation by which we can distill sometimes maximally entangled