arXiv:2104.14058v4 [quant-ph] 21 Dec 2022 ON A CLASS OF k-ENTANGLEMENT WITNESSES TOMASZ M LYNIK, HIROYUKI OSAKA, AND MARCIN MARCINIAK Abstract. Recently, Yang et al. showed that each 2-positive map acting from M 3 into itself is decomposable. It is equivalent to the statement that each PPT state on C 3 ⊗C 3 has Schmidt number at most 2. It is a generalization of Perez- Horodecki criterion which states that each PPT state on C 2 ⊗ C 2 or C 2 ⊗ C 3 has Schmidt rank 1 i.e. is separable. Natural question arises whether the result of Yang at al. stays true for PPT states on C 3 ⊗ C 4 . This question can be considered also in higher dimensions. Motivated by these results and questions we construct a family of positive maps between matrix algebras of different dimensions depending on a parameter with the property that their k-positivity can be easily controlled. The estimate bounds on the parameter a are better than those derived from the spectral conditions considered by Chru´ sci´ nski and Kossakowski. We found that in case where dimensions are differ by one we can give explicit analytic formula for parameter a that guarantee k-positivity. 1. Introduction Characterizing the mixed states of composed quantum systems to determine whether they exhibits quantum correlation or in other words if the state is entangled or sep- arable [1, 2] is a key problem in quantum information theory. In particular, it is important in private and quantum communication problems [3]. Most experiments related to communication issues are performed on the basis of entanglement be- tween qubits. For low-dimensional complex systems, i.e. qubit-qubit, qubit-qutrit, the Peres-Horodecki criterion [4] completely characterizes separable states. It as- serts that a state is separable if and only if its partial transpose is positive. However, thanks to advances in technology, it is now possible to experimentally control com- posed systems with a larger number of degrees of freedom [5, 6, 7, 8]. For the high-dimensional systems there is no complete separability condition. Recall that a state is separable if and only if its Schmidt number [9] is equal to one. Therefore, one can look at the Schmidt number as an indicator of how far away from separabil- ity it is. Having in mind the Peres-Horodecki criterion for low dimensions one may ask how large the Schmidt number can be for the PPT states in higher dimensions [10, 11, 12, 13]. In [12], a family of PPT states in M m ⊗M m is constructed whose Schmidt number is between 1 and  m-1 4  . This result is improved in [13, 35], where PPT states with a range of Schmidt number from 1 to  m-1 2  are described. Regarding the low-dimensional case, in [14] it is proved that every 2-positive linear map from M 3 to M 3 is decomposable, which is a natural continuation of [15, 16, 17]. This result can be restated dually [18, 19, 20]: the Schmidt number of each two- qutrit PPT state is at most 2. In view of this result the natural question arise: Date : December 22, 2022. Key words and phrases. ¡entanglement¿, entanglement witness, positive maps. 1