Unextendible maximally entangled bases in 
                
                  
                
                $${\mathbb {C}}^{pd}\otimes {\mathbb {C}}^{qd}$$
                
                  
                    
                      
                        
                          C
                        
                        
                          pd
                        
                      
                      ⊗

Zhang, Guijun; Tao, Yuan-Hong; Han, Yi-Fan; Yong, Xin-Lei; Fei, Shao-Ming

doi:10.1007/s11128-018-2094-4

Cited by 10 publications

(2 citation statements)

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“…There they proved that no UMEB exists in two qubits system and presented examples of UMEBs in C 3 ⊗ C 3 and C 4 ⊗ C 4 . Since then, the UMEB was further studied by several researchers [14][15][16][17][18][19][20][21]. Lots of the works paid attention to the UMEBs for general quantum systems C d ⊗ C d′ .…”

Section: Introductionmentioning

confidence: 99%

Constructions of Unextendible Special Entangled Bases

Wang

2022

Front. Phys.

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Unextendible product basis (UPB), a set of incomplete orthonormal product states whose complementary space has no product state, is very useful for constructing bound entangled states. Naturally, instead of considering the set of product states, Bravyi and Smolin considered the set of maximally entangled states. They introduced the concept of unextendible maximally entangled basis (UMEB), a set of incomplete orthonormal maximally entangled states whose complementary space contains no maximally entangled state [Phys. Rev. A 84, 042,306 (2011)]. An entangled state whose nonzero Schmidt coefficients are all equal to 1/k is called a special entangled state of “type k”. In this paper, we introduce a concept named special unextendible entangled basis of “type k” which generalizes both UPB and UMEB. A special unextendible entangled basis of “type k” (SUEBk) is a set of incomplete orthonormal special entangled states of “type k” whose complementary space has no special entangled state of “type k”. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by k while the other one can be generated by two kinds of the special entangled states of “type k”. This method is very effective when k = pm ≥ 3 where p is a prime number. For these cases, we can obtain sets of SUEBk with continuous integer cardinality when the local dimensions are large.

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Section: Introductionmentioning

confidence: 99%

Constructions of Unextendible Special Entangled Bases

Wang

2022

Front. Phys.

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“…Similarly, if the product state is replaced by the maximally entangled state in the previous definition, we can get the definition of the unextendable maximally entangled basis. The UMEB was introduced by Bravyi et al [18], and was further studied in [19,20,[22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Construction of special entangled basis based on generalized weighing matrices

Wang

2019

J. Phys. A: Math. Theor.

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A special entangled basis (SEBk) is a set with orthonormal entangled pure states in whose nonzero Schmidt coefficients are all equal to . When k is equal to the minimum of d and , we get a maximally entangled basis. In this paper, we present how to construct a special entangled basis via some special matrices which are known as weighing matrices. Specifically, using a skew Hadamard matrix of order k + 1, we derive a weighing matrix that is useful for constructing SEBk in whenever . These results are further progress of those studied by Guo et al (2015 J. Phys. A: Math. Theor. 48 245301). We also disprove two conjectures proposed by them.

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