Mutually Unbiased Property of Maximally Entangled Bases and Product Bases in ℂ d ⊗ ℂ d $\mathbb {C}^{d}\otimes \mathbb {C}^{d}$

Abstract

We investigate mutually unbiased property between maximally entangled bases and product bases in bipartite systems $\mathbb {C}^{d} \otimes \mathbb {C}^{d}$ . We first visualize the description of $p_{1}^{a_{1}}-1$ -member mutually unbiased maximally entangled bases(MUMEBs) in $\mathbb {C}^{d} \otimes \mathbb {C}^{d}\ $ , while $d=p_{1}^{a_{1}}p_{2}^{a_{2}}...p_{s}^{a_{s}}$ , $3\leq p_{1}^{a_{1}}\leq p_{2}^{a_{2}}\leq ...\leq p_{s}^{a_{s}}$ , $p_{1}^{a_{1}},...,p_{s}^{a_{s}}$ are distinct primes, which was proposed by Liu et al. (Quantum Inf. Process. 16(6), 159, 2017). We then establish two more mutually unbiased product bases which are also mutually unbiased to the above $p_{1}^{a_{1}}-1$ MUMEBs, thus we present $p_{1}^{a_{1}}+ 1$ mutually unbiased bases(MUBs) in $\mathbb {C}^{d} \otimes \mathbb {C}^{d}\ $ . We also show the concrete construction of those MUBs in bipartite systems $\mathbb {C}^{3} \otimes \mathbb {C}^{3}\ $ , $\mathbb {C}^{4} \otimes \mathbb {C}^{4}\ $ , $\mathbb {C}^{5} \otimes \mathbb {C}^{5}\ $ and $\mathbb {C}^{12} \otimes \mathbb {C}^{12}$ .