Constructing 2 × 2 × 4 and 4 × 4 unextendible product bases and positive-partial-transpose entangled states

Wang, Kai; Chen, Lin; Yi, Shen; Sun, Yize; Zhao, Lei

doi:10.1080/03081087.2019.1588849

Cited by 16 publications

(6 citation statements)

References 28 publications

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“…The studies of UPBs begin with the size of them [5]. Since then, the existence of UPBs with distinct sizes has received extensive attentions [13][14][15]. It is shown that all 4-qubit UPBs are constructed [16].…”

Section: Introductionmentioning

confidence: 99%

The construction and local distinguishability of multiqubit unextendible product bases

Sun,

Chen

2021

Preprint

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An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we construct a 7-qubit UPB of size 11. It solves an open problem in [Quantum Information Processing 19:185 (2020)]. Next, we graphtheoretically show that the UPB is locally indistinguishable in the bipartite systems of two qubits and five qubits, respectively. It turns out that the UPB corresponds to a complete graph with 11 vertices constructed by three sorts of nonisomorphic graphs. Taking the graphs as product vectors, we show that they are in three different orbits up to local unitary equivalence. Moreover, we also present the number of sorts of nonisomorphic graphs of complete graphs of some known UPBs and their orbits.

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Section: Introductionmentioning

confidence: 99%

The construction and local distinguishability of multiqubit unextendible product bases

Sun,

Chen

2021

Preprint

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“…Hence, studying the six-dimensional CHM in terms of Schmidt rank connects the MUB problem and bipartite unitary operations. Besides, the technique employed in our results relate the MUB problem to other fundamental notions like multiqubit entangled states, unextendible product basis and Birkhoff matrices [23][24][25]. Very recently, bipartite operator of Schmidt rank three has been applied to the study of entanglement distillability of three bipartite reduced density operators from the same tripartite state [26].…”

Section: Introductionmentioning

confidence: 99%

Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three

Chen

Sun

2020

Proc. R. Soc. A.

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Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I2 ⊗ V and I2 ⊗ W . We show that V and W have no zero entry, and apply it to exclude examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.

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“…Nonlocality is a fundamental feature of quantum mechanics [1,2]. It is also a key resource in information processing [3][4][5][6], and is related to various topics in quantum information theory such as the understanding of classical and quantum boundary [7,8], the entangling power of nonlocal unitary operations [9][10][11], and the efficient decomposition for realization in quantum circuits [12], unextendible product basis [13], and positivepartial-transpose entangled states [14].…”

Section: Introductionmentioning

confidence: 99%

“…Since S(Ψ abc ) = S(Ψ abd ) = S(Ψ acd ) = S(Ψ bcd ) ≤ 4 cos 2θ for the state Ψ abcd = |ψ abcd ψ abcd |, one gets the inequality(13).The equality holds in(13)when cos θ a cos θ c −cos θ a cos θ c = 2, ω = θ d = 0, θ d = π/2, namely, S max (Ψ abc ) = S max (Ψ abd ) = S max (Ψ acd ) = S max (Ψ bcd ) = 4 cos 2θ ≤ 4. It means that in this case all the reduced states of GGHZ state do not violate the Svetlichny inequality.…”

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confidence: 99%

Trade-off relation among genuine three-qubit nonlocalities in four-qubit systems

et al. 2019

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We study the trade-off relations satisfied by the genuine tripartite nonlocality in multipartite quantum systems. From the reduced three-qubit density matrices of the four-qubit generalized GHZ states and W states, we find that there exists a trade-off relation among the mean values of the Svetlichny operators associated with these reduced states. Namely, the genuine three-qubit nonlocalities are not independent. For four-qubit generalized GHZ states and W states, the summation of all their three-qubit maximal (squared) mean values of the Svetlichny operator is upper bounded. And this bound is superior to the one derived from the upper bounds of individual three-qubit mean values of the Svetlichny operator. Detailed examples are presented to illustrate the trade-off relation among the three-qubit nonlocalities.

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Constructing 2 × 2 × 4 and 4 × 4 unextendible product bases and positive-partial-transpose entangled states

Cited by 16 publications

References 28 publications

The construction and local distinguishability of multiqubit unextendible product bases

The construction and local distinguishability of multiqubit unextendible product bases

Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three

Trade-off relation among genuine three-qubit nonlocalities in four-qubit systems

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