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Dragomir Z. Djokovic

Publications and source records attributed to Dragomir Z. Djokovic.

At least 19 recordsLinked to original sources
arXiv2023

The existence of distinguishable bases in three-dimensional subspaces of qutrit-qudit systems under one-way local operations and classical communication

We show that every three-dimensional subspace of qutrit-qudit complex or real systems has a distinguishable basis under one-way local operations and classical communication (LOCC). In particular this solves an open problem proposed in [J. Phys. A, 40, 7937, 2007]. We construct a three-dimensional space whose locally distinguishable basis is unique and apply the uniqueness property to the task of state transformation. We also construct a three-dimensional locally distinguishable multipartite space assisted with entanglement. On the other hand, we show that four-dimensional indistinguishable bipartite subspaces under one-way LOCC exist. Further, we show that the environment-assisted classical capacity of every channel with a three-dimensional environment is at least $\log_2 3$, and the environment-assisting classical capacity of any qutrit channel is $\log_2 3$. We also show that every two-qutrit state can be converted into a generalized classical state near the quantum-classical boundary by an entanglement-breaking channel.

quant-ph↗
arXiv2018

A SAT+CAS Approach to Finding Good Matrices: New Examples and Counterexamples

We enumerate all circulant good matrices with odd orders divisible by 3 up to order 70. As a consequence of this we find a previously overlooked set of good matrices of order 27 and a new set of good matrices of order 57. We also find that circulant good matrices do not exist in the orders 51, 63, and 69, thereby finding three new counterexamples to the conjecture that such matrices exist in all odd orders. Additionally, we prove a new relationship between the entries of good matrices and exploit this relationship in our enumeration algorithm. Our method applies the SAT+CAS paradigm of combining computer algebra functionality with modern SAT solvers to efficiently search large spaces which are specified by both algebraic and logical constraints.

cs.LO↗
arXiv2018

The unextendible product bases of four qubits: Hasse diagrams

We consider the unextendible product bases (UPBs) of fixed cardinality $m$ in quantum systems of $n$ qubits. These UPBs are divided into finitely many equivalence classes with respect to an equivalence relation introduced by N. Johnston. There is a natural partial order `$\leq$' on the set of these equivalence classes for fixed $m$, and we use this partial order to study the topological closure of an equivalence class of UPBs. In the case of four qubits, for $m=8,9,10$, we construct explicitly the Hasse diagram of this partial order.

quant-ph↗
arXiv2018

Algorithms for difference families in finite abelian groups

Our main objective is to show that the computational methods that we previously developed to search for difference families in cyclic groups can be fully extended to the more general case of arbitrary finite abelian groups. In particular the power density PSD-test and the method of compression can be used to help the search.

math.CO↗
arXiv2018

Multiqubit UPB: The method of formally orthogonal matrices

We use formal matrices whose entries we view as vector variables taking unit vectors values in one-qubit Hilbert spaces of a multiqubit quantum system. We construct many unextendible product bases (UPBs) of new sizes in such systems and provide a new construction of UPBs of $n$ qubits of cardinality $n+1$ when $n\equiv 3 \pmod{4}$. We also give a new method of constructing multiqubit entangled states with all partial transposes positive.

quant-ph↗
arXiv2018

Goethals--Seidel difference families with symmetric or skew base blocks

We single out a class of difference families which is widely used in some constructions of Hadamard matrices and which we call Goethals--Seidel (GS) difference families. They consist of four subsets (base blocks) of a finite abelian group of order $v$, which can be used to construct Hadamard matrices via the well-known Goethals--Seidel array. We consider the special class of these families in cyclic groups, where each base block is either symmetric or skew. We omit the well-known case where all four blocks are symmetric. By extending previous computations by several authors, we complete the classification of GS-difference families of this type for odd $v<50$. In particular, we have constructed the first examples of so called good matrices, G-matrices and best matrices of order 43, and good matrices and G-matrices of order 45. We also point out some errors in one of the cited references.

math.CO↗
arXiv2016

A class of cyclic $(v;k_1,k_2,k_3;λ)$ difference families with $v \equiv 3 \pmod{4}$ a prime

We construct several cyclic $(v;k_1,k_2,k_3;λ)$ difference families with $v\equiv3 \pmod{4}$ a prime and $λ=k_1+k_2+k_3-(3v-1)/4$. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order $4v$. Our main result is that we have constructed for the first time the example of skew-Hadamard matrices of orders $4\cdot239=956$ and $4\cdot331=1324$.

math.CO↗
arXiv2016

On two-distillable Werner states

We consider bipartite mixed states in a $d\otimes d$ quantum system. We say that $ρ$ is PPT if its partial transpose $1 \otimes T (ρ)$ is positive semidefinite, and otherwise $ρ$ is NPT. The well-known Werner states are divided into three types: (a) the separable states (the same as the PPT states); (b) the one-distillable states (necessarily NPT); and (c) the NPT states which are not one-distillable. We give several different formulations and provide further evidence for validity of the conjecture that the Werner states of type (c) are not two-distillable.

quant-ph↗
arXiv2016

Generalization of Scarpis's theorem on Hadamard matrices

A $\{1,-1\}$-matrix $H$ of order $m$ is a Hadamard matrix if $HH^T=mI_m$, where $T$ is the transposition operator and $I_m$ the identity matrix of order $m$. J. Hadamard published his paper on Hadamard matrices in 1893. Five years later, Scarpis showed how one can use a Hadamard matrix of order $n=1+p$, $p\equiv 3 \pmod{4}$ a prime, to construct a bigger Hadamard matrix of order $pn$. In this note we show that Scarpis's construction can be extended to the more general case where $p$ is replaced by a prime power $q$.

math.CO↗
arXiv2015

Symmetric Hadamard matrices of order 116 and 172 exist

We construct new symmetric Hadamard matrices of orders $92,116$, and $172$. While the existence of those of order $92$ was known since 1978, the orders $116$ and $172$ are new. Our construction is based on a recent new combinatorial array discovered by N. A. Balonin and J. Seberry. For order $116$ we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem. We also point out that a new infinite series of symmetric Hadamard matrices arises by plugging into the GP array the matrices constructed by Xia, Xia, Seberry, and Wu in 2005.

math.CO↗
arXiv2015

Boundary of the set of separable states

Motivated by the separability problem in quantum systems $2\otimes4$, $3\otimes3$ and $2\otimes2\otimes2$, we study the maximal (proper) faces of the convex body, $S_1$, of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space $H=H_1\otimes H_2\otimes\cdots\otimes H_n$. To any subspace $V$ of $H$ we associate a face $F_V$ of $S_1$ consisting of all states $ρ\in S_1$ whose range is contained in $V$. We prove that $F_V$ is a maximal face if and only if $V$ is a hyperplane. If $V$ is the hyperplane orthogonal to a product vector, we prove that $\dim F_V=d^2-1-\prod(2d_i-1)$, where $d_i$ is the dimension of $H_i$ and $d=\prod d_i$. We classify the maximal faces of $S_1$ in the cases $2\otimes2$ and $2\otimes3$. In particular we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for $2\otimes2$, and 20 and 24 for $2\otimes3$. The boundary of $S_1$ is the union of all maximal faces. When $d>6$ we prove that there exist full states $ρ$ on the boundary, i.e., such that all partial transposes of $ρ$ (including $ρ$ itself) have rank $d$. K.-C. Ha and S.-K. Kye have recently constructed explicit such states in $2\times4$ and $3\otimes3$. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter $b>0$, $b\ne1$. Each face in the family is a 9-dimensional simplex and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses (OEW) for these faces and analyze the three limiting cases $b=0,1,\infty$.

quant-ph↗
arXiv2015

Dimension formula for induced maximal faces of separable states and genuine entanglement

The normalized separable states of a finite-dimensional multipartite quantum system, represented by its Hilbert space ${\cal H}$, form a closed convex set ${\cal S}_1$. The set ${\cal S}_1$ has two kinds of faces, induced and non-induced. An induced face, $F$, has the form $F=Γ(F_V)$, where $V$ is a subspace of ${\cal H}$, $F_V$ is the set of $ρ\in{\cal S}_1$ whose range is contained in $V$, and $Γ$ is a partial transposition operator. Such $F$ is a maximal face if and only if $V$ is a hyperplane. We give a simple formula for the dimension of any induced maximal face. We also prove that the maximum dimension of induced maximal faces is equal to $d(d-2)$ where $d$ is the dimension of ${\cal H}$. The equality $\dimΓ(F_V)=d(d-2)$ holds if and only if $V^\perp$ is spanned by a genuinely entangled vector.

quant-ph↗
arXiv2015

D-optimal matrices of orders 118, 138, 150, 154 and 174

We construct supplementary difference sets (SDS) with parameters $(59;28,22;21)$, $(69;31,27;24)$, $(75;36,29;28)$, $(77;34,31;27)$ and $(87;38,36;31)$. These SDSs give D-optimal designs (DO-designs) of two-circulant type of orders 118,138,150,154 and 174. Until now, no DO-designs of orders 138,154 and 174 were known. While a DO-design (not of two-circulant type) of order 150 was constructed previously by Holzmann and Kharaghani, no such design of two-circulant type was known. The smallest undecided order for DO-designs is now 198. We use a novel property of the compression map to speed up some computations.

math.CO↗
arXiv2015

Periodic Golay pairs of length 72

We construct supplementary difference sets (SDS) with parameters $(72;36,30;30)$. These SDSs give periodic Golay pairs of length 72. No periodic Golay pair of length 72 was known previously. The smallest undecided order for periodic Golay pairs is now 90. The periodic Golay pairs constructed here are the first examples having length divisible by a prime congruent to 3 modulo 4. The main tool employed is a recently introduced compression method. We observe that Turyn's multiplication of Golay pairs can be also used to multiply a Golay pair and a periodic Golay pair.

math.CO↗
arXiv2014

Some new periodic Golay pairs

Periodic Golay pairs are a generalization of ordinary Golay pairs. They can be used to construct Hadamard matrices. A positive integer $v$ is a (periodic) Golay number if there exists a (periodic) Golay pair of length $v$. Taking into the account the results obtained in this note and an yet unpublished new result, there are only seven known periodic Golay numbers which are definitely not Golay numbers, namely 34,50,58,68,72,74,82. We construct here periodic Golay pairs of lengths 74,122,164,202,226. It is apparently unknown whether 122,164,202,226 are Golay numbers. The smallest length for which the existence of periodic Golay pairs is undecided is now 90.

math.CO↗
arXiv2013

Four-qubit pure states as fermionic states

The embedding of the $n$-qubit space into the $n$-fermion space with $2n$ modes is a widely used method in studying various aspects of these systems. This simple mapping raises a crucial question: does the embedding preserve the entanglement structure? It is known that the answer is affirmative for $n=2$ and $n=3$. That is, under either local unitary (LU) operations or with respect to stochastic local operations and classical communication (SLOCC), there is a one-to-one correspondence between the 2- (or 3)-qubit orbits and the 2- (or 3)-fermion orbits with 4 (or 6) modes. However these results do not generalize as the mapping from the $n$-qubit orbits to the $n$-fermion orbits with $2n$ modes is no longer surjective for $n>3$. Here we consider the case of $n=4$. We show that surprisingly, the orbit mapping from qubits to fermions remains injective under SLOCC, and a similar result holds under LU for generic orbits. As a byproduct, we obtain a complete answer to the problem of SLOCC equivalence of pure 4-qubit states.

quant-ph↗
arXiv2013

Proof of the Gour-Wallach conjecture

The absolute value of the hyperdeterminant of four qubits is a useful measure of genuine entanglement. We prove a recent conjecture of Gour and Wallach describing the pure maximally entangled four-qubit states with respect to this measure.

quant-ph↗