A semidefinite program for distillable entanglement

A new additive and semidefinite programming (SDP) computable entanglement measure is introduced to upper bound the amount of distillable entanglement in bipartite quantum states by operations completely preserving the positivity of partial transpose (PPT). This quantity is always smaller than or equal to the logarithmic negativity, the previously best known SDP bound on distillable entanglement, and the inequality is strict in general. Furthermore, a succinct SDP characterization of the one-copy PPT deterministic distillable entanglement for any given state is also obtained, which provides a simple but useful lower bound on the PPT distillable entanglement. Remarkably, there is a genuinely mixed state of which both bounds coincide with the distillable entanglement while being strictly less than the logarithmic negativity.

We address an open question about the existence of entangled continuous-variable (CV) Werner states with positive partial transpose (PPT). We prove that no such state exists by showing that all PPT CV Werner states are separable. The separability follows by observing that these CV Werner states can be approximated by truncating the states into a finite-dimensional convex mixture of product states. In addition, the constituents of the product states comprise a generalized non-Gaussian measurement which gives, rather surprisingly, a strictly tighter upper bound on quantum discord than photon counting. These results uncover the presence of only negative partial transpose entanglement and illustrate the complexity of more general non-classical correlations in this paradigmatic class of genuine non-Gaussian quantum states.

The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics. The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve. In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties. We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension. We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity. As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned. This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point. We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement. Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes. We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

Exploiting special structure in semidefinite programming: A survey of theory and applications

Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information. In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence. Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps. We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.

I show that two distant parties can transform pure entangled states to arbitrary pure states by stochastic local operations and classical communication (SLOCC) at the single copy level, if they share bound entangled states. This is the effect of bound entanglement since this entanglement processing is impossible by SLOCC alone. A similar effect of bound entanglement exists in three qubits where two incomparable entangled states of GHZ (Greenberger, Horne, and Zeilinger) and W can be interconverted. In general, multipartite settings composed by N distant parties, all N-partite pure entangled states are interconvertible by SLOCC with the assistance of bound entangled states with positive partial transpose.

Quantifying coherence and entanglement is extremely important in quantum information processing. Here, we present numerical and analytical results for the geometric measure of coherence, and also present numerical results for the geometric measure of entanglement. On the one hand, we first provide a semidefinite algorithm to numerically calculate geometric measure of coherence for arbitrary finite-dimensional mixed states. Based on this semidefinite algorithm, we test randomly generated single-qubit states, single-qutrit states, and a special kind of d-dimensional mixed states. Moreover, we also obtain an analytical solution of geometric measure of coherence for a special kind of mixed states. On the other hand, another algorithm is proposed to calculate the geometric measure of entanglement for arbitrary two-qubit and qubit-qutrit states, and some special kinds of higher dimensional mixed states. For other states, the algorithm can get a lower bound of the geometric measure of entanglement. Randomly generated two-qubit states, the isotropic states and the Werner states are tested. Furthermore, we compare our numerical results with some analytical results, which coincide with each other.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

Yannakakis [27, 26] showed that the matching problem does not have a small symmetric linear program. Rothvos [23] recently proved that any, not necessarily symmetric, linear program also has exponential size. It is natural to ask whether the matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization. We answer this question negatively for symmetric SDPs: any symmetric SDP for the matching problem has exponential size.We also show that an O(k)-round Lasserre SDP relaxation for the asymmetric metric traveling salesperson problem yields at least as good an approximation as any symmetric SDP relaxation of size nk.The key technical ingredient underlying both these results is an upper bound on the degree needed to derive polynomial identities that hold over the space of matchings or traveling salesperson tours.

Improving the linear relaxation of maximum k-cut with semidefinite-based constraints

We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as functions of a scalar parameter. We present a unifying geometric framework that subsumes the concept of the optimal partition in linear programming (LP) and semidefinite programming (SDP) and extends it to conic optimization. Similar to the optimal partition approach to sensitivity analysis in LP and SDP, the range of perturbations for which the optimal partition remains constant can be computed by solving two conic optimization problems. Under a weaker notion of nondegeneracy, this range is simply given by a minimum ratio test. We discuss briefly the properties of the optimal value function under such perturbations.

Conic Linear Programming, hereafter CLP, is a natural extension of Linear programming (LP). In LP, the variables form a vector which is required to be componentwise nonnegative, while in CLP they are points in a pointed convex cone (see Appendix B.1) of an Euclidean space, such as vectors as well as matrices of finite dimensions. For example, Semidefinite programming (SDP) is a kind of CLP, where the variable points are symmetric matrices constrained to be positive semidefinite. Both types of problems may have linear equality constraints as well. Although CLPs have long been known to be convex optimization problems, no efficient solution algorithm was known until about two decades ago, when it was discovered that interior-point algorithms for LP discussed in Chap. 5, can be adapted to solve certain CLPs with both theoretical and practical efficiency. During the same period, it was discovered that CLP, especially SDP, is representative of a wide assortment of applications, including combinatorial optimization, statistical computation, robust optimization, Euclidean distance geometry, quantum computing, optimal control, etc. CLP is now widely recognized as a powerful mathematical computation model of general importance.

One of the downsides of the massive multiple-input-multiple-output (M-MIMO) system is its computational complexity. Considering that techniques and different algorithms proposed in the literature applied to conventional MIMO may not be well suited or readily applicable to M-MIMO systems, in this paper, the application of different formulations inside the convex optimization framework is investigated. This paper is divided into two parts. In the first part, linear programming, quadratic programming (QP), and semidefinite programming are explored in an M-MIMO environment with high-order modulation and under realistic channel conditions, i.e., considering spatial correlation, error in the channel estimation, as well as different system loading. The bit error rate is evaluated numerically through Monte Carlo simulations. In the second part, algorithms to solve the QP formulation are explored, and computational complexity in terms of floating-point operations (flops) is compared with linear detectors. Those algorithms have interesting aspects when applied to our specific problem (M-MIMO detection formulated as QP), such as the exploitation of the structure of the problem ( simple constraints ) and the improvement of the rate of convergence due to the well-conditioned Gram matrix (channel hardening). The number of iterations is higher when the number of users $K$ becomes similar to the number of base station antennas $M$ (i.e., $K\approx M$ ) than the case $K\ll M$ ; the number of iterations increases slowly as the number of users $K$ and base station antennas $M$ increases while keeping a low system loading. The QP with projected algorithms presented better performance than minimum mean square error detector when $K\approx M$ and promising computational complexity for scenarios with increasing $K$ and low system loading.

Quantum entanglement is the core resource in quantum information processing and quantum computing. It is a significant challenge to effectively characterize the entanglement of quantum states. Recently, elegant separability criterion is presented by Elben et al. (Phys Rev Lett 125:200501, 2020) based on the first three partially transposed (PT) moments of density matrices. Then, Yu et al. (Phys Rev Lett 127:060504, 2021) proposed two general powerful criteria based on the PT moments. In this paper, based on the realignment operations of matrices we propose entanglement detection criteria in terms of such realignment moments. We show by detailed example that the realignment moments can also be used to identify quantum entanglement.