Classical complexity and quantum entanglement

Physical Review A

We study families of positive and completely positive maps acting on a bipartite system C M ⊗ C N (with M ≤ N). The maps have a property that when applied to any state (of a given entanglement class) they result in a separable state, or more generally a state of another certain entanglement class (e.g., Schmidt number ≤ k). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary M, N, with a special emphasis on M = 2. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to "weakly" entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure state projectors, partially transposed and locally transformed pure state projectors.

Physical Review A

We describe a direct method for experimental determination of the negativity of an arbitrary twoqubit state with 11 measurements performed on multiple copies of the two-qubit system. Our method is based on the experimentally accessible sequences of singlet projections performed on up to four qubit pairs. In particular, our method permits the application of the Peres-Horodecki separability criterion to an arbitrary two-qubit state. We explicitly demonstrate that measuring entanglement in terms of negativity requires three measurements more than detecting two-qubit entanglement. The reported minimal set of interferometric measurements provides a complete description of bipartite quantum entanglement in terms of two-photon interference. This set is smaller than the set of 15 measurements needed to perform a complete quantum state tomography of an arbitrary twoqubit system. Finally, we demonstrate that the set of 9 Makhlin's invariants needed to express the negativity can be measured by performing 13 multicopy projections. We demonstrate that these invariants are both a useful theoretical concept for designing specialized quantum interferometers and that their direct measurement within the framework of linear optics does not require performing complete quantum state tomography.

Physical Review A, 2018

We study the problem of optimal preparation of a bipartite entangled state, which remains entangled the longest time under action of local qubit noises. We show that for unital noises such a state is always maximally entangled, whereas for nonunital noises, it is not. We develop a decomposition technique relating nonunital and unital qubit channels, based on which we find the explicit form of the ultimately robust state for general local noises. We illustrate our findings by amplitude damping processes at finite temperature, for which the ultimately robust state remains entangled up to two times longer than conventional maximally entangled states.

Foundations of Computational Mathematics, 2019

In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over Q is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time [IQS17] (whether or not randomization is allowed). The algorithm efficiently solves the "word problem" for the free skew field, and the identity testing problem for arithmetic formulae with division over noncommuting variables, two problems which had only exponential-time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits [Gur04], who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits [Gur04]). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp-Lieb inequalities). We also extend the algorithm to compute not only singularity, but actually the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation of the commutative rank. This naturally raises a relaxation of the commutative PIT problem to achieving better deterministic approximation of the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular we explain how some of these sources played an important role in the development of Gurvits' algorithm and in our analysis of it here.

2021

Bang-Hai Wang1,∗ Si-Qi Zhou2,3,† Zhihao Ma2,3,‡ and Shao-Ming Fei4,5§ School of Computer Science and Technology, Guangdong University of Technology, Guangzhou 510006, China School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

Lobachevskii Journal of Mathematics, 2020

Phase covariant qubit dynamics describes an evolution of a two-level system under simultaneous action of pure dephasing, energy dissipation, and energy gain with time-dependent rates γz(t), γ−(t), and γ+(t), respectively. Non-negative rates correspond to completely positive divisible dynamics, which can still exhibit such peculiarities as non-monotonicity of populations for any initial state. We find a set of quantum channels attainable in the completely positive divisible phase covariant dynamics and show that this set coincides with the set of channels attainable in semigroup phase covariant dynamics. We also construct new examples of eternally indivisible dynamics with γz(t) < 0 for all t > 0 that is neither unital nor commutative. Using the quantum Sinkhorn theorem, we for the first time derive a restriction on the decoherence rates under which the dynamics is positive divisible, namely, γ±(t) ≥ 0, γ+(t)γ−(t) + 2γz(t) > 0. Finally, we consider phase covariant convolution master equations and find a class of admissible memory kernels that guarantee complete positivity of the dynamical map.

Annals of Physics

Entanglement detection problem is one of the important problem in quantum information theory. Gurvit [1] showed that this problem is NP complete and thus this may be the possible reason that only one criterion is not sufficient to detect all entangled states. There are some powerful entanglement detection criterion such as partial transposition criterion, realignment criterion that it may not be possible to implement them successfully in the experiment. But this situation can be avoided if the entanglement is detected through the construction of witness operator method. This method involves the calculation of expectation values of the Hermitian operator W and if it comes out to be negative with respect to the state ρ which is to be detected then we definitely conclude that the state ρ is entangled. The construction of witness operator to detect entanglement is advantageous in two ways: (i) the entanglement can be detected experimentally if we have prior partial information about the state which is under investigation and (ii) It can detect both negative partial transpose entangled states (NPTES) and positive partial transpose entangled states (PPTES). In this work, we take an analytical approach to construct a witness operator. To achieve this task, we first construct a linear map using partial transposition and realignment operation. Then we find some conditions on the parameters of the map for which the map represent a positive map. Further, we have constructed a Choi matrix corresponding to the map and have shown that it is not positive semi-definite, which clearly tells the fact that the constructed map is not completely positive. We then construct an operator, which is based on the linear combination of the function of Choi matrix and the identity matrix and show that it satisfies all the properties of the witness operator. Furthermore, we show that the constructed witness operator can detect both NPTES and PPTES. Finally, we prove its efficiency by detecting several bipartite bound entangled states which were previously undetected by some well-known separability criteria. We also compared the detection power of our witness operator with three well-known powerful entanglement detection criteria, namely, dV criterion, CCNR criterion and the separability criteria based on correlation tensor (CT) proposed by Sarbicki et. al. and find that our witness operator detect more entangled states than these criterion.

Physical Review A

We present the variational separability verifier (VSV), which is a variational quantum algorithm that determines the closest separable state (CSS) of an arbitrary quantum state with respect to the Hilbert-Schmidt distance (HSD). We first assess the performance of the VSV by investigating the convergence of the optimization procedure for Greenberger-Horne-Zeilinger states of up to seven qubits, using both state-vector and shot-based simulations. We also numerically determine the CSS of maximally entangled mixed X states, and subsequently use the results of the algorithm to surmise the analytical form of the aforementioned CSS. Our results indicate that current noisy intermediate-scale quantum devices may be useful in addressing the NP-hard full separability problem using the VSV, due to the shallow quantum circuit imposed by employing the destructive SWAP test to evaluate the HSD. The VSV may also possibly lead to the characterization of multipartite quantum states, once the algorithm is adapted and improved to obtain the closest k-separable state of a multipartite entangled state.

Entropy, 2018

We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those inva...

SIAM Journal on Computing, 2021

In the 1970's, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings (FCT 1979). A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS 2017). In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex c-coloring problem reduce to the maximum isotropic space problem and the isotropic c-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.

Scientific Reports

Quantum entanglement is one of the essential resources involved in quantum information processing tasks. However, its detection for usage remains a challenge. The Bell-type inequality for relative entropy of coherence serves as an entanglement witness for pure entangled states. However, it does not perform reliably for mixed entangled states. This paper constructs a classifier by employing the relationship between coherence and entanglement for supervised machine learning methods. This method encodes multiple Bell-type inequalities for the relative entropy of coherence into an artificial neural network to detect the entangled and separable states in a quantum dataset.

Geometric and Functional Analysis, 2018

The celebrated Brascamp-Lieb (BL) inequalities [BL76, Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry and information theory, with many used in computer science. While their structural theory is very well understood, far less is known about computing their main parameters (which we later define below). Prior to this work, the best known algorithms for any of these optimization tasks required at least exponential time. In this work, we give polynomial time algorithms to compute: (1) Feasibility of BL-datum, (2) Optimal BLconstant, (3) Weak separation oracle for BL-polytopes. What is particularly exciting about this progress, beyond the better understanding of BLinequalities, is that the objects above naturally encode rich families of optimization problems which had no prior efficient algorithms. In particular, the BL-constants (which we efficiently compute) are solutions to non-convex optimization problems, and the BL-polytopes (for which we provide efficient membership and separation oracles) are linear programs with exponentially many facets. Thus we hope that new combinatorial optimization problems can be solved via reductions to the ones above, and make modest initial steps in exploring this possibility. Our algorithms are obtained by a simple efficient reduction of a given BL-datum to an instance of the Operator Scaling problem defined by [Gur04]. To obtain the results above, we utilize the two (very recent and different) algorithms for the operator scaling problem [GGOW16, IQS15]. Our reduction implies algorithmic versions of many of the known structural results on BL-inequalities, and in some cases provide proofs that are different or simpler than existing ones. Further, the analytic properties of the [GGOW16] algorithm provide new, effective bounds on the magnitude and continuity of BL-constants; prior work relied on compactness, and thus provided no bounds. On a higher level, our application of operator scaling algorithm to BL-inequalities further connects analysis and optimization with the diverse mathematical areas used so far to motivate and solve the operator scaling problem, which include commutative invariant theory, non-commutative algebra, computational complexity and quantum information theory.

Physical Review Letters

Machine Learning

In this work, we study the optimal transport (OT) problem between symmetric positive definite (SPD) matrix-valued measures. We formulate the above as a generalized optimal transport problem where the cost, the marginals, and the coupling are represented as block matrices and each component block is a SPD matrix. The summation of row blocks and column blocks in the coupling matrix are constrained by the given block-SPD marginals. We endow the set of such block-coupling matrices with a novel Riemannian manifold structure. This allows to exploit the versatile Riemannian optimization framework to solve generic SPD matrix-valued OT problems. We illustrate the usefulness of the proposed approach in several applications.

Journal of Mathematical Physics, 2021

Several techniques of generating random quantum channels, which act on the set of d-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show they are mathematically equivalent. We discuss under which conditions they give the uniform, Lebesgue measure on the convex set of quantum operations and compare their advantages and computational complexity, and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity and the 2-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed and their spectral properties are studied using the Bloch representation of a classical probability vector.