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Presented by The Aspen Center for Physics Scott Aaronson is Schlumberger Chair of Computer Science and Director, Quantum Information Center, Department of Computer Science at The University of Texas at Austin. In summer 2022 he announced he would be working for a year at OpenAI on theoretical foundations of AI safety. Know Aspen. Watch GrassRoots. GrassRoots TV programming is entirely funded by direct contributions from members of the Roaring Fork Valley community, and through grants from the City of Aspen Community Non-Profit Grant Program, Pitkin County Healthy Community Fund, The Town of Snowmass Village and The Aspen Thrift Shop. Founded by citizens of Aspen, Colorado in 1972, GrassRoots TV is the first and oldest community operated and programmed open-source media center and television station in the United States. Virtually all of the programming is created by local residents, occasionally with world renown guests. In partnership with the American Archive of Public Media and the Library of Congress, GrassRoots is currently cataloguing, restoring and digitally archiving 500 of the most at-risk original video programs out of its 45,000 title Community Video Archive. Help sustain this community treasure by donating now! Go to: https://bit.ly/GRTVContribute Don't forget to SUBSCRIBE, HIT LIKE and leave a COMMENT to let us know if you enjoyed this video, it is important to us and the community for you to become part of the conversation. Thanks for tuning in! Download the GrassRoots Community Network iOS and Android apps Subscribe for more videos: https://bit.ly/2Ycpi4P Facebook: https://www.facebook.com/GrassRootsCommunityNetwork/ Twitter: https://twitter.com/grassrootstv Web: http://www.grassrootstv.org/

We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial isoperimetric function. The embedding can be chosen in such a way that $G$ has bounded distortion in $H$.

We study the problem of optimal multi-robot path planning on graphs (MPP) over four distinct minimization objectives: the total arrival time, the makespan (last arrival time), the total distance, and the maximum (single-robot traveled) distance. On the structure side, we show that each pair of these four objectives induces a Pareto front and cannot always be optimized simultaneously. Then, through reductions from 3-SAT, we further establish that computation over each objective is an NP-hard task, providing evidence that solving MPP optimally is generally intractable. Nevertheless, in a related paper, we design complete algorithms and efficient heuristics for optimizing all four objectives, capable of solving MPP optimally or near-optimally for hundreds of robots in challenging setups.

Some elementary rigorous remark about the replica formalism in the Statistical Physics' approach to threshold phenomena in Computational Complexity Theory is presented.

The Sarkar-Wang algorithm computes the hat version of the Heegaard Floer homology of a closed oriented three manifold. This paper analyzes the computational complexity of the Sarkar-Wang algorithm; then the algorithm is modified to obtain a better bound. Then the computational complexity of calculating HFK hat from a Heegaard diagram by means of the modified Sarkar-Wang algorithm is also analyzed. Under certain assumptions it is shown that the modified Sarkar-Wang algorithm is faster than the Manolescu-Ozsvath-Sarkar algorithm.

The Sarkar-Wang algorithm computes the hat version of the Heegaard Floer homology of a closed oriented three manifold. This paper analyzes the computational complexity of the Sarkar-Wang algorithm; then the algorithm is modified to obtain a better bound. Then the computational complexity of calculating HFK hat from a Heegaard diagram by means of the modified Sarkar-Wang algorithm is also analyzed. Under certain assumptions it is shown that the modified Sarkar-Wang algorithm is faster than the Manolescu-Ozsvath-Sarkar algorithm.

The Sarkar-Wang algorithm computes the hat version of the Heegaard Floer homology of a closed oriented three manifold. This paper analyzes the computational complexity of the Sarkar-Wang algorithm; then the algorithm is modified to obtain a better bound. Then the computational complexity of calculating HFK hat from a Heegaard diagram by means of the modified Sarkar-Wang algorithm is also analyzed. Under certain assumptions it is shown that the modified Sarkar-Wang algorithm is faster than the Manolescu-Ozsvath-Sarkar algorithm.

The Sarkar-Wang algorithm computes the hat version of the Heegaard Floer homology of a closed oriented three manifold. This paper analyzes the computational complexity of the Sarkar-Wang algorithm; then the algorithm is modified to obtain a better bound. Then the computational complexity of calculating HFK hat from a Heegaard diagram by means of the modified Sarkar-Wang algorithm is also analyzed. Under certain assumptions it is shown that the modified Sarkar-Wang algorithm is faster than the Manolescu-Ozsvath-Sarkar algorithm.

The Sarkar-Wang algorithm computes the hat version of the Heegaard Floer homology of a closed oriented three manifold. This paper analyzes the computational complexity of the Sarkar-Wang algorithm; then the algorithm is modified to obtain a better bound. Then the computational complexity of calculating HFK hat from a Heegaard diagram by means of the modified Sarkar-Wang algorithm is also analyzed. Under certain assumptions it is shown that the modified Sarkar-Wang algorithm is faster than the Manolescu-Ozsvath-Sarkar algorithm.

Modern financial networks exhibit a high degree of interconnectedness and determining the causes of instability and contagion in financial networks is necessary to inform policy and avoid future financial collapse. In the American Economic Review, Elliott, Golub and Jackson proposed a simple model for capturing the dynamics of complex financial networks. In Elliott, Golub and Jackson's model, each institution in the network can buy underlying assets or percentage shares in other institutions (cross-holdings) and if any institution's value drops below a critical threshold value, its value suffers an additional failure cost. This work shows that even in simple model put forward by Elliott, Golub and Jackson there are fundamental barriers to understanding the risks that are inherent in a network. First, if institutions are not required to maintain a minimum amount of self-holdings, an $\epsilon$ change in investments by a single institution can have an arbitrarily magnified influence on the net worth of the institutions in the system. This sensitivity result shows that if institutions have small self-holdings, then estimating the market value of an institution requires almost perfect information about every cross-holding in the system. Second, we show that even if a regulator has complete information about all cross-holdings in the system, it may be computationally intractable to even estimate the number of failures that could be caused by an arbitrarily small shock to the system. Together, these results show that any uncertainty in the cross-holdings or values of the underlying assets can be amplified by the network to arbitrarily large uncertainty in the valuations of institutions in the network.

We consider some well-known families of two-player, zero-sum, perfect information games that can be viewed as special cases of Shapley's stochastic games. We show that the following tasks are polynomial time equivalent: - Solving simple stochastic games. - Solving stochastic mean-payoff games with rewards and probabilities given in unary. - Solving stochastic mean-payoff games with rewards and probabilities given in binary.

We analyse the resilience of the quantum search algorithm in the presence of quantum noise modelled as trace preserving completely positive maps. We study the influence of noise on computational complexity of the quantum search algorithm. We show that only for small amounts of noise the quantum search algorithm is still more efficient than any classical algorithm.

This is an open letter that we sent to S. Ilie, G. Soederlind and R.M. Corless in August 2008.

This is an open letter that we sent to S. Ilie, G. Soederlind and R.M. Corless in August 2008.

This is an open letter that we sent to S. Ilie, G. Soederlind and R.M. Corless in August 2008.

We introduce a prime number generator in the form of a stochastic algorithm. The character of such algorithm gives rise to a continuous phase transition which distinguishes a phase where the algorithm is able to reduce the whole system of numbers into primes and a phase where the system reaches a frozen state with low prime density. In this paper we firstly pretend to give a broad characterization of this phase transition, both in terms of analytical and numerical analysis. Critical exponents are calculated, and data collapse is provided. Further on we redefine the model as a search problem, fitting it in the hallmark of computational complexity theory. We suggest that the system belongs to the class NP. The computational cost is maximal around the threshold, as common in many algorithmic phase transitions, revealing the presence of an easy-hard-easy pattern. We finally relate the nature of the phase transition to an average-case classification of the problem.

We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem for a ball or cone is NP-hard if and only if the corresponding problem for the dual ball or cone is NP-hard. In a similar vein, we show that computation of the dual norm of a given norm is polynomial-time reducible to computation of the given norm. This extends to convex functions satisfying a polynomial growth condition: for such a given function, computation of its Fenchel dual/conjugate is polynomial-time reducible to computation of the given function. Hence the computation of a norm or a convex function of polynomial-growth is NP-hard if and only if the computation of its dual norm or Fenchel dual is NP-hard. We discuss implications of these results on the weak membership problem for a symmetric convex body and its polar dual, the polynomial approximability of Mahler volume, and the weak membership problem for the epigraph of a convex function with polynomial growth and that of its Fenchel dual.

We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem for a ball or cone is NP-hard if and only if the corresponding problem for the dual ball or cone is NP-hard. In a similar vein, we show that computation of the dual norm of a given norm is polynomial-time reducible to computation of the given norm. This extends to convex functions satisfying a polynomial growth condition: for such a given function, computation of its Fenchel dual/conjugate is polynomial-time reducible to computation of the given function. Hence the computation of a norm or a convex function of polynomial-growth is NP-hard if and only if the computation of its dual norm or Fenchel dual is NP-hard. We discuss implications of these results on the weak membership problem for a symmetric convex body and its polar dual, the polynomial approximability of Mahler volume, and the weak membership problem for the epigraph of a convex function with polynomial growth and that of its Fenchel dual.

We consider the problem of Data Flow Analysis over monotone data flow frameworks with a finite lattice. The problem of computing the Maximum Fixed Point (MFP) solution is shown to be P-complete even when the lattice has just four elements. This shows that the problem is unlikely to be efficiently parallelizable. It is also shown that the problem of computing the Meet Over all Paths (MOP) solution is NL-complete (and hence efficiently parallelizable) when the lattice is finite even for non-monotone data flow frameworks. These results appear in contrast with the fact that when the lattice is not finite, solving the MOP problem is undecidable and hence significantly harder than the MFP problem which is polynomial time computable for lattices of finite height.

This paper describes a method for computing the Discrete Fourier Transform (DFT) of a sequence of n elements over a finite field GF (p to the mth power) with a number of bit operations 0(nm log (nm). P(q)) where P(q) is the number of bit operations required to multiply two g-bit integers and g approx. = 2 log sub 2 + 4 log sub 2m + 4 log sub 2p. This method is uniformly applicable to all instances and its order of complexity is not inferior to that of methods whose success depends upon the existence of certain primes.

We prove that one-way quantum computations have the same computational power as quantum circuits with unbounded fan-out. It demonstrates that the one-way model is not only one of the most promising models of physical realisation, but also a very powerful model of quantum computation. It confirms and completes previous results which have pointed out, for some specific problems, a depth separation between the one-way model and the quantum circuit model. Since one-way model has the same computational power as unbounded quantum fan-out circuits, the quantum Fourier transform can be approximated in constant depth in the one-way model, and thus the factorisation can be done by a polytime probabilistic classical algorithm which has access to a constant-depth one-way quantum computer. The extra power of the one-way model, comparing with the quantum circuit model, comes from its classical-quantum hybrid nature. We show that this extra power is reduced to the capability to perform unbounded classical parity gates in constant depth.

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability.

Technology trends are making the cost of data movement increasingly dominant, both in terms of energy and time, over the cost of performing arithmetic operations in computer systems. The fundamental ratio of aggregate data movement bandwidth to the total computational power (also referred to the machine balance parameter) in parallel computer systems is decreasing. It is there- fore of considerable importance to characterize the inherent data movement requirements of parallel algorithms, so that the minimal architectural balance parameters required to support it on future systems can be well understood. In this paper, we develop an extension of the well-known red-blue pebble game to develop lower bounds on the data movement complexity for the parallel execution of computational directed acyclic graphs (CDAGs) on parallel systems. We model multi-node multi-core parallel systems, with the total physical memory distributed across the nodes (that are connected through some interconnection network) and in a multi-level shared cache hierarchy for processors within a node. We also develop new techniques for lower bound characterization of non-homogeneous CDAGs. We demonstrate the use of the methodology by analyzing the CDAGs of several numerical algorithms, to develop lower bounds on data movement for their parallel execution.

The low-temperature driven or thermally activated motion of several condensed matter systems is often modeled by the dynamics of interfaces (co-dimension-1 elastic manifolds) subject to a random potential. Two characteristic quantitative features of the energy landscape of such a many-degree-of-freedom system are the ground-state energy and the magnitude of the energy barriers between given configurations. While the numerical determination of the former can be accomplished in time polynomial in the system size, it is shown here that the problem of determining the latter quantity is NP-complete. Exact computation of barriers is therefore (almost certainly) much more difficult than determining the exact ground states of interfaces.

The problem is to calculate an approximate solution of an initial value problem for a scalar autonomous differential equation. A generalized notion of a nonlinear Runge-Kutta (NRK) method is defined. We show that the order of any s-stage NRK method cannot exceed 2s-1; hence, the family of NRK methods due to Brent has the maximal order possible. Using this result, we derive complexity bounds on the problem of finding an approximate solution with error not exceeding epsilon. We also compute the order which minimizes these bounds, and show that this optimal order increases as epsilon decreases, tending to infinity as epsilon tends to zero. (Author)

The theory of computational complexity focuses on functions and, hence, studies programs whose interactive behavior is reduced to a simple question/answer pattern. We propose a broader theory whose ultimate goal is expressing and analyzing the intrinsic difficulty of fully general interactive behaviors. To this extent, we use standard tools from concurrency theory, including labelled transition systems (formalizing behaviors) and their asynchronous extension (providing causality information). Behaviors are implemented by means of a multiprocessor machine executing CCS-like processes. The resulting theory is shown to be consistent with the classical definitions: when we restrict to functional behaviors (i.e., question/answer patterns), we recover several standard computational complexity classes.

In this note, we extend the result of \cite{PoulyG16} about the complexity of solving polynomial differential equations over unbounded domains to work with non-rational input. In order to deal with arbitrary input, we phrase the result in framework of Conputable Analysis \cite{Ko91}. As a side result, we also get a uniform result about complexity of the operator, and not just about the solution.

We demonstrate how the physics of multiboson correlation interference leads to the computational complexity of linear optical interferometers based on correlation measurements in the degrees of freedom of the input bosons. In particular, we address the task of MultiBoson Correlation Sampling (MBCS) from the probability distribution associated with polarization- and time-resolved detections at the output of random linear optical networks. We show that the MBCS problem is fundamentally hard to solve classically even for nonidentical input photons, regardless of the color of the photons, making it also very appealing from an experimental point of view. These results fully manifest the quantum computational supremacy inherent to the fundamental nature of quantum interference.

A precise estimation of the computational complexity in Shor's factoring algorithm under the condition that the large integer we want to factorize is composed by the product of two prime numbers, is derived by the results related to number theory. Compared with Shor's original estimation, our estimation shows that one can obtain the solution under such a condition, by less computational complexity.

Title from cover

This review project aims to systematically examine and synthesize the recent advances in computational modeling (CM) of chronic stress (CS) and its relationship to disease progression (DP), with a particular focus on the concept of allostatic load (AL). The overarching goal is to provide a comprehensive overview of how computational approaches have been utilized to elucidate the physiological mechanisms linking chronic stress to disease, identify methodological trends, and highlight gaps in the current literature.

conway game the infernal joke

torunska rewelacja

In this paper we prove that the avalanche problem for Kadanoff sandpile model (KSPM) is P-complete for two-dimensions. Our proof is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with configurations in KSPM. The proof is also related to the known prediction problem for sandpile which is in NC for one-dimensional sandpiles and is P-complete for dimension 3 or greater. The computational complexity of the prediction problem remains open for two-dimensional sandpiles.

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational "phase transitions". Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems.

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational "phase transitions". Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems.

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.

Mean filters are among the most commonly used filters in the field of digital signal processing in general and image processing in particular. Therefore, the research and improvement of the filter technology will have a positive impact on both the hardware and the software. Our studies in this paper have shown a new implementation solution, which greatly improves computational complexity, thus reducing the time taken for implementation to be superior to that of the original technique. in experimental results in the form of software. In particular, these suggestions can naturally expand into the n-dimensional space.

Mean filters are among the most commonly used filters in the field of digital signal processing in general and image processing in particular. Therefore, the research and improvement of the filter technology will have a positive impact on both the hardware and the software. Our studies in this paper have shown a new implementation solution, which greatly improves computational complexity, thus reducing the time taken for implementation to be superior to that of the original technique. in experimental results in the form of software. In particular, these suggestions can naturally expand into the n-dimensional space.

Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a typical system state or history starting from simple initial conditions. The properties of depth are discussed and it is compared to other complexity measures. Depth can only be large for systems with embedded computation.

We study the Poincare-Bendixson theorem for two-dimensional continuous dynamical systems in compact domains from the point of view of computation, seeking algorithms for finding the limit cycle promised by this classical result. We start by considering a discrete analogue of this theorem and show that both finding a point on a limit cycle, and determining if a given point is on one, are PSPACE-complete. For the continuous version, we show that both problems are uncomputable in the real complexity sense; i.e., their complexity is arbitrarily high. Subsequently, we introduce a notion of an "approximate cycle" and prove an "approximate" Poincar\'e-Bendixson theorem guaranteeing that some orbits come very close to forming a cycle in the absence of approximate fixpoints; surprisingly, it holds for all dimensions. The corresponding computational problem defined in terms of arithmetic circuits is PSPACE-complete.

Quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables. It generalizes, and on 1-dimensional Hilbert space coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic terms. It turns out that in dimension two both are also NP-complete. For higher-dimensional spaces R^d and C^d with d>2 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory; and extends the still relatively scarce family of NP_R-complete problems with one perhaps closest in spirit to the classical Cook-Levin Theorem. Our investigations on the dimensions a term is weakly/strongly satisfiable in lead to satisfiability problems in indefinite finite and finally in infinite dimension. Here, strong satisfiability turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations

Some graphs admit drawings in the Euclidean k-space in such a (natu- ral) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent vertices are drawn in the same point, we say that the representation is degenerate. The dimension (the Euclidean dimension) of a graph is defined to be the minimum integer k needed that a given graph has non-degenerate k dimensional unit distance representation (with the property that non-adjacent vertices are mapped to points, that are not distance one appart). It is proved that deciding if an input graph is homomorphic to a graph with dimension k >= 2 (with the Euclidean dimension k >= 2) are NP-hard problems.

Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem'' when applied to fermions - causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.

This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of quantum complexity classes based on these notions, such as BQP, QMA, and QIP, are presented. Other topics in quantum complexity, including quantum advice, space-bounded quantum computation, and bounded-depth quantum circuits, are also discussed.

Consider an undirected graph modeling a social network, where the vertices represent users, and the edges do connections among them. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her opinion, and then it spreads along the edges in the graphs. The objective of every player is to maximize the number of vertices the opinion infects. In this paper, we investigate a computational problem of asking whether a pure Nash equilibrium exists in the competitive diffusion game on unweighed and weighted graphs, and present several negative and positive results. We first prove that the problem is W[1]-hard when parameterized by the number of players even for unweighted graphs. We also show that the problem is NP-hard even for series-parallel graphs with positive integer weights, and is NP-hard even for forests with arbitrary integer weights. Furthermore, we show that the problem for forest of paths with arbitrary weights is solvable in pseudo-polynomial time; and it is solvable in quadratic time if a given graph is unweighted. We also prove that the problem for chain, cochain, and threshold graphs with arbitrary integer weights is solvable in polynomial time.

Consider an undirected graph modeling a social network, where the vertices represent users, and the edges do connections among them. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her opinion, and then it spreads along the edges in the graphs. The objective of every player is to maximize the number of vertices the opinion infects. In this paper, we investigate a computational problem of asking whether a pure Nash equilibrium exists in the competitive diffusion game on unweighed and weighted graphs, and present several negative and positive results. We first prove that the problem is W[1]-hard when parameterized by the number of players even for unweighted graphs. We also show that the problem is NP-hard even for series-parallel graphs with positive integer weights, and is NP-hard even for forests with arbitrary integer weights. Furthermore, we show that the problem for forest of paths with arbitrary weights is solvable in pseudo-polynomial time; and it is solvable in quadratic time if a given graph is unweighted. We also prove that the problem for chain, cochain, and threshold graphs with arbitrary integer weights is solvable in polynomial time.

We survey research that studies the connection between the computational complexity of optimization problems on the one hand, and the duality gap between the primal and dual optimization problems on the other. To our knowledge, this is the first survey that connects the two very important areas. We further look at a similar phenomenon in finite model theory relating to complexity and optimization.

We shed new light on entanglement measures in multipartite quantum systems by taking a computational-complexity approach toward quantifying quantum entanglement with two familiar notions--approximability and distinguishability. Built upon the formal treatment of partial separability, we measure the complexity of an entangled quantum state by determining (i) how hard to approximate it from a fixed classical state and (ii) how hard to distinguish it from all partially separable states. We further consider the Kolmogorovian-style descriptive complexity of approximation and distinction of partial entanglement.

We shed new light on entanglement measures in multipartite quantum systems by taking a computational-complexity approach toward quantifying quantum entanglement with two familiar notions--approximability and distinguishability. Built upon the formal treatment of partial separability, we measure the complexity of an entangled quantum state by determining (i) how hard to approximate it from a fixed classical state and (ii) how hard to distinguish it from all partially separable states. We further consider the Kolmogorovian-style descriptive complexity of approximation and distinction of partial entanglement.