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how to compute hardtocompute matrix norms  http://www.njohnston.ca/2016/01/howtocomputehardtocomputematrixnorms/ 
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norms of matrices and operators  https://en.wikipedia.org/wiki/matrix_norm 
schatten norms  https://en.wikipedia.org/wiki/schatten_norm 
ky fan norms  http://www.njohnston.ca/2009/08/kyfannormsschattennormsandeverythinginbetween/ 
induced pnorms  https://en.wikipedia.org/wiki/matrix_norm#induced_norm 
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power method  https://en.wikipedia.org/wiki/power_iteration 
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inducedmatrixnorm  http://www.qetlab.com/inducedmatrixnorm 
schatten pnorm  http://www.njohnston.ca/2009/08/kyfannormsschattennormsandeverythinginbetween/ 
hölder inequality for schatten norms  https://en.wikipedia.org/wiki/schatten_class_operator 
hilbert–schmidt inner product  https://en.wikipedia.org/wiki/hilbert%e2%80%93schmidt_operator 
inducedschattennorm  http://www.qetlab.com/inducedschattennorm 
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schmidt rank  https://en.wikipedia.org/wiki/schmidt_decomposition 
computation of matrix norms with applications to robust optimization  http://www2.isye.gatech.edu/~nemirovs/daureen.pdf 
arxiv:0908.1397  http://arxiv.org/abs/0908.1397 
arxiv:quantph/0411077  http://arxiv.org/abs/quantph/0411077 
arxiv:quantph/0212030  http://arxiv.org/abs/quantph/0212030 
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choi map  http://www.qetlab.com/choimap 
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certain function  http://en.wikipedia.org/wiki/entropy_of_entanglement 
hilbert–schmidt inner product  http://en.wikipedia.org/wiki/hilbert%e2%80%93schmidt_operator 
product states  http://en.wikipedia.org/wiki/product_state 
partial traces  http://en.wikipedia.org/wiki/partial_trace 
arxiv:quantph/0303055  http://arxiv.org/abs/quantph/0303055 
arxiv:quantph/0205017  http://arxiv.org/abs/quantph/0205017 
arxiv:quantph/0212047  http://arxiv.org/abs/quantph/0212047 
arxiv:0709.3766  http://arxiv.org/abs/0709.3766 
arxiv:0803.0757  http://arxiv.org/abs/0803.0757 
arxiv:1311.7275  http://arxiv.org/abs/1311.7275 
arxiv:1405.3634  http://arxiv.org/abs/1405.3634 
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a combinatorial problem  http://projecteuclid.org/euclid.mmj/1028989731 
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how to compute hardtocompute matrix norms  http://www.njohnston.ca/2016/01/howtocomputehardtocomputematrixnorms/ 
introducing qetlab: a matlab toolbox for quantum entanglement  http://www.njohnston.ca/2015/04/introducingqetlabamatlabtoolboxforquantumentanglement/ 
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all minimal superpermutations on five symbols have been found  http://www.njohnston.ca/2014/08/allminimalsuperpermutationsonfivesymbolshavebeenfound/ 
what the operatorschmidt decomposition tells us about entanglement  http://www.njohnston.ca/2014/06/whattheoperatorschmidtdecompositiontellsusaboutentanglement/ 
counting the possible orderings of pairwise multiplication  http://www.njohnston.ca/2014/02/countingthepossibleorderingsofpairwisemultiplication/ 
in search of a 4by11 matrix  http://www.njohnston.ca/2013/10/insearchofa4by11matrix/ 
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nathaniel johnston assistant professor mount allison university sackville, new brunswick, canadanathaniel@njohnston.ca home cv publications my websites contact how to compute hardtocompute matrix norms january 11th, 2016 no comments there are a wide variety of different norms of matrices and operators that are useful in many different contexts. some matrix norms, such as the schatten norms and ky fan norms, are easy to compute thanks to the singular value decomposition. however, the computation of many other norms, such as the induced pnorms (when p ≠ 1, 2, ∞), is nphard. in this post, we will look at a general method for getting quite good estimates of almost any matrix norm. the basic idea is that every norm can be written as a maximization of a convex function over a convex set (in particular, every norm can be written as a maximization over the unit ball of the dual norm). however, this maximization is often difficult to deal with or solve analytically, so instead it can help to write the norm as a maximization over two or more simpler sets, each of which can be solved individually. to illustrate how this works, let’s start with the induced matrix norms. induced matrix norms the induced p → q norm of a matrix b is defined as follows: where is the vector pnorm. there are three special cases of these norms that are easy to compute: when p = q = 2, this is the usual operator norm of b (i.e., its largest singular value). when p = q = 1, this is the maximum absolute column sum: . when p = q = ∞, this is the maximum absolute row sum: . however, outside of these three special cases (and some other special cases, such as when b only has real entries that are nonnegative [1]), this norm is much messier. in general, its computation is nphard [2], so how can we get a good idea of its value? well, we rewrite the norm as the following double maximization: where is the positive real number such that (and we take if , and viceversa). the idea is then to maximize over and one at a time, alternately. start by setting and fixing a randomlychosen vector , scaled so that . compute keeping fixed, and let be the vector attaining this maximum. by hölder’s inequality, we know that this maximum value is exactly equal to . furthermore, the equality condition of hölder’s inequality tells us that the vector attaining this maximum is the one with complex phases that are the same as those of , and whose magnitudes are such that is a multiple of (here the notation means we take the absolute value and the qth power of every entry of the vector). compute keeping fixed, and let be the vector attaining this maximum. by an argument almost identical to that of step 2, this maximum is equal to , where is the positive real number such that . furthermore, the vector attaining this maximum is the one with complex phases that are the same as those of , and whose magnitudes are such that is a multiple of . increment by 1 and return to step 2. repeat until negligible gains are made after each iteration. this algorithm is extremely quick to run, since hölder’s inequality tells us exactly how to solve each of the two maximizations separately, so we’re left only performing simple vector calculations at each step. the downside of this algorithm is that, even though it will always converge to some local maximum, it might converge to a value that is smaller than the true induced p → q norm. however, in practice this algorithm is fast enough that it can be run several thousand times with different (randomlychosen) starting vectors to get an extremely good idea of the value of . it is worth noting that this algorithm is essentially the same as the one presented in [3], and reduces to the power method for finding the largest singular value when p = q = 2. this algorithm has been implemented in the qetlab package for matlab as the inducedmatrixnorm function. induced schatten superoperator norms there is a natural family of induced norms on superoperators (i.e., linear maps ) as well. first, for a matrix , we define its schatten pnorm to be the pnorm of its vector of singular values: three special cases of the schatten pnorms include: p = 1, which is often called the “trace norm” or “nuclear norm”, p = 2, which is often called the “frobenius norm” or “hilbert–schmidt norm”, and p = ∞, which is the usual operator norm. the schatten norms themselves are easy to compute (since singular values are easy to compute), but their induced counterparts are not. given a superoperator , its induced schatten p → q norm is defined as follows: these induced schatten norms were studied in some depth in [4], and crop up fairly frequently in quantum information theory (especially when p = q = 1) and operator theory (especially when p = q = ∞). the fact that they are nphard to compute in general is not surprising, since they reduce to the induced matrix norms (discussed earlier) in the case when only acts on the diagonal entries of and just zeros out the offdiagonal entries. however, it seems likely that this norm’s computation is also difficult even in the special cases p = q = 1 and p = q = ∞ (however, it is straightforward to compute when p = q = 2). nevertheless, we can obtain good estimates of this norm’s value numerically using essentially the same method as discussed in the previous section. we start by rewriting the norm as a double maximization, where each maximization individually is easy to deal with: where is again the positive real number (or infinity) satisfying . we now maximize over and , one at a time, alternately, just as before: start by setting and fixing a randomlychosen matrix , scaled so that . compute keeping fixed, and let be the matrix attaining this maximum. by the hölder inequality for schatten norms, we know that this maximum value is exactly equal to . furthermore, the matrix attaining this maximum is the one with the same left and right singular vectors as , and whose singular values are such that there is a constant so that for all (i.e., the vector of singular values of , raised to the power, is a multiple of the vector of singular values of , raised to the power). compute keeping fixed, and let be the matrix attaining this maximum. by essentially the same argument as in step 2, we know that this maximum value is exactly equal to , where is the map that is dual to in the hilbert–schmidt inner product. furthermore, the matrix attaining this maximum is the one with the same left and right singular vectors as , and whose singular values are such that there is a constant so that for all . increment by 1 and return to step 2. repeat until negligible gains are made after each iteration. the above algorithm is almost identical to the algorithm presented for induced matrix norms, but with absolute values and complex phases of the vectors and replaced by the singular values and singular vectors of the matrices and , respectively. the entire algorithm is still extremely quick to run, since each step just involves computing one singular value decomposition. the downside of this algorithm, as with the induced matrix norm algorithm, is that we have no guarantee that this method will actually converge to the induced schatten p → q norm; only that it will converge to some lower bound of it. however, the algorithm works pretty well in practice, and is fast enough that we can simply run it a few thousand times to get a very good idea of what the norm actually is. if you’re interested in making use of this algorithm, it has been implemented in qetlab as the inducedschattennorm function. entanglement norms the central idea used for the previous two families of norms can also be used to get lower bounds on the following norm on that comes up from time to time when dealing with quantum entanglement: (as a side note: this norm, and some other ones like it, were the central focus on my thesis.) this norm is already written for us as a double maximization, so the idea presented in the previous two sections is somewhat clearer from the start: we fix randomlygenerated vectors and and then maximize over all vectors and , which can be done simply by computing the left and right singular vectors associated with the maximum singular value of the operator we then fix and as those singular vectors and then maximize over all vectors and (which is again a singular value problem), and we iterate back and forth until we converge to some value. as with the previouslydiscussed norms, this algorithm always converges, and it converges to a lower bound of , but perhaps not its exact value. if you want to take this algorithm out for a spin, it has been implemented in qetlab as the sk_iterate function. it’s also worth mentioning that this algorithm generalizes straightforwardly in several different directions. for example, it can be used to find lower bounds on the norms where we maximize on the left and right by pure states with schmidt rank not larger than k rather than separable pure states, and it can be used to find lower bounds on the geometric measure of entanglement [5]. references: d. steinberg. computation of matrix norms with applications to robust optimization. research thesis. technion – israel university of technology, 2005. j. m. hendrickx and a. olshevsky. matrix pnorms are nphard to approximate if p ≠ 1,2,∞. 2009. eprint: arxiv:0908.1397 d. w. boyd. the power method for ℓp norms. linear algebra and its applications, 9:95–101, 1974. j. watrous. notes on superoperator norms induced by schatten norms. quantum information & computation, 5(1):58–68, 2005. eprint: arxiv:quantph/0411077 t.c. wei and p. m. goldbart. geometric measure of entanglement and applications to bipartite and multipartite quantum states. physical review a, 68:042307, 2003. eprint: arxiv:quantph/0212030 tags: coding, matrix analysis, norms, qetlab, research introducing qetlab: a matlab toolbox for quantum entanglement april 14th, 2015 1 comment after over two and a half years in various stages of development, i am happy to somewhat “officially” announce a matlab package that i have been developing: qetlab (quantum entanglement theory laboratory). this announcement is completely arbitrary, since people started finding qetlab via google about a year ago, and a handful of papers have made use of it already, but i figured that i should at least acknowledge its existence myself at some point. i’ll no doubt be writing some posts in the near future that highlight some of its more advanced features, but i will give a brief rundown of what it’s about here. the basics first off, qetlab has a variety of functions for dealing with “simple” things like tensor products, schmidt decompositions, random pure and mixed states, applying superoperators to quantum states, computing choi matrices and kraus operators, and so on, which are fairly standard daily tasks for quantum information theorists. these sorts of functions are somewhat standard, and are also found in a few other matlab packages (such as toby cubitt’s nice quantinf package and géza tóth’s qubit4matlab package), so i won’t really spend any time discussing them here. mixed state separability the “motivating problem” for qetlab is the separability problem, which asks us to (efficiently / operationally / practically) determine whether a given mixed quantum state is separable or entangled. the (by far) most wellknown tool for this job is the positive partial transpose (ppt) criterion, which says that every separable state remains positive semidefinite when the partial transpose map is applied to it. however, this is just a quickanddirty oneway test, and going beyond it is much more difficult. the qetlab function that tries to solve this problem is the isseparable function, which goes through several separability criteria in an attempt to prove the given state separable or entangled, and provides a journal reference to the paper that contains the separability criteria that works (if one was found). as an example, consider the “tiles” state, introduced in [1], which is an example of a quantum state that is entangled, but is not detected by the simple ppt test for entanglement. we can construct this state using qetlab’s upb function, which lets the user easily construct a wide variety of unextendible product bases, and then verify its entanglement as follows: >> u = upb('tiles'); % generates the "tiles" upb >> rho = eye(9)  u*u'; % rho is the projection onto the orthogonal complement of the upb >> rho = rho/trace(rho); % we are now done constructing the bound entangled state >> isseparable(rho) determined to be entangled via the realignment criterion. reference: k. chen and l.a. wu. a matrix realignment method for recognizing entanglement. quantum inf. comput., 3:193202, 2003. ans = 0 and of course more advanced tests for entanglement, such as those based on symmetric extensions, are also checked. generally, quick and easy tests are done first, and slow but powerful tests are only performed if the script has difficulty finding an answer. alternatively, if you want to check individual tests for entanglement yourself, you can do that too, as there are standalone functions for the partial transpose, the realignment criterion, the choi map (a specific positive map in 3dimensional systems), symmetric extensions, and so on. symmetry of subsystems one problem that i’ve come across repeatedly in my work is the need for robust functions relating to permuting quantum systems that have been tensored together, and dealing with the symmetric and antisymmetric subspaces (and indeed, this type of thing is quite common in quantum information theory). some very basic functionality of this type has been provided in other matlab packages, but it has never been as comprehensive as i would have liked. for example, qubit4matlab has a function that is capable of computing the symmetric projection on two systems, or on an arbitrary number of 2 or 3dimensional systems, but not on an arbitrary number of systems of any dimension. qetlab’s symmetricprojection function fills this gap. similarly, there are functions for computing the antisymmetric projection, for permuting different subsystems, and for constructing the unitary swap operator that implements this permutation. nonlocality and bell inequalities qetlab also has a set of functions for dealing with quantum nonlocality and bell inequalities. for example, consider the chsh inequality, which says that if and are valued measurement settings, then the following inequality holds in classical physics (where denotes expectation): however, in quantummechanical settings, this inequality can be violated, and the quantity on the left can take on a value as large as (this is tsirelson’s bound). finally, in nosignalling theories, the quantity on the left can take on a value as large as . all three of these quantities can be easily computed in qetlab via the bellinequalitymax function: >> coeffs = [1 1;1 1]; % coefficients of the terms
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of entanglement  0.04% (3) to find  0.04% (3) that every  0.04% (3) ). this  0.04% (3) realignment criterion.  0.04% (3) integer sequences  0.04% (3) good idea  0.04% (3) quantum entanglement,  0.04% (3) been implemented  0.04% (3) previous two  0.04% (3) complex phases  0.04% (3) variety of  0.04% (3) from a  0.04% (3) us about  0.04% (3) decomposition tells  0.04% (3) more than  0.04% (3) so there  0.04% (3) bound of  0.04% (3) decomposition of  0.04% (3) mixed quantum  0.04% (3) in its  0.04% (3) rows and  0.04% (3) with operatorschmidt  0.04% (3) to check  0.04% (3) is often  0.04% (3) counting the  0.04% (3) coefficients in  0.04% (3) since is  0.04% (3) of singular  0.04% (3) in terms  0.04% (3) entangled state  0.04% (3) difficult to  0.04% (3) are decreasing.  0.04% (3) this matrix  0.04% (3) wasted characters,  0.04% (3) place the  0.04% (3) of and  0.04% (3) part of  0.04% (3) the rows  0.04% (3) the uppertriangular  0.04% (3) and columns  0.04% (3) induced schatten  0.04% (3) for entanglement  0.04% (3) columns of  0.04% (3) where each  0.04% (3) separable or  0.04% (3) arbitrary number  0.04% (3) only if  0.03% (2) imply “true”:  0.03% (2) contained within  0.03% (2) are orthonormal  0.03% (2) all and  0.03% (2) on five  0.03% (2) will use  0.03% (2) symbols have  0.03% (2) if then  0.03% (2) comes from  0.03% (2) which completes  0.03% (2) case. i  0.03% (2) of pairwise  0.03% (2) entanglement in  0.03% (2) some matrix  0.03% (2) detects the  0.03% (2) witness that  0.03% (2) entanglement of  0.03% (2) toolbox for  0.03% (2) a similar  0.03% (2) bound is  0.03% (2) seem to  0.03% (2) verify that  0.03% (2) case, and  0.03% (2) 7 case  0.03% (2) write a  0.03% (2) pham. enumeration  0.03% (2) however, multiplying  0.03% (2) be phrased  0.03% (2) witness, we  0.03% (2) integer sequences,  0.03% (2) decomposition. however,  0.03% (2) orderings are  0.03% (2) decomposition is  0.03% (2) anything fancy:  0.03% (2) an entanglement  0.03% (2) is indeed  0.03% (2) where the  0.03% (2) form orthonormal  0.03% (2) improved the  0.03% (2) does not  0.03% (2) real scalar  0.03% (2) in these  0.03% (2) the equality  0.03% (2) the proof.  0.03% (2) entanglement detection  0.03% (2) a more  0.03% (2) can write  0.03% (2) so on.  0.03% (2) upper bounds  0.03% (2) shown in  0.03% (2) matrix is  0.03% (2) semidefinite, and  0.03% (2) 1, which  0.03% (2) all states  0.03% (2) how the  0.03% (2) have used  0.03% (2) entries that  0.03% (2) negative semidefinite  0.03% (2) exactly the  0.03% (2) if they  0.03% (2) are done.  0.03% (2) semidefinite, we  0.03% (2) both negative  0.03% (2) that either  0.03% (2) bound to  0.03% (2) some .  0.03% (2) separable. proof. if  0.03% (2) matrix in  0.03% (2) to sometimes  0.03% (2) can also  0.03% (2) eigenvalues of  0.03% (2) this, we  0.03% (2) can use  0.03% (2) separable, which  0.03% (2) criterion. phys.  0.03% (2) d. cariello.  0.03% (2) state in  0.03% (2) the possible  0.03% (2) pairwise multiplication  0.03% (2) are entangled  0.03% (2) real numbers  0.03% (2) by this  0.03% (2) in has  0.03% (2) is separable,  0.03% (2) possible products  0.03% (2) under the  0.03% (2) between these  0.03% (2) case that  0.03% (2) since each  0.03% (2) the minimum  0.03% (2) standard continuity  0.03% (2) and notice  0.03% (2) define a  0.03% (2) how many  0.03% (2) such orderings  0.03% (2) quantum entanglement:  0.03% (2) of such  0.03% (2) sometimes prove  0.03% (2) that and  0.03% (2) realignment map  0.03% (2) gives ,  0.03% (2) part are  0.03% (2) beyond the  0.03% (2) that then  0.03% (2) of generality  0.03% (2) without loss  0.03% (2) have the  0.03% (2) following three  0.03% (2) inequalities: ,  0.03% (2) follows from  0.03% (2) multiplying the  0.03% (2) gauge function  0.03% (2) that uppertriangular  0.03% (2) which contradicts  0.03% (2) functions of  0.03% (2) detected by  0.03% (2) entangled based  0.03% (2) operatorschmidt decomposition.  0.03% (2) these two  0.03% (2) formula for  0.03% (2) bound on  0.03% (2) entangled. the  0.03% (2) norm (i.e.,  0.03% (2) we see  0.03% (2) state with  0.03% (2) the integers  0.03% (2) lefthand side  0.03% (2) to conclude  0.03% (2) criterion, which  0.03% (2) let have  0.03% (2) cases where  0.03% (2) formulation of  0.03% (2) of that  0.03% (2) completes the  0.03% (2) be done  0.03% (2) have which  0.03% (2) different ways  0.03% (2) entangled if  0.03% (2) above on  0.03% (2) are 12  0.03% (2) the interval  0.03% (2) value in  0.03% (2) any value  0.03% (2) with and  0.03% (2) matrix above  0.03% (2) such example.  0.03% (2) is one  0.03% (2) the state  0.03% (2) gives us  0.03% (2) better upper  0.03% (2) 4 case,  0.03% (2) in disguise  0.03% (2) nathaniel johnston  0.03% (2) is just  0.03% (2) , raised  0.03% (2) made after  0.03% (2) gains are  0.03% (2) until negligible  0.03% (2) 2. repeat  0.03% (2) to step  0.03% (2) and return  0.03% (2) . increment  0.03% (2) a constant  0.03% (2) singular values are  0.03% (2) same left  0.03% (2) hilbert–schmidt inner  0.03% (2) the matrix attaining  0.03% (2) raised to  0.03% (2) that for  0.03% (2) almost identical  0.03% (2) constant so  0.03% (2) values are such  0.03% (2) whose singular  0.03% (2) matrix attaining  0.03% (2) matrix attaining this  0.03% (2) that . compute  0.03% (2) scaled so  0.03% (2) as before:  0.03% (2) a time,  0.03% (2) one at  0.03% (2) is again  0.03% (2) to deal  0.03% (2) double maximization,  0.03% (2) we start  0.03% (2) each iteration.  0.03% (2) values and  0.03% (2) diagonal entries  0.03% (2) a lower  0.03% (2) to quantum  0.03% (2) i will  0.03% (2) more advanced  0.03% (2) happy to  0.03% (2) matlab toolbox  0.03% (2) qetlab: a  0.03% (2) quantum states.  0.03% (2) are nphard  0.03% (2) geometric measure  0.03% (2) where we  0.03% (2) find lower  0.03% (2) in qetlab  0.03% (2) out for  0.03% (2) you want  0.03% (2) over all  0.03% (2) , respectively.  0.03% (2) then maximize  0.03% (2) and and  0.03% (2) a double  0.03% (2) with quantum  0.03% (2) the central  0.03% (2) making use  0.03% (2) of what  0.03% (2) and is  0.03% (2) as with  0.03% (2) this algorithm,  0.03% (2) downside of  0.03% (2) each step  0.03% (2) run, since  0.03% (2) quick to  0.03% (2) good estimates  0.03% (2) nphard to  0.03% (2) whether a  0.03% (2) maximization over  0.03% (2) setting and  0.03% (2) well, we  0.03% (2) is nphard  0.03% (2) some other  0.03% (2) . when  0.03% (2) maximum absolute  0.03% (2) largest singular  0.03% (2) the usual  0.03% (2) is defined  0.03% (2) induced p →  0.03% (2) start with  0.03% (2) to write  0.03% (2) deal with  0.03% (2) the unit  0.03% (2) a convex  0.03% (2) , scaled  0.03% (2) norm can  0.03% (2) the basic  0.03% (2) estimates of  0.03% (2) a general  0.03% (2) look at  0.03% (2) norms, such  0.03% (2) computation of  0.03% (2) however, the  0.03% (2) value decomposition.  0.03% (2) matrix norms,  0.03% (2) many different  0.03% (2) a wide  0.03% (2) no comments  0.03% (2) hardtocompute matrix  0.03% (2) fixing a  0.03% (2) so that .  0.03% (2) and operator  0.03% (2) extremely quick  0.03% (2) theory (especially  0.03% (2) defined as  0.03% (2) called the  0.03% (2) 2, which  0.03% (2) norm” or  0.03% (2) often called  0.03% (2) the qetlab  0.03% (2) presented in  0.03% (2) get an  0.03% (2) thousand times  0.03% (2) fast enough  0.03% (2) is smaller  0.03% (2) the downside  0.03% (2) to run,  0.03% (2) after each  0.03% (2) inequality tells  0.03% (2) are made  0.03% (2) negligible gains  0.03% (2) repeat until  0.03% (2) step 2.  0.03% (2) return to  0.03% (2) increment by  0.03% (2) whose magnitudes  0.03% (2) those of  0.03% (2) step 2,  0.03% (2) identical to  0.03% (2) we take  0.03% (2) magnitudes are  0.03% (2) phases that  0.03% (2) with complex  0.03% (2) other matlab  0.03% (2) or entangled.  0.03% (2) and form  0.03% (2) the symbols  0.03% (2) in such  0.03% (2) to append  0.03% (2) no way  0.03% (2) add the  0.03% (2) the character  0.03% (2) we append  0.03% (2) append the  0.03% (2) new superpermutations  0.03% (2) the six  0.03% (2) the second  0.03% (2) the eight  0.03% (2) this case,  0.03% (2) length 153,  0.03% (2) five symbols  0.03% (2) depthfirst search  0.03% (2) all minimal  0.03% (2) tags: combinatorics,  0.03% (2) problem is  0.03% (2) due to  0.03% (2) the conjectured  0.03% (2) that is,  0.03% (2) n ≥ 6.  0.03% (2) out that  0.03% (2) it turns  0.03% (2) people had  0.03% (2) problem was  0.03% (2) or less  0.03% (2) seemed to  0.03% (2) (n1)! +  0.03% (2) consecutive “wasted”  0.03% (2) discussed in  0.03% (2) this for  0.03% (2) 120 permutations  0.03% (2) scalar and  0.03% (2) a real  0.03% (2) each is  0.03% (2) decomposition theorem  0.03% (2) of minimal  0.03% (2) houston has  0.03% (2) 2014]: robin  0.03% (2) than one  0.03% (2) has more  0.03% (2) property that  0.03% (2) has found  0.03% (2) 18, 2014]:  0.03% (2) update [august  0.03% (2) available here.  0.03% (2) the results  0.03% (2) known superpermutation  0.03% (2) up the  0.03% (2) to speed  0.03% (2) we find  0.03% (2) strings that  0.03% (2) speed up  0.03% (2) 2 wasted  0.03% (2) 10 permutations  0.03% (2) are allowed  0.03% (2) can find  0.03% (2) wasted characters.  0.03% (2) trying to  0.03% (2) problem of  0.03% (2) must be  0.03% (2) permutations of  0.03% (2) (see a  0.03% (2) since there  0.03% (2) when the  0.03% (2) quantum inf.  0.03% (2) set of  0.03% (2) bell inequalities  0.03% (2) similarly, there  0.03% (2) an arbitrary  0.03% (2) not on  0.03% (2) projection on  0.03% (2) the symmetric  0.03% (2) of computing  0.03% (2) function that  0.03% (2) this type  0.03% (2) that i’ve  0.03% (2) realignment criterion,  0.03% (2) tests are  0.03% (2) and of  0.03% (2) recognizing entanglement.  0.03% (2) example, consider  0.03% (2) matrix realignment  0.03% (2) and l.a.  0.03% (2) k. chen  0.03% (2) >> rho  0.03% (2) wide variety  0.03% (2) construct a  0.03% (2) function, which  0.03% (2) state that  0.03% (2) the separability  0.03% (2) prove the  0.03% (2) separability criteria  0.03% (2) just a  0.03% (2) it. however,  0.03% (2) map is  0.03% (2) and bell  0.03% (2) quantity on  0.03% (2) + (n1)!  0.03% (2) through the  0.03% (2) additional character  0.03% (2) n!, since  0.03% (2) the right  0.03% (2) the length  0.03% (2) some lower  0.03% (2) not difficult  0.03% (2) immediately gives  0.03% (2) repeat this  0.03% (2) with this  0.03% (2) computer search  0.03% (2) of “123”  0.03% (2) was so  0.03% (2) this conjecture  0.03% (2) the origin  0.03% (2) “true”: the  0.03% (2) value as  0.03% (2) not imply  0.03% (2) “obvious” does  0.03% (2) you have  0.03% (2) a computer  0.03% (2) nosignalling value  0.03% (2) also a  0.03% (2) npa hierarchy  0.03% (2) inequality is  0.03% (2) the classical  0.03% (2) % values  0.03% (2) [1 1];  0.03% (2) measurements >>  0.03% (2) [0 0];  0.03% (2) via the  0.03% (2) seemed like  0.03% (2)  the realignment criterion  0.17% (13) the number of  0.13% (10) the fact that  0.12% (9) the minimal superpermutation  0.12% (9) there is a  0.1% (8) the operatorschmidt decomposition  0.09% (7) superpermutation of length  0.09% (7) that is entangled  0.09% (7) a state is  0.08% (6) that we can  0.08% (6) is straightforward to  0.08% (6) we know that  0.07% (5) that there is  0.07% (5) it can be  0.07% (5) it is straightforward  0.07% (5) singular value decomposition  0.07% (5) from the fact  0.07% (5) quantum information theory  0.07% (5) in this case  0.07% (5) this maximum is  0.07% (5) the operatorschmidt rank  0.07% (5) possible orderings of  0.07% (5) there is no  0.07% (5) can be written  0.07% (5) conjecture is false  0.07% (5) the upper bound  0.07% (5) minimal superpermutation conjecture  0.07% (5) fixed, and let  0.05% (4) be used to  0.05% (4) lower bounds on  0.05% (4) the bell inequality  0.05% (4) the value of  0.05% (4) conclude that is  0.05% (4) . furthermore, the  0.05% (4) of the realignment  0.05% (4) on the left  0.05% (4) that we have  0.05% (4) right singular vectors  0.05% (4) that is separable  0.05% (4) p → q norm  0.05% (4) in a string  0.05% (4) maximum is the  0.05% (4) the sets and  0.05% (4) permutations that we  0.05% (4) and right singular  0.05% (4) to show that  0.05% (4) this maximum. by  0.05% (4) minimal superpermutations have  0.05% (4) are both positive  0.05% (4) vector attaining this  0.05% (4) number of orderings  0.05% (4) the vector attaining  0.05% (4) the one with  0.05% (4) that is a  0.05% (4) the previous section  0.05% (4) , and whose  0.05% (4) are easy to  0.05% (4) unextendible product bases  0.05% (4) is as follows:  0.05% (4) induced matrix norms  0.05% (4) is positive semidefinite,  0.05% (4) keeping fixed, and  0.05% (4) let be the  0.05% (4) that is entangled,  0.04% (3) a superpermutation of  0.04% (3) the shortest string  0.04% (3) uppertriangular part of  0.04% (3) it is not  0.04% (3) we have found  0.04% (3) an nsymbol superpermutation  0.04% (3) number of possible  0.04% (3) here is that  0.04% (3) corresponds to the  0.04% (3) value of the  0.04% (3) b_coe, a_val, b_val,  0.04% (3) of golomb rulers  0.04% (3) >> bellinequalitymax(coeffs, a_coe,  0.04% (3) bellinequalitymax(coeffs, a_coe, b_coe,  0.04% (3) a_coe, b_coe, a_val,  0.04% (3) values of the  0.04% (3) superpermutation conjecture is  0.04% (3) bell inequality >>  0.04% (3) in the bell  0.04% (3) % coefficients of  0.04% (3) as large as  0.04% (3) what the operatorschmidt  0.04% (3) have length at  0.04% (3) can fit in  0.04% (3) there exists a  0.04% (3) since is positive  0.04% (3) in terms of  0.04% (3) in its operatorschmidt  0.04% (3) if the operatorschmidt  0.04% (3) rank of is  0.04% (3) proving a state  0.04% (3) the singular values  0.04% (3) . since is  0.04% (3) and are both  0.04% (3) of is then  0.04% (3) of the coefficients  0.04% (3) has operatorschmidt rank  0.04% (3) be written in  0.04% (3) in this post,  0.04% (3) the rows and  0.04% (3) that for all  0.04% (3) and the sets  0.04% (3) in quantum information  0.04% (3) tells us about  0.04% (3) a mixed state  0.04% (3) we can fit  0.04% (3) of permutations that  0.04% (3) the maximum number  0.04% (3) that the rows  0.04% (3) that a state  0.04% (3) and columns of  0.04% (3) the uppertriangular part  0.04% (3) maximum number of  0.04% (3) it follows that  0.04% (3) decomposition tells us  0.04% (3) vector of singular  0.04% (3) the same as  0.04% (3) good idea of  0.04% (3) lower bound of  0.04% (3) converge to some  0.04% (3) the positive real  0.04% (3) this maximum value  0.04% (3) essentially the same  0.04% (3) is exactly equal  0.04% (3) tells us that  0.04% (3) the previous two  0.04% (3) is a multiple  0.04% (3) is the positive  0.04% (3) exactly equal to  0.04% (3) maximum value is  0.04% (3) know that this  0.04% (3) the induced matrix  0.04% (3) smaller than the  0.04% (3) three special cases  0.04% (3) has been implemented  0.04% (3) the norm as  0.04% (3) as a maximization  0.04% (3) orderings of the  0.03% (2) “true”: the minimal  0.03% (2) side of theorem  0.03% (2) we can use  0.03% (2) to sometimes prove  0.03% (2) until negligible gains  0.03% (2) prove that a  0.03% (2) we will use  0.03% (2) step 2. repeat  0.03% (2) is the number  0.03% (2) and return to  0.03% (2) increment by 1  0.03% (2) magnitudes are such  0.03% (2) orderings of pairwise  0.03% (2) then is separable.  0.03% (2) then is entangled.  0.03% (2) then we can  0.03% (2) as those of  0.03% (2) positive semidefinite, it  0.03% (2) real number such  0.03% (2) if they are  0.03% (2) we are done.  0.03% (2) both negative semidefinite  0.03% (2) it turns out  0.03% (2) with operatorschmidt rank  0.03% (2) does not imply  0.03% (2) is separable. proof. if  0.03% (2) to , where  0.03% (2) then it can  0.03% (2) but i will  0.03% (2) operatorschmidt decomposition if  0.03% (2) and so on.  0.03% (2) state with and  0.03% (2) detected by the  0.03% (2) if we can  0.03% (2) value of .  0.03% (2) entangled based on  0.03% (2) is fast enough  0.03% (2) be the case  0.03% (2) loss of generality  0.03% (2) exists a separable  0.03% (2) value in the  0.03% (2) separable state with  0.03% (2) and for all  0.03% (2) : the state  0.03% (2) is one such  0.03% (2) for all :  0.03% (2) of the matrices  0.03% (2) the state is  0.03% (2) one such example.  0.03% (2) furthermore, it is  0.03% (2) we have which  0.03% (2) see that we  0.03% (2) use the value  0.03% (2) of to conclude  0.03% (2) such as the  0.03% (2) downside of this  0.03% (2) each of the  0.03% (2) quick to run,  0.03% (2) are made after  0.03% (2) trace norm (i.e.,  0.03% (2) post, we will  0.03% (2) a matlab toolbox  0.03% (2) and whose magnitudes  0.03% (2) the usual operator  0.03% (2) ways to place  0.03% (2) part of a  0.03% (2) possible orderings in  0.03% (2) columns of that  0.03% (2) uppertriangular part are  0.03% (2) matrix above on  0.03% (2) to the case  0.03% (2) the matrix above  0.03% (2) which contradicts the  0.03% (2) a maximization over  0.03% (2) are in fact  0.03% (2) is defined as  0.03% (2) formula for the  0.03% (2) number of such  0.03% (2) immediately gives us  0.03% (2) number such that  0.03% (2) special cases of  0.03% (2) bound on the  0.03% (2) the maximum absolute  0.03% (2) to place the  0.03% (2) is the maximum  0.03% (2) that uppertriangular part  0.03% (2) the above matrix  0.03% (2) problem here is  0.03% (2) the following three  0.03% (2) inequalities: , ,  0.03% (2) multiplying the first  0.03% (2) gives , so  0.03% (2) , which contradicts  0.03% (2) in the uppertriangular  0.03% (2) maximize over and  0.03% (2) of the terms  0.03% (2) criterion. phys. rev.  0.03% (2) and notice that  0.03% (2) and only if  0.03% (2) are the same  0.03% (2) complex phases that  0.03% (2) . thus and  0.03% (2) pham. enumeration of  0.03% (2) we define a  0.03% (2) notice that for  0.03% (2) follows that is  0.03% (2) that is separable,  0.03% (2) all states with  0.03% (2) t. pham. enumeration  0.03% (2) states with operatorschmidt  0.03% (2) [quantph] d. cariello.  0.03% (2) by setting and  0.03% (2) the possible orderings  0.03% (2) of pairwise multiplication  0.03% (2) mean by this  0.03% (2) and it is  0.03% (2) this case, and  0.03% (2) the case that  0.03% (2) so there are  0.03% (2) orderings in this  0.03% (2) upper bound to  0.03% (2) so that . compute  0.03% (2) this quantity is  0.03% (2) this matrix is  0.03% (2) 1, which is  0.03% (2) fixing a randomlychosen  0.03% (2) to . furthermore,  0.03% (2) all vectors and  0.03% (2) its operatorschmidt decomposition.  0.03% (2) and whose singular  0.03% (2) making use of  0.03% (2) 1]; % values  0.03% (2) extremely quick to  0.03% (2) almost identical to  0.03% (2) the npa hierarchy  0.03% (2) if you have  0.03% (2) “obvious” does not  0.03% (2) imply “true”: the  0.03% (2) for all .  0.03% (2) . there is  0.03% (2) constant so that  0.03% (2) is the shortest  0.03% (2) string that contains  0.03% (2) values are such that  0.03% (2) that it is  0.03% (2) dealing with quantum  0.03% (2) 1 additional character  0.03% (2) that this problem  0.03% (2) this problem was  0.03% (2) values of ,  0.03% (2) is no way  0.03% (2) + (n1)! +  0.03% (2) raised to the  0.03% (2) + n!, since  0.03% (2) vectors as ,  0.03% (2) must have length  0.03% (2) , where is  0.03% (2) the length of  0.03% (2) the matrix attaining  0.03% (2) the same left  0.03% (2) with this property.  0.03% (2) [1 1]; %  0.03% (2) 0]; % coefficients  0.03% (2) so that for  0.03% (2) method for recognizing  0.03% (2) find lower bounds  0.03% (2) then maximize over  0.03% (2) of the operator  0.03% (2) used to find  0.03% (2) the left and  0.03% (2) qetlab: a matlab  0.03% (2) toolbox for quantum  0.03% (2) i am happy  0.03% (2) of functions for  0.03% (2) separable or entangled.  0.03% (2) which says that  0.03% (2) the realignment criterion.  0.03% (2) and l.a. wu.  0.03% (2) a matrix realignment  0.03% (2) entanglement. quantum inf.  0.03% (2) and then maximize  0.03% (2) quantity on the  0.03% (2) over all vectors  0.03% (2) a value as  0.03% (2) can take on  0.03% (2) can be done  0.03% (2) on a value  0.03% (2) left can take  0.03% (2) says that if  0.03% (2) if you want  0.03% (2) example, consider the  0.03% (2) for dealing with  0.03% (2) arbitrary number of  0.03% (2) an arbitrary number  0.03% (2) the realignment criterion,  0.03% (2) tests for entanglement  0.03% (2) turns out that  0.03% (2) for all n ≥  0.03% (2) the coefficients in  0.03% (2) often called the  0.03% (2) in the form  0.03% (2) where each is  0.03% (2) a real scalar  0.03% (2) good estimates of  0.03% (2) and form orthonormal  0.03% (2) each is a  0.03% (2) real scalar and  0.03% (2) that they are  0.03% (2) form orthonormal bases of  0.03% (2) theory (especially when  0.03% (2) that the singular  0.03% (2) mixed state is  0.03% (2) (especially when p  0.03% (2) is the usual  0.03% (2) 2, which is  0.03% (2) that . compute keeping  0.03% (2) cases of the  0.03% (2) is a natural  0.03% (2) it has been  0.03% (2) that is entangled.  0.03% (2) the trace norm  0.03% (2) formulation of the  0.03% (2) completes the proof.  0.03% (2) that the sets  0.03% (2) decomposition if then  0.03% (2) which is often  0.03% (2) for all and  0.03% (2) entanglement witness, we  0.03% (2) to see that  0.03% (2) detects the entanglement  0.03% (2) entanglement witness that  0.03% (2) as a double  0.03% (2) be the matrix attaining  0.03% (2) is a constant  0.03% (2) to add the  0.03% (2) this problem is  0.03% (2) all minimal superpermutations  0.03% (2) on five symbols  0.03% (2) have been found  0.03% (2) as a contiguous  0.03% (2) the symbols “1”  0.03% (2) such that there  0.03% (2) of minimal superpermutations  0.03% (2) a wide variety  0.03% (2) am happy to  0.03% (2) 153, and there  0.03% (2) the one that  0.03% (2) the six new  0.03% (2) then we append  0.03% (2) such a way  0.03% (2) is available here.  0.03% (2) matrix attaining this  0.03% (2) has more than  0.03% (2) found a superpermutation  0.03% (2) robin houston has  0.03% (2) 2 wasted characters  0.03% (2) to speed up  0.03% (2) if we are  0.03% (2) we are allowed  0.03% (2) discussed in the  0.03% (2) a string if  0.03% (2) one with the  0.03% (2) same left and  0.03% (2) whose singular values are  0.03% (2) superpermutations have length  0.03% (2) we can find  0.03% (2) to compute hardtocompute  0.03% (2) 