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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 
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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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realignment criterion  http://www.qetlab.com/realignment 
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/ 
“obvious” does not imply “true”: the minimal superpermutation conjecture is false  http://www.njohnston.ca/2014/08/obviousdoesnotimplytruetheminimalsuperpermutationconjectureisfalse/ 
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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quantum  0.04% (3) decomposition of  0.04% (3) more than  0.04% (3) us about  0.04% (3) decomposition tells  0.04% (3) a_coe, b_coe,  0.04% (3) for n ≤  0.04% (3) of entanglement  0.04% (3) are also  0.04% (3) good idea  0.04% (3) vectors as  0.04% (3) induced schatten  0.04% (3) found in  0.04% (3) to find  0.04% (3) arbitrary number  0.04% (3) this post,  0.04% (3) start by  0.04% (3) vector of  0.04% (3) a multiple  0.04% (3) for computing  0.04% (3) entangled state  0.04% (3) of singular  0.04% (3) norm as  0.04% (3) for entanglement  0.04% (3) of functions  0.04% (3) to check  0.04% (3) for quantum  0.04% (3) essentially the  0.04% (3) implemented in  0.04% (3) norm is  0.04% (3) that every  0.04% (3) three special  0.04% (3) idea is  0.04% (3) coefficients of  0.04% (3) complex phases  0.04% (3) variety of  0.04% (3) large as  0.04% (3) inequality >>  0.04% (3) us that  0.04% (3) separable or  0.04% (3) of and  0.04% (3) same as  0.04% (3) been implemented  0.04% (3) can take  0.04% (3) exactly equal  0.04% (3) difficult to  0.04% (3) bound of  0.04% (3) where each  0.04% (3) previous two  0.04% (3) multiple of  0.04% (3) if and  0.04% (3) is often  0.04% (3) however, in  0.04% (3) % coefficients  0.04% (3) return to  0.03% (2) that either  0.03% (2) separable. proof. if  0.03% (2) extremely quick  0.03% (2) after each  0.03% (2) step 2.  0.03% (2) are made  0.03% (2) negligible gains  0.03% (2) some .  0.03% (2) repeat until  0.03% (2) measurements >>  0.03% (2) both negative  0.03% (2) step 2,  0.03% (2) the minimum  0.03% (2) standard continuity  0.03% (2) and notice  0.03% (2) define a  0.03% (2) magnitudes are  0.03% (2) under the  0.03% (2) this, we  0.03% (2) we take  0.03% (2) identical to  0.03% (2) so on.  0.03% (2) semidefinite, we  0.03% (2) those of  0.03% (2) shown in  0.03% (2) whose magnitudes  0.03% (2) the downside  0.03% (2) all states  0.03% (2) can write  0.03% (2) negative semidefinite  0.03% (2) increment by  0.03% (2) if they  0.03% (2) are done.  0.03% (2) to run,  0.03% (2) cases where  0.03% (2) to sometimes  0.03% (2) detected by  0.03% (2) that then  0.03% (2) of generality  0.03% (2) without loss  0.03% (2) called the  0.03% (2) follows from  0.03% (2) defined as  0.03% (2) theory (especially  0.03% (2) and operator  0.03% (2) gauge function  0.03% (2) nphard to  0.03% (2) functions of  0.03% (2) entangled based  0.03% (2) norm” or  0.03% (2) operatorschmidt decomposition.  0.03% (2) diagonal entries  0.03% (2) these two  0.03% (2) entangled. the  0.03% (2) norm (i.e.,  0.03% (2) realignment map  0.03% (2) a more  0.03% (2) the proof.  0.03% (2) which completes  0.03% (2) good estimates  0.03% (2) we start  0.03% (2) 2, which  0.03% (2) often called  0.03% (2) can also  0.03% (2) get an  0.03% (2) sometimes prove  0.03% (2) can use  0.03% (2) is smaller  0.03% (2) that and  0.03% (2) fast enough  0.03% (2) lefthand side  0.03% (2) criterion, which  0.03% (2) let have  0.03% (2) separable, which  0.03% (2) formulation of  0.03% (2) thousand times  0.03% (2) completes the  0.03% (2) state with  0.03% (2) have which  0.03% (2) entangled if  0.03% (2) to conclude  0.03% (2) presented in  0.03% (2) the interval  0.03% (2) value in  0.03% (2) any value  0.03% (2) the qetlab  0.03% (2) with and  0.03% (2) such example.  0.03% (2) is one  0.03% (2) the state  0.03% (2) since each  0.03% (2) , scaled  0.03% (2) is separable,  0.03% (2) multiplying the  0.03% (2) write a  0.03% (2) bound is  0.03% (2) look at  0.03% (2) a general  0.03% (2) we see  0.03% (2) estimates of  0.03% (2) bound on  0.03% (2) formula for  0.03% (2) which contradicts  0.03% (2) gives ,  0.03% (2) inequalities: ,  0.03% (2) seem to  0.03% (2) following three  0.03% (2) have the  0.03% (2) the basic  0.03% (2) part are  0.03% (2) that uppertriangular  0.03% (2) the integers  0.03% (2) are 12  0.03% (2) 4 case,  0.03% (2) better upper  0.03% (2) gives us  0.03% (2) norm can  0.03% (2) verify that  0.03% (2) norms, such  0.03% (2) a convex  0.03% (2) many different  0.03% (2) is 9080jdarc  0.03% (2) hardtocompute matrix  0.03% (2) no comments  0.03% (2) a wide  0.03% (2) of pairwise  0.03% (2) symbols have  0.03% (2) on five  0.03% (2) imply “true”:  0.03% (2) does not  0.03% (2) toolbox for  0.03% (2) integer sequences,  0.03% (2) however, multiplying  0.03% (2) matrix norms,  0.03% (2) pham. enumeration  0.03% (2) value decomposition.  0.03% (2) case, and  0.03% (2) however, the  0.03% (2) 7 case  0.03% (2) improved the  0.03% (2) in these  0.03% (2) case. i  0.03% (2) computation of  0.03% (2) orderings are  0.03% (2) matrix above  0.03% (2) above on  0.03% (2) in has  0.03% (2) fixing a  0.03% (2) quantum entanglement:  0.03% (2) such orderings  0.03% (2) how many  0.03% (2) case that  0.03% (2) between these  0.03% (2) possible products  0.03% (2) by this  0.03% (2) setting and  0.03% (2) real numbers  0.03% (2) pairwise multiplication  0.03% (2) the possible  0.03% (2) of such  0.03% (2) comes from  0.03% (2) d. cariello.  0.03% (2) so that .  0.03% (2) criterion. phys.  0.03% (2) beyond the  0.03% (2) entanglement detection  0.03% (2) inequality tells  0.03% (2) with complex  0.03% (2) state in  0.03% (2) phases that  0.03% (2) are entangled  0.03% (2) well, we  0.03% (2) is nphard  0.03% (2) maximization over  0.03% (2) the usual  0.03% (2) different ways  0.03% (2) be done  0.03% (2) of that  0.03% (2) the unit  0.03% (2) matrix in  0.03% (2) deal with  0.03% (2) to write  0.03% (2) start with  0.03% (2) bound to  0.03% (2) induced p →  0.03% (2) is defined  0.03% (2) largest singular  0.03% (2) exactly the  0.03% (2) maximum absolute  0.03% (2) . when  0.03% (2) entries that  0.03% (2) have used  0.03% (2) how the  0.03% (2) 1, which  0.03% (2) semidefinite, and  0.03% (2) matrix is  0.03% (2) upper bounds  0.03% (2) some other  0.03% (2) eigenvalues of  0.03% (2) are orthonormal  0.03% (2) as before:  0.03% (2) the equality  0.03% (2) problem was  0.03% (2) whether a  0.03% (2) or entangled.  0.03% (2) that is,  0.03% (2) n ≥ 6.  0.03% (2) when the  0.03% (2) out that  0.03% (2) it turns  0.03% (2) map is  0.03% (2) it. however,  0.03% (2) people had  0.03% (2) just a  0.03% (2) the conjectured  0.03% (2) separability criteria  0.03% (2) prove the  0.03% (2) or less  0.03% (2) the separability  0.03% (2) state that  0.03% (2) seemed to  0.03% (2) (n1)! +  0.03% (2) (see a  0.03% (2) this for  0.03% (2) function, which  0.03% (2) construct a  0.03% (2) other matlab  0.03% (2) due to  0.03% (2) + (n1)!  0.03% (2) geometric measure  0.03% (2) you want  0.03% (2) new superpermutations  0.03% (2) the six  0.03% (2) out for  0.03% (2) in qetlab  0.03% (2) the second  0.03% (2) find lower  0.03% (2) the eight  0.03% (2) this case,  0.03% (2) where we  0.03% (2) length 153,  0.03% (2) are nphard  0.03% (2) problem is  0.03% (2) quantum states.  0.03% (2) qetlab: a  0.03% (2) matlab toolbox  0.03% (2) the symbols  0.03% (2) happy to  0.03% (2) more advanced  0.03% (2) five symbols  0.03% (2) i will  0.03% (2) all minimal  0.03% (2) to quantum  0.03% (2) tags: combinatorics,  0.03% (2) since there  0.03% (2) wide variety  0.03% (2) append the  0.03% (2) “obvious” does  0.03% (2) was so  0.03% (2) this conjecture  0.03% (2) similarly, there  0.03% (2) the origin  0.03% (2) through the  0.03% (2) bell inequalities  0.03% (2) set of  0.03% (2) and bell  0.03% (2) example, consider  0.03% (2) “true”: the  0.03% (2) not imply  0.03% (2) quantity on  0.03% (2) not on  0.03% (2) value as  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) via the  0.03% (2) inequality is  0.03% (2) the classical  0.03% (2) [0 0];  0.03% (2) % values  0.03% (2) an arbitrary  0.03% (2) of “123”  0.03% (2) >> rho  0.03% (2) and of  0.03% (2) additional character  0.03% (2) n!, since  0.03% (2) k. chen  0.03% (2) and l.a.  0.03% (2) matrix realignment  0.03% (2) recognizing entanglement.  0.03% (2) the right  0.03% (2) quantum inf.  0.03% (2) the length  0.03% (2) some lower  0.03% (2) not difficult  0.03% (2) tests are  0.03% (2) projection on  0.03% (2) seemed like  0.03% (2) immediately gives  0.03% (2) realignment criterion,  0.03% (2) that i’ve  0.03% (2) this type  0.03% (2) function that  0.03% (2) repeat this  0.03% (2) with this  0.03% (2) computer search  0.03% (2) of computing  0.03% (2) the symmetric  0.03% (2) a lower  0.03% (2) over all  0.03% (2) where the  0.03% (2) only if  0.03% (2) anything fancy:  0.03% (2) form orthonormal  0.03% (2) the matrix attaining  0.03% (2) real scalar  0.03% (2) hilbert–schmidt inner  0.03% (2) same left  0.03% (2) singular values are  0.03% (2) a similar  0.03% (2) contained within  0.03% (2) a constant  0.03% (2) . increment  0.03% (2) and return  0.03% (2) decomposition. however,  0.03% (2) entanglement of  0.03% (2) to step  0.03% (2) some matrix  0.03% (2) in disguise  0.03% (2) 2. repeat  0.03% (2) is just  0.03% (2) and form  0.03% (2) until negligible  0.03% (2) scalar and  0.03% (2) a real  0.03% (2) each is  0.03% (2) decomposition is  0.03% (2) raised to  0.03% (2) decomposition theorem  0.03% (2) entanglement in  0.03% (2) double maximization,  0.03% (2) to deal  0.03% (2) is again  0.03% (2) all and  0.03% (2) one at  0.03% (2) witness, we  0.03% (2) an entanglement  0.03% (2) is indeed  0.03% (2) a time,  0.03% (2) will use  0.03% (2) [1 1];  0.03% (2) detects the  0.03% (2) , raised  0.03% (2) witness that  0.03% (2) scaled so  0.03% (2) if then  0.03% (2) that . compute  0.03% (2) matrix attaining this  0.03% (2) matrix attaining  0.03% (2) be phrased  0.03% (2) whose singular  0.03% (2) values are such  0.03% (2) constant so  0.03% (2) that for  0.03% (2) gains are  0.03% (2) made after  0.03% (2) then maximize  0.03% (2) the central  0.03% (2) as with  0.03% (2) and is  0.03% (2) of what  0.03% (2) making use  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) known superpermutation  0.03% (2) discussed in  0.03% (2) 10 permutations  0.03% (2) depthfirst search  0.03% (2) consecutive “wasted”  0.03% (2) with quantum  0.03% (2) in such  0.03% (2) to append  0.03% (2) no way  0.03% (2) a double  0.03% (2) and and  0.03% (2) add the  0.03% (2) the character  0.03% (2) we append  0.03% (2) are allowed  0.03% (2) this algorithm,  0.03% (2) each iteration.  0.03% (2) available here.  0.03% (2) almost identical  0.03% (2) values and  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) , respectively.  0.03% (2) 2 wasted  0.03% (2) 120 permutations  0.03% (2) quick to  0.03% (2) the results  0.03% (2) up the  0.03% (2) to speed  0.03% (2) we find  0.03% (2) run, since  0.03% (2) each step  0.03% (2) downside of  0.03% (2) strings that  0.03% (2) speed up  0.03% (2) nathaniel johnston  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) superpermutation of length  0.09% (7) the operatorschmidt decomposition  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) this maximum is  0.07% (5) that there is  0.07% (5) in this case  0.07% (5) we know that  0.07% (5) the operatorschmidt rank  0.07% (5) the upper bound  0.07% (5) conjecture is false  0.07% (5) it can be  0.07% (5) minimal superpermutation conjecture  0.07% (5) can be written  0.07% (5) it is straightforward  0.07% (5) from the fact  0.07% (5) possible orderings of  0.07% (5) singular value decomposition  0.07% (5) quantum information theory  0.07% (5) there is no  0.07% (5) be used to  0.05% (4) to show that  0.05% (4) p → q norm  0.05% (4) maximum is the  0.05% (4) right singular vectors  0.05% (4) the previous section  0.05% (4) number of orderings  0.05% (4) . furthermore, the  0.05% (4) is positive semidefinite,  0.05% (4) that is a  0.05% (4) fixed, and let  0.05% (4) , and whose  0.05% (4) the bell inequality  0.05% (4) minimal superpermutations have  0.05% (4) this maximum. by  0.05% (4) the value of  0.05% (4) unextendible product bases  0.05% (4) lower bounds on  0.05% (4) keeping fixed, and  0.05% (4) let be the  0.05% (4) vector attaining this  0.05% (4) are easy to  0.05% (4) conclude that is  0.05% (4) the sets and  0.05% (4) induced matrix norms  0.05% (4) that is separable  0.05% (4) of the realignment  0.05% (4) in a string  0.05% (4) that we have  0.05% (4) is as follows:  0.05% (4) the vector attaining  0.05% (4) the one with  0.05% (4) and right singular  0.05% (4) are both positive  0.05% (4) permutations that we  0.05% (4) on the left  0.05% (4) values of the  0.04% (3) a mixed state  0.04% (3) and are both  0.04% (3) in the bell  0.04% (3) bell inequality >>  0.04% (3) that for all  0.04% (3) % coefficients of  0.04% (3) as large as  0.04% (3) of is then  0.04% (3) has operatorschmidt rank  0.04% (3) be written in  0.04% (3) since is positive  0.04% (3) . since is  0.04% (3) we have found  0.04% (3) a_coe, b_coe, a_val,  0.04% (3) in terms of  0.04% (3) it follows that  0.04% (3) of the coefficients  0.04% (3) in this post,  0.04% (3) the singular values  0.04% (3) and the sets  0.04% (3) proving a state  0.04% (3) in quantum information  0.04% (3) tells us about  0.04% (3) we can fit  0.04% (3) of permutations that  0.04% (3) the maximum number  0.04% (3) can fit in  0.04% (3) maximum number of  0.04% (3) there exists a  0.04% (3) bellinequalitymax(coeffs, a_coe, b_coe,  0.04% (3) that is entangled,  0.04% (3) have length at  0.04% (3) an nsymbol superpermutation  0.04% (3) that a state  0.04% (3) it is not  0.04% (3) the shortest string  0.04% (3) a superpermutation of  0.04% (3) in its operatorschmidt  0.04% (3) if the operatorschmidt  0.04% (3) rank of is  0.04% (3) value of the  0.04% (3) b_coe, a_val, b_val,  0.04% (3) >> bellinequalitymax(coeffs, a_coe,  0.04% (3) the previous two  0.04% (3) the game “entanglement”  0.04% (3) the induced matrix  0.04% (3) vector of singular  0.04% (3) decomposition tells us  0.04% (3) the same as  0.04% (3) essentially the same  0.04% (3) superpermutation conjecture is  0.04% (3) has been implemented  0.04% (3) the rows and  0.04% (3) this maximum value  0.04% (3) is exactly equal  0.04% (3) tells us that  0.04% (3) the uppertriangular part  0.04% (3) is the positive  0.04% (3) corresponds to the  0.04% (3) is a multiple  0.04% (3) exactly equal to  0.04% (3) maximum value is  0.04% (3) know that this  0.04% (3) of golomb rulers  0.04% (3) uppertriangular part of  0.04% (3) the positive real  0.04% (3) what the operatorschmidt  0.04% (3) and columns of  0.04% (3) maximum score in  0.04% (3) game “entanglement” is  0.04% (3) smaller than the  0.04% (3) as a maximization  0.04% (3) here is that  0.04% (3) converge to some  0.04% (3) score in the  0.04% (3) lower bound of  0.04% (3) on the maximum  0.04% (3) number of possible  0.04% (3) in the game  0.04% (3) the maximum score  0.04% (3) that the rows  0.04% (3) good idea of  0.04% (3) three special cases  0.04% (3) the norm as  0.04% (3) one such example.  0.03% (2) are in fact  0.03% (2) for all :  0.03% (2) the state is  0.03% (2) t. pham. enumeration  0.03% (2) furthermore, it is  0.03% (2) we have which  0.03% (2) see that we  0.03% (2) this case, and  0.03% (2) which contradicts the  0.03% (2) use the value  0.03% (2) possible orderings in  0.03% (2) of to conclude  0.03% (2) upper bound to  0.03% (2) state with and  0.03% (2) orderings in this  0.03% (2) does not imply  0.03% (2) is one such  0.03% (2) detected by the  0.03% (2) that the sets  0.03% (2) completes the proof.  0.03% (2) 9080jdarc on the  0.03% (2) formulation of the  0.03% (2) the trace norm  0.03% (2) that is entangled.  0.03% (2) “entanglement” is 9080jdarc  0.03% (2) the coefficients in  0.03% (2) its operatorschmidt decomposition.  0.03% (2) orderings of pairwise  0.03% (2) if we can  0.03% (2) : the state  0.03% (2) entangled based on  0.03% (2) be the case  0.03% (2) loss of generality  0.03% (2) “true”: the minimal  0.03% (2) trace norm (i.e.,  0.03% (2) a matlab toolbox  0.03% (2) exists a separable  0.03% (2) value in the  0.03% (2) pham. enumeration of  0.03% (2) separable state with  0.03% (2) and for all  0.03% (2) orderings of the  0.03% (2) , which contradicts  0.03% (2) of the matrices  0.03% (2) for all and  0.03% (2) with operatorschmidt rank  0.03% (2) columns of that  0.03% (2) is separable. proof. if  0.03% (2) then it can  0.03% (2) part of a  0.03% (2) and so on.  0.03% (2) of the terms  0.03% (2) and notice that  0.03% (2) and only if  0.03% (2) in the uppertriangular  0.03% (2) . thus and  0.03% (2) we define a  0.03% (2) notice that for  0.03% (2) that is separable,  0.03% (2) both negative semidefinite  0.03% (2) 1, which is  0.03% (2) all states with  0.03% (2) this matrix is  0.03% (2) this quantity is  0.03% (2) states with operatorschmidt  0.03% (2) criterion. phys. rev.  0.03% (2) [quantph] d. cariello.  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) the case that  0.03% (2) so there are  0.03% (2) it turns out  0.03% (2) uppertriangular part are  0.03% (2) operatorschmidt decomposition if  0.03% (2) the above matrix  0.03% (2) then is entangled.  0.03% (2) but i will  0.03% (2) side of theorem  0.03% (2) we can use  0.03% (2) to sometimes prove  0.03% (2) gives , so  0.03% (2) multiplying the first  0.03% (2) prove that a  0.03% (2) we will use  0.03% (2) inequalities: , ,  0.03% (2) the following three  0.03% (2) problem here is  0.03% (2) is the number  0.03% (2) that uppertriangular part  0.03% (2) we are done.  0.03% (2) to place the  0.03% (2) then is separable.  0.03% (2) then we can  0.03% (2) bound on the  0.03% (2) immediately gives us  0.03% (2) ways to place  0.03% (2) number of such  0.03% (2) positive semidefinite, it  0.03% (2) formula for the  0.03% (2) the matrix above  0.03% (2) to the case  0.03% (2) matrix above on  0.03% (2) if they are  0.03% (2) follows that is  0.03% (2) to compute hardtocompute  0.03% (2) entanglement witness, we  0.03% (2) is a constant  0.03% (2) vectors as ,  0.03% (2) the same left  0.03% (2) the matrix attaining  0.03% (2) , where is  0.03% (2) raised to the  0.03% (2) values of ,  0.03% (2) so that for  0.03% (2) such that there  0.03% (2) values are such that  0.03% (2) whose singular values are  0.03% (2) same left and  0.03% (2) one with the  0.03% (2) matrix attaining this  0.03% (2) be the matrix attaining  0.03% (2) that . compute keeping  0.03% (2) as a double  0.03% (2) and whose singular  0.03% (2) constant so that  0.03% (2) that they are  0.03% (2) of the operator  0.03% (2) toolbox for quantum  0.03% (2) qetlab: a matlab  0.03% (2) used to find  0.03% (2) find lower bounds  0.03% (2) it has been  0.03% (2) all vectors and  0.03% (2) then maximize over  0.03% (2) the left and  0.03% (2) for all .  0.03% (2) can be done  0.03% (2) over all vectors  0.03% (2) and then maximize  0.03% (2) dealing with quantum  0.03% (2) making use of  0.03% (2) extremely quick to  0.03% (2) almost identical to  0.03% (2) good estimates of  0.03% (2) theory (especially when  0.03% (2) of functions for  0.03% (2) number such that  0.03% (2) are the same  0.03% (2) complex phases that  0.03% (2) to . furthermore,  0.03% (2) so that . compute  0.03% (2) fixing a randomlychosen  0.03% (2) by setting and  0.03% (2) maximize over and  0.03% (2) the maximum absolute  0.03% (2) to , where  0.03% (2) is the maximum  0.03% (2) the usual operator  0.03% (2) special cases of  0.03% (2) is defined as  0.03% (2) a maximization over  0.03% (2) post, we will  0.03% (2) such as the  0.03% (2) and whose magnitudes  0.03% (2) real number such  0.03% (2) (especially when p  0.03% (2) is fast enough  0.03% (2) is the usual  0.03% (2) often called the  0.03% (2) 2, which is  0.03% (2) which is often  0.03% (2) cases of the  0.03% (2) is a natural  0.03% (2) value of .  0.03% (2) downside of this  0.03% (2) as those of  0.03% (2) each of the  0.03% (2) quick to run,  0.03% (2) are made after  0.03% (2) until negligible gains  0.03% (2) step 2. repeat  0.03% (2) and return to  0.03% (2) increment by 1  0.03% (2) magnitudes are such  0.03% (2) i am happy  0.03% (2) separable or entangled.  0.03% (2) to see that  0.03% (2) the one that  0.03% (2) superpermutations have length  0.03% (2) we can find  0.03% (2) discussed in the  0.03% (2) such a way  0.03% (2) to add the  0.03% (2) a wide variety  0.03% (2) the six new  0.03% (2) 153, and there  0.03% (2) we are allowed  0.03% (2) am happy to  0.03% (2) of minimal superpermutations  0.03% (2) the symbols “1”  0.03% (2) as a contiguous  0.03% (2) have been found  0.03% (2) on five symbols  0.03% (2) all minimal superpermutations  0.03% (2) a string if  0.03% (2) if we are  0.03% (2) for all n ≥  0.03% (2) each is a  0.03% (2) detects the entanglement  0.03% (2) entanglement witness that  0.03% (2) decomposition if then  0.03% (2) mixed state is  0.03% (2) that the singular  0.03% (2) form orthonormal bases of  0.03% (2) real scalar and  0.03% (2) and form orthonormal  0.03% (2) to speed up  0.03% (2) a real scalar  0.03% (2) where each is  0.03% (2) in the form  0.03% (2) is available here.  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) this problem is  0.03% (2) turns out that  0.03% (2) which says that  0.03% (2) an arbitrary number  0.03% (2) on a value  0.03% (2) left can take  0.03% (2) quantity on the  0.03% (2) says that if  0.03% (2) example, consider the  0.03% (2) for dealing with  0.03% (2) arbitrary number of  0.03% (2) the realignment criterion,  0.03% (2) a value as  0.03% (2) tests for entanglement  0.03% (2) if you want  0.03% (2) entanglement. quantum inf.  0.03% (2) method for recognizing  0.03% (2) a matrix realignment  0.03% (2) and l.a. wu.  0.03% (2) the realignment criterion.  0.03% (2) can take on  0.03% (2) 0]; % coefficients  0.03% (2) that this problem  0.03% (2) with this property.  0.03% (2) this problem was  0.03% (2) is no way  0.03% (2) + (n1)! +  0.03% (2) 1 additional character  0.03% (2) + n!, since  0.03% (2) must have length  0.03% (2) the length of  0.03% (2) that it is  0.03% (2) [1 1]; %  0.03% (2) string that contains  0.03% (2) is the shortest  0.03% (2) . there is  0.03% (2) imply “true”: the  0.03% (2) “obvious” does not  0.03% (2) if you have  0.03% (2) the npa hierarchy  0.03% (2) 1]; % values  0.03% (2) then we append  0.03% (2) 