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nathaniel johnston  mailto:nathaniel@njohnston.ca 
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how to compute hardtocompute matrix norms  http://www.njohnston.ca/2016/01/howtocomputehardtocomputematrixnorms/ 
no comments  http://www.njohnston.ca/2016/01/howtocomputehardtocomputematrixnorms/#respond 
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 
hölder’s inequality  https://en.wikipedia.org/wiki/h%c3%b6lder%27s_inequality 
power method  https://en.wikipedia.org/wiki/power_iteration 
qetlab package  http://www.qetlab.com/main_page 
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 
my thesis  http://www.njohnston.ca/publications/normsandconesinthetheoryofquantumentanglement/ 
sk_iterate  http://www.qetlab.com/sk_iterate 
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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upb  http://www.qetlab.com/upb 
isseparable  http://www.qetlab.com/isseparable 
partial transpose  http://www.qetlab.com/partialtranspose 
realignment criterion  http://www.qetlab.com/realignment 
choi map  http://www.qetlab.com/choimap 
symmetric extensions  http://www.qetlab.com/symmetricextension 
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tsirelson’s bound  http://en.wikipedia.org/wiki/tsirelson%27s_bound 
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bellinequalitymax  http://www.qetlab.com/bellinequalitymax 
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bellinequalitymax  http://www.qetlab.com/bellinequalitymax 
npahierarchy  http://www.qetlab.com/npahierarchy 
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here  http://www.notatt.com/permutations.pdf 
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singular value decomposition  http://en.wikipedia.org/wiki/singular_value_decomposition 
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 
quantum entanglement  http://www.njohnston.ca/tag/quantumentanglement/ 
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arxiv:quantph/0502170  http://arxiv.org/abs/quantph/0502170 
a combinatorial problem  http://projecteuclid.org/euclid.mmj/1028989731 
arxiv:1110.6154  http://arxiv.org/abs/1110.6154 
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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/ 
the spectrum of the partial transpose of a density matrix  http://www.njohnston.ca/2013/07/thespectrumofthepartialtranspose/ 
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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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0.04% (3) previous two  0.04% (3) problem here  0.04% (3) bound of  0.04% (3) to find  0.04% (3) essentially the  0.04% (3) of entanglement  0.04% (3) a multiple  0.04% (3) for quantum  0.04% (3) of functions  0.04% (3) are also  0.04% (3) found in  0.04% (3) positive semidefinite.  0.04% (3) has operatorschmidt  0.04% (3) corresponds to  0.04% (3) complex phases  0.04% (3) separable or  0.04% (3) rows and  0.04% (3) place the  0.04% (3) counting the  0.04% (3) are decreasing.  0.04% (3) where each  0.04% (3) this matrix  0.04% (3) part of  0.04% (3) the uppertriangular  0.04% (3) columns of  0.04% (3) vectors as  0.04% (3) so there  0.04% (3) the rows  0.04% (3) induced schatten  0.04% (3) and columns  0.04% (3) vector of  0.04% (3) orderings is  0.04% (3) of singular  0.04% (3) implemented in  0.04% (3) from a  0.04% (3) with operatorschmidt  0.04% (3) ). this  0.04% (3) of possible  0.04% (3) as large  0.04% (3) arbitrary number  0.04% (3) same as  0.04% (3) since is  0.04% (3) multiple of  0.04% (3) entangled state  0.04% (3) for entanglement  0.04% (3) for computing  0.04% (3) to check  0.04% (3) us that  0.04% (3) of theorem  0.04% (3) exactly equal  0.04% (3) entanglement in  0.03% (2) of such  0.03% (2) different ways  0.03% (2) will use  0.03% (2) which contradicts  0.03% (2) detects the  0.03% (2) witness that  0.03% (2) quantum entanglement:  0.03% (2) be done  0.03% (2) if then  0.03% (2) integer sequences,  0.03% (2) exactly the  0.03% (2) completes the  0.03% (2) witness, we  0.03% (2) is indeed  0.03% (2) possible products  0.03% (2) where the  0.03% (2) pairwise multiplication  0.03% (2) case, and  0.03% (2) real numbers  0.03% (2) above on  0.03% (2) pham. enumeration  0.03% (2) by this  0.03% (2) have which  0.03% (2) an entanglement  0.03% (2) between these  0.03% (2) case that  0.03% (2) all and  0.03% (2) how many  0.03% (2) toolbox for  0.03% (2) such orderings  0.03% (2) separable. proof. if  0.03% (2) eigenvalues of  0.03% (2) to sometimes  0.03% (2) of that  0.03% (2) that and  0.03% (2) entries that  0.03% (2) can use  0.03% (2) real scalar  0.03% (2) cases where  0.03% (2) a similar  0.03% (2) contained within  0.03% (2) conway's game  0.03% (2) let have  0.03% (2) anything fancy:  0.03% (2) of life  0.03% (2) bound to  0.03% (2) criterion, which  0.03% (2) only if  0.03% (2) score in  0.03% (2) the game  0.03% (2) entanglement of  0.03% (2) form orthonormal  0.03% (2) decomposition is  0.03% (2) does not  0.03% (2) imply “true”:  0.03% (2) we see  0.03% (2) bound on  0.03% (2) can also  0.03% (2) be phrased  0.03% (2) upper bounds  0.03% (2) matrix is  0.03% (2) semidefinite, and  0.03% (2) on five  0.03% (2) have used  0.03% (2) symbols have  0.03% (2) of pairwise  0.03% (2) 1, which  0.03% (2) sometimes prove  0.03% (2) matrix in  0.03% (2) how the  0.03% (2) decomposition. however,  0.03% (2) the equality  0.03% (2) some .  0.03% (2) the possible  0.03% (2) formula for  0.03% (2) without loss  0.03% (2) however, multiplying  0.03% (2) this, we  0.03% (2) orderings are  0.03% (2) under the  0.03% (2) some matrix  0.03% (2) bound is  0.03% (2) follows from  0.03% (2) write a  0.03% (2) define a  0.03% (2) and notice  0.03% (2) any value  0.03% (2) standard continuity  0.03% (2) case. i  0.03% (2) have the  0.03% (2) both negative  0.03% (2) of generality  0.03% (2) semidefinite, we  0.03% (2) the minimum  0.03% (2) shown in  0.03% (2) the state  0.03% (2) negative semidefinite  0.03% (2) such example.  0.03% (2) can write  0.03% (2) all states  0.03% (2) with and  0.03% (2) verify that  0.03% (2) state with  0.03% (2) that then  0.03% (2) so on.  0.03% (2) gives ,  0.03% (2) multiplying the  0.03% (2) inequalities: ,  0.03% (2) following three  0.03% (2) seem to  0.03% (2) if they  0.03% (2) are done.  0.03% (2) gauge function  0.03% (2) functions of  0.03% (2) comes from  0.03% (2) the interval  0.03% (2) a more  0.03% (2) better upper  0.03% (2) the proof.  0.03% (2) which completes  0.03% (2) value in  0.03% (2) gives us  0.03% (2) entanglement detection  0.03% (2) 7 case  0.03% (2) 4 case,  0.03% (2) beyond the  0.03% (2) are orthonormal  0.03% (2) to conclude  0.03% (2) criterion. phys.  0.03% (2) entangled if  0.03% (2) matrix above  0.03% (2) d. cariello.  0.03% (2) is one  0.03% (2) realignment map  0.03% (2) are 12  0.03% (2) detected by  0.03% (2) operatorschmidt decomposition.  0.03% (2) since each  0.03% (2) separable, which  0.03% (2) is separable,  0.03% (2) entangled based  0.03% (2) in has  0.03% (2) that either  0.03% (2) are entangled  0.03% (2) in these  0.03% (2) state in  0.03% (2) part are  0.03% (2) these two  0.03% (2) entangled. the  0.03% (2) improved the  0.03% (2) norm (i.e.,  0.03% (2) that uppertriangular  0.03% (2) the integers  0.03% (2) lefthand side  0.03% (2) formulation of  0.03% (2) nathaniel johnston  0.03% (2) in disguise  0.03% (2) , raised  0.03% (2) each iteration.  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) values and  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) almost identical  0.03% (2) , respectively.  0.03% (2) diagonal entries  0.03% (2) you want  0.03% (2) other matlab  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) a lower  0.03% (2) quick to  0.03% (2) over all  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) good estimates  0.03% (2) nphard to  0.03% (2) or entangled.  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) whether a  0.03% (2) when the  0.03% (2) is just  0.03% (2) the symbols  0.03% (2) consecutive “wasted”  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) discussed in  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) depthfirst search  0.03% (2) known superpermutation  0.03% (2) this for  0.03% (2) available here.  0.03% (2) and form  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) 120 permutations  0.03% (2) permutations of  0.03% (2) the results  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) (see a  0.03% (2) since there  0.03% (2) map is  0.03% (2) and of  0.03% (2) and bell  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) quantum inf.  0.03% (2) quantity on  0.03% (2) recognizing entanglement.  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) example, consider  0.03% (2) value as  0.03% (2) + (n1)!  0.03% (2) the origin  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) seemed like  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) through the  0.03% (2) via the  0.03% (2) “true”: the  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) “entanglement” is  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) that is entangled  0.09% (7) the operatorschmidt decomposition  0.09% (7) that we can  0.08% (6) is straightforward to  0.08% (6) a state is  0.08% (6) it can be  0.07% (5) we know that  0.07% (5) conjecture is false  0.07% (5) singular value decomposition  0.07% (5) it is straightforward  0.07% (5) there is no  0.07% (5) minimal superpermutation conjecture  0.07% (5) the operatorschmidt rank  0.07% (5) quantum information theory  0.07% (5) this maximum is  0.07% (5) that there is  0.07% (5) the upper bound  0.07% (5) in this case  0.07% (5) possible orderings of  0.07% (5) from the fact  0.07% (5) can be written  0.07% (5) let be the  0.05% (4) this maximum. by  0.05% (4) that is separable  0.05% (4) induced matrix norms  0.05% (4) that we have  0.05% (4) minimal superpermutations have  0.05% (4) fixed, and let  0.05% (4) the bell inequality  0.05% (4) vector attaining this  0.05% (4) be used to  0.05% (4) number of orderings  0.05% (4) p → q norm  0.05% (4) and right singular  0.05% (4) conclude that is  0.05% (4) to show that  0.05% (4) in a string  0.05% (4) on the left  0.05% (4) is as follows:  0.05% (4) the previous section  0.05% (4) of the realignment  0.05% (4) keeping fixed, and  0.05% (4) is positive semidefinite,  0.05% (4) maximum is the  0.05% (4) the sets and  0.05% (4) the value of  0.05% (4) the vector attaining  0.05% (4) the one with  0.05% (4) are both positive  0.05% (4) unextendible product bases  0.05% (4) . furthermore, the  0.05% (4) right singular vectors  0.05% (4) lower bounds on  0.05% (4) permutations that we  0.05% (4) are easy to  0.05% (4) , and whose  0.05% (4) that is a  0.05% (4) converge to some  0.04% (3) >> bellinequalitymax(coeffs, a_coe,  0.04% (3) bellinequalitymax(coeffs, a_coe, b_coe,  0.04% (3) values of the  0.04% (3) rank of is  0.04% (3) a_coe, b_coe, a_val,  0.04% (3) lower bound of  0.04% (3) maximum number of  0.04% (3) good idea of  0.04% (3) that for all  0.04% (3) and are both  0.04% (3) of is then  0.04% (3) have length at  0.04% (3) has operatorschmidt rank  0.04% (3) that is entangled,  0.04% (3) be written in  0.04% (3) since is positive  0.04% (3) in terms of  0.04% (3) there exists a  0.04% (3) we have found  0.04% (3) as large as  0.04% (3) . since is  0.04% (3) % coefficients of  0.04% (3) a mixed state  0.04% (3) in the bell  0.04% (3) bell inequality >>  0.04% (3) an nsymbol superpermutation  0.04% (3) the previous two  0.04% (3) b_coe, a_val, b_val,  0.04% (3) of permutations that  0.04% (3) that the rows  0.04% (3) this maximum value  0.04% (3) the singular values  0.04% (3) here is that  0.04% (3) and the sets  0.04% (3) is the positive  0.04% (3) is a multiple  0.04% (3) that a state  0.04% (3) of golomb rulers  0.04% (3) tells us that  0.04% (3) is exactly equal  0.04% (3) it follows that  0.04% (3) it is not  0.04% (3) superpermutation conjecture is  0.04% (3) a superpermutation of  0.04% (3) the positive real  0.04% (3) three special cases  0.04% (3) in this post,  0.04% (3) what the operatorschmidt  0.04% (3) decomposition tells us  0.04% (3) the norm as  0.04% (3) as a maximization  0.04% (3) of the coefficients  0.04% (3) number of possible  0.04% (3) in quantum information  0.04% (3) essentially the same  0.04% (3) the maximum number  0.04% (3) and columns of  0.04% (3) the uppertriangular part  0.04% (3) value of the  0.04% (3) the shortest string  0.04% (3) the rows and  0.04% (3) can fit in  0.04% (3) if the operatorschmidt  0.04% (3) exactly equal to  0.04% (3) maximum value is  0.04% (3) know that this  0.04% (3) corresponds to the  0.04% (3) tells us about  0.04% (3) we can fit  0.04% (3) the induced matrix  0.04% (3) in its operatorschmidt  0.04% (3) uppertriangular part of  0.04% (3) vector of singular  0.04% (3) proving a state  0.04% (3) has been implemented  0.04% (3) the same as  0.04% (3) smaller than the  0.04% (3) side of theorem  0.03% (2) but i will  0.03% (2) then is entangled.  0.03% (2) to compute hardtocompute  0.03% (2) operatorschmidt decomposition if  0.03% (2) value in the  0.03% (2) that the sets  0.03% (2) completes the proof.  0.03% (2) formulation of the  0.03% (2) the trace norm  0.03% (2) that is entangled.  0.03% (2) the coefficients in  0.03% (2) its operatorschmidt decomposition.  0.03% (2) detected by the  0.03% (2) if we can  0.03% (2) entangled based on  0.03% (2) be the case  0.03% (2) loss of generality  0.03% (2) exists a separable  0.03% (2) separable state with  0.03% (2) of the matrices  0.03% (2) and for all  0.03% (2) : the state  0.03% (2) is one such  0.03% (2) state with and  0.03% (2) for all :  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) trace norm (i.e.,  0.03% (2) we can use  0.03% (2) notice that for  0.03% (2) to sometimes prove  0.03% (2) to place the  0.03% (2) multiplying the first  0.03% (2) inequalities: , ,  0.03% (2) the following three  0.03% (2) problem here is  0.03% (2) the above matrix  0.03% (2) that uppertriangular part  0.03% (2) bound on the  0.03% (2) , which contradicts  0.03% (2) immediately gives us  0.03% (2) ways to place  0.03% (2) number of such  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) gives , so  0.03% (2) orderings of the  0.03% (2) columns of that  0.03% (2) “true”: the minimal  0.03% (2) score in the  0.03% (2) on the maximum  0.03% (2) in the game  0.03% (2) the maximum score  0.03% (2) game of life  0.03% (2) orderings of pairwise  0.03% (2) does not imply  0.03% (2) are in fact  0.03% (2) a matlab toolbox  0.03% (2) pham. enumeration of  0.03% (2) t. pham. enumeration  0.03% (2) this case, and  0.03% (2) upper bound to  0.03% (2) possible orderings in  0.03% (2) which contradicts the  0.03% (2) uppertriangular part are  0.03% (2) part of a  0.03% (2) prove that a  0.03% (2) it turns out  0.03% (2) and notice that  0.03% (2) of the terms  0.03% (2) and so on.  0.03% (2) then it can  0.03% (2) is separable. proof. if  0.03% (2) with operatorschmidt rank  0.03% (2) both negative semidefinite  0.03% (2) . thus and  0.03% (2) we are done.  0.03% (2) if they are  0.03% (2) positive semidefinite, it  0.03% (2) then we can  0.03% (2) then is separable.  0.03% (2) is the number  0.03% (2) we will use  0.03% (2) and only if  0.03% (2) we define a  0.03% (2) in the uppertriangular  0.03% (2) mean by this  0.03% (2) 1, which is  0.03% (2) this matrix is  0.03% (2) this quantity is  0.03% (2) orderings in this  0.03% (2) so there are  0.03% (2) the case that  0.03% (2) and it is  0.03% (2) of pairwise multiplication  0.03% (2) entanglement witness, we  0.03% (2) the possible orderings  0.03% (2) [quantph] d. cariello.  0.03% (2) criterion. phys. rev.  0.03% (2) states with operatorschmidt  0.03% (2) all states with  0.03% (2) that is separable,  0.03% (2) follows that is  0.03% (2) for all and  0.03% (2) the one that  0.03% (2) to see that  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) detects the entanglement  0.03% (2) a wide variety  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) then we append  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) and form orthonormal  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) each is a  0.03% (2) a real scalar  0.03% (2) to speed up  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) game “entanglement” is  0.03% (2) 