On the dimension of subspaces with bounded Schmidt rank
aa r X i v : . [ qu a n t - ph ] J un On the dimension of subspaces with bounded Schmidt rank
Toby Cubitt, Ashley Montanaro, and Andreas Winter
1, 3 Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK Department of Computer Science, University of Bristol, Woodland Road, Bristol, BS8 1UB, UK Quantum Information Technology Lab, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Dated: 30 May 2007)We consider the question of how large a subspace of a given bipartite quantum system can bewhen the subspace contains only highly entangled states. This is motivated in part by results ofHayden et al. , which show that in large d × d –dimensional systems there exist random subspaces ofdimension almost d , all of whose states have entropy of entanglement at least log d − O (1). It is alsorelated to results due to Parthasarathy on the dimension of completely entangled subspaces, whichhave connections with the construction of unextendible product bases. Here we take as entanglementmeasure the Schmidt rank, and determine, for every pair of local dimensions d A and d B , and every r , the largest dimension of a subspace consisting only of entangled states of Schmidt rank r orlarger. This exact answer is a significant improvement on the best bounds that can be obtainedusing random subspace techniques. We also determine the converse: the largest dimension of asubspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspacescontaining only states with Schmidt equal to r . Introduction.
Entanglement is at the heart of quantuminformation theory, and this property of quantum sys-tems is ultimately responsible for new information taskssuch as teleportation [1], quantum key agreement [2, 3]or quantum computational speedup [4]. Consequently,a theory of measuring and comparing the entanglementcontent of quantum states has emerged [5], which at-tempts to classify states according to their non-classicalcapabilities. It is, however, remarkable how large anumber of entanglement measures have been put for-ward [5, 6], indicating that the structure of entanglementis not one that can be captured by a single number. Oneparticular measure is the Schmidt rank of a pure bipar-tite state | ψ i , i.e. the number of non-zero coefficients λ i in the — essentially unique — Schmidt form: | ψ i = X i λ i | e i i A | f i i B . This measure has even been extended to mixed states,as the maximum Schmidt rank in an optimal pure statedecomposition [7], but the convex hull construction couldalso be considered. For pure states, the Schmidt rank isindeed the unique invariant under the class of stochasticlocal operations and classical communication (SLOCC).Here we ask, and answer completely, the question:what is the maximum dimension of a subspace S in a d A × d B bipartite system such that every state in S hasSchmidt rank at least r ? This is trivial for r = 1, sowe can assume r ≥
2; also, the Schmidt rank can be atmost min { d A , d B } , which we will assume without loss ofgenerality to be d A .There are two extreme cases. The first, r = 2 (i.e. asubspace that contains no product state), is addressedin [8]; the answer is ( d A − d B − r = d A = d B =: d , has an elementary solution: theanswer is 1 (take any one-dimensional subspace spannedby a vector of maximum Schmidt rank d ). To show this, consider any two-dimensional subspace spanned by unitvectors | ϕ i , | ψ i ∈ C d ⊗ C d . We want to show that atleast one superposition | φ x i = | ϕ i + x | ψ i has Schmidtrank less than d . The crucial observation is that we canarrange the coefficients of a state vector | φ i in the compu-tational basis {| i i | j i} i,j =1 ,...,d , into a d × d matrix M ( φ ),and that the Schmidt rank of the state vector equals thelinear rank of the associated matrix. In other words,the statement that | φ x i has Schmidt rank less than r iscaptured by the vanishing of the determinant det M ( φ x ).But the latter is a non-constant polynomial in x of degree d . Hence, it must have a root in the complex field, andthe corresponding | φ x i has Schmidt rank r − r restson the same matrix representation, and the character-isation of Schmidt rank via vanishing of certain deter-minants again plays a crucial role. It involves, however,much deeper algebraic geometry machinery, extendingthe above use of the fundamental theorem of algebra. Notation and Terminology.
We will denote theSchmidt rank of a bipartite pure state | ψ i AB (nonzero,but not necessarily normalised) by Sch( | ψ i AB ). We saythat a subspace S of the bipartite space in question hasSchmidt rank ≥ r if all its nonzero vectors have Schmidtrank ≥ r (analogously for ≤ r and = r ). If M is a ma-trix, R is a set of indices for rows of M , and C is a set ofindices for columns, then M { R,C } denotes the submatrixformed by deleting all rows and columns other than thosein R and C .The projective space of dimension d is denoted by P d .This is the space of lines (one-dimensional subspaces) of C d +1 ; i.e. it is obtained from the nonzero elements of C d +1 by identifying collinear vectors. If a subset S ⊆ C d +1 is a union of lines, this identification associates toit a natural projectification of S , denoted by P ( S ) ⊆ P d ;see e.g. [9, 10, 11, 12] for this and related notions fromalgebraic geometry.For a state vector | ψ i ∈ C d A ⊗ C d B , with fixed lo-cal bases of the two Hilbert spaces, such that | ψ i = P i,j c ij | i i | j i , define the d A × d B matrix M ( ψ ) =( c ij ) i =1 ,...,d A ,j =1 ,...,d B . This identifies C d A ⊗ C d B withthe space M ( d A , d B ) of d A × d B matrices. Bounding the dimension of highly entangled sub-spaces.
We first state some preparatory lemmas relat-ing bipartite states to matrices, whose proofs are widelyknown and do not warrant repetition here.
Lemma 1
The set of (unnormalised) states in C d A ⊗ C d B with Schmidt rank r is isomorphic to the set of d A × d B complex matrices with rank r . Proof
Obvious from the standard proof of the Schmidtdecomposition via the singular value decomposition. (cid:3)
Lemma 2
A matrix M has rank M < r iff all its order– r minors (the determinants of r × r submatrices) are zero. Proof
See [13, p.13]. (cid:3)
This means that a geometric characterisation of sub-spaces of Schmidt rank ≥ r is to say that the linearspace M ( S ) of associated matrices doesn’t intersect theset of common zeros of all order– r minors (except in thezero vector). Such common zeroes of sets of multivariatepolynomials are called (algebraic) varieties, and the onein question has been studied in the mathematical litera-ture [20]. Definition 3 (Determinantal variety)
The affinedeterminantal variety D r ( d A , d B ) over the (algebraicallyclosed) field F in the space F d A d B is the variety definedby the vanishing of all order– r minors of a d A × d B matrix, whose elements are considered as independentvariables in F . (Of course, in quantum theory we are mostly interestedin the case F = C .)The basic idea is now essentially parameter counting: ifthe dimension of S plus that of the variety D r ( d A , d B ) islarger than d A d B , then the polynomial equations defin-ing the order– r minors have roots in M ( S ). To makethis heuristic rigorous, we need to go to the correspond-ing projective spaces: since the polynomials defined bythe minors of a matrix are homogeneous, a determinan-tal variety can also be thought of as a projective variety P ( D r ( d A , d B )) in the space P d A d B − . The same is truefor the subspace S , so it also has a projectification P ( S ). Lemma 4 (Dimension of determinantal varieties)
The dimension of an affine determinantal variety isgiven by dim D r ( d A , d B ) = d A d B − ( d A − r +1)( d B − r +1) . Proof
See e.g. [10, Proposition 12.2, p. 151]. (cid:3)
The corresponding projective determinantal variety has,of course, dimension one less: dim P ( D r ( d A , d B )) = d A d B − ( d A − r +1)( d B − r +1) −
1. Likewise, the dimensionof P ( S ) is dim S − Lemma 5 (Intersection of projective varieties) If V and W are projective varieties in P d such that dim V + dim W ≥ d , then V ∩ W = ∅ . Proof
See e.g. [11, Theorem 6, p. 76] or [10, Exer-cise 11.38, p. 148]. (cid:3)
Proposition 6
For any subspace S ⊆ C d A ⊗ C d B of di-mension dim S > ( d A − r + 1)( d B − r + 1) , there exists atleast one state in the subspace with Schmidt rank strictlyless than r . Proof
The set of all (unnormalised) states in the bipar-tite space C d A ⊗ C d B forms a projective space P d A d B − over the complex field. From Lemmas 1 and 2, andDefinition 3, the subset of those states with Schmidtrank less than r then forms a projective determinan-tal variety P ( D r ( d A , d B )) in that space. The subspace S corresponds to the projective variety P ( S ) (a projec-tive linear subspace), which has dimension dim P ( S ) > ( d A − r + 1)( d B − r + 1) − P ( D r ( d A , d B )) + dim P ( S ) ≥ d A d B −
1= dim P d A d B − . Thus by Lemma 5, P ( S ) and P ( D r ( d A , d B )) have a non-empty intersection, i.e. the subspace S contains at leastone state with Schmidt rank less than r . (cid:3) Construction of highly entangled subspaces.
Wewill now give an explicit construction of a subspacewith bounded Schmidt rank that saturates the boundof Proposition 6, based on totally non-singular matrices.(Note that we can not simply take the complement of P ( D r ( d A , d B )) in P d A d B − , since it is by no means clearthat this is a projective linear variety, i.e. a subspace.) Definition 7 (Totally non-singular matrix)
A ma-trix is said to be totally non-singular if all of its minorsare non-zero.
Lemma 8
There exist totally non-singular matrices ofany dimension.
Proof
The n × n Vandermonde matrix generated by0 < λ < λ < · · · < λ n is totally positive (i.e. all itsminors are strictly positive, see [14]), therefore is also to-tally non-singular. Alternatively, it is also clear that ageneric complex matrix will be totally non-singular, asthe vanishing of a minor defines a set of matrices of mea-sure 0. (cid:3) Lemma 9
Let M be an m × m totally non-singular ma-trix, with m ≥ n . Let v be any linear combination of n of the columns of M . Then v contains at most n − zeroelements. Proof
Assume for contradiction that there exists a linearcombination of n columns of M containing n or more zeroelements. Let R be the set of indices of n of those zero ele-ments and C be the set of indices of the n columns. Sincethere is a linear combination of the columns of M suchthat the elements indexed by R are all zero, the columnsof the submatrix M { R,C } are linearly dependent, thus theminor det M { R,C } is zero and we have a contradiction. (cid:3) The construction of the subspace is based on the setsof vectors introduced in Lemma 9.
Proposition 10
Every bipartite system C d A ⊗ C d B hasa subspace S of Schmidt rank ≥ r , and of dimension dim S = ( d A − r + 1)( d B − r + 1) . Proof
Since the bipartite states with Schmidt rankbounded by r are isomorphic to d A × d B matrices whoserank is at least r (Lemma 1), and a matrix has rankgreater than or equal to r iff at least one of its order– r minors is non-zero (Lemma 2), it is sufficient to constructa set of linearly independent matrices S of cardinality | S | = ( d A − r + 1)( d B − r + 1) such that any linear combi-nation of them has at least one non-zero order– r minor,since these then define a basis for a subspace with thedesired properties.Label the diagonals of a d A × d B matrix by integers k ,with k increasing from lower-left to upper-right, and de-note the length of the k th diagonal by | k | . From Lemma 9,there exist sets of t = | k | − r + 1 linearly independentvectors of length | k | such that any linear combination ofthem has at most t − | k | − ( t −
1) = r non-zero elements.For each diagonal with length | k | ≥ r , construct aset of linearly independent matrices S k of cardinality | S k | = | k | − r + 1 by putting these vectors down the k th diagonal. By construction, any linear combination ofthese will have at least r non-zero elements down thatdiagonal. Since the determinant of the r × r submatrixwith these r non-zero elements down its main diagonalis clearly non-zero, any linear combination of matricesin S k has at least one non-zero order– r minor, thus hasrank at least r .Now define the set S = S k S k . Since matrices fromdifferent S k have elements down different diagonals, thematrices in S are linearly independent. It remains toshow that any linear combination of matrices from dif-ferent S k still has rank at least r . Let M be a matrixgiven by some linear combination of matrices in S , andlet κ be the maximum k for which the linear combinationincludes matrices from S k . It is still true that the κ th di-agonal of M must contain at least r non-zero elements.As κ labels the top-rightmost diagonal of M that con-tains any non-zero elements, the r × r submatrix of M with those r non-zero elements down its main diagonal islower-triangular, so has non-zero determinant. Thus M has at least one non-zero order– r minor, so has rank atleast r .Assume for convenience that d B ≥ d A . To determinethe cardinality of S , i.e. the dimension of the subspace,note that a d A × d B matrix has 1 + d B − d A diagonals oflength d A , and 2 diagonals of each length less than d A . Then the cardinality of S is given by | S | = X k | S k | = X k ( | k | − r + 1)= (cid:0) d B − d A (cid:1)(cid:0) d A − r + 1 (cid:1) + 2 d A − X i = r ( i − r + 1)= ( d A − r + 1)( d B − r + 1) , which matches the claimed dimension of the subspace. (cid:3) Subspaces with bounded Schmidt rank.
Puttingtogether Propositions 6 and 10, we obtain our main re-sult:
Theorem 11
The maximum dimension of a subspace S ⊆ C d A ⊗ C d B of Schmidt rank ≥ r is given by ( d A − r + 1)( d B − r + 1) . (cid:3) One could instead ask for the converse: subspaces ofSchmidt rank ≤ r . Note that geometrically this corre-sponds to a linear subspace lying within the determinan-tal variety D r +1 ( d A , d B ). There is a simple construction, S = R ⊗ C d B , for any subspace R ⊆ C d A of dimension r , which achieves dim S = rd B . This is clearly tight if r = 1 or r = d A . In fact, one can show that this con-struction is optimal in general, which is immediate fromthe following theorem due to Flanders [15]: Theorem 12 (Flanders)
Let S be a subspace of thespace of d A × d B matrices, where d A ≤ d B . Let r be themaximum rank of any element of S . Then dim S ≤ rd B . (cid:3) Another interesting variant is to ask what are the sub-spaces which have Schmidt rank exactly r . For exam-ple, our construction above yields subspaces of dimension d B − d A + 1 in C d A ⊗ C d B of Schmidt rank equal to d A . Adifferent example is given by the three-dimensional com-pletely antisymmetric subspace of C ⊗ C , which hasSchmidt rank equal to 2. This question has been thesubject of a remarkably long-running study in the lin-ear algebra literature and, as far as we are aware, thegeneral case remains unsolved. The best existing resultsare summarised in the following theorem, which can befound in [16]: Theorem 13 (Westwick)
Let S be the largest subspaceof the space of d A × d B matrices, with d B ≥ d A , suchthat the rank of every non-zero element of S is r . Thenin general, d B − r + 1 ≤ dim S ≤ d A + d B − r + 1 . Furthermore, if d B − r +1 does not divide ( d A − / ( r − ,then dim S = d B − r + 1 . If d A = r + 1 , d B = 2 r − , then dim S = r + 1 . (cid:3) For sufficiently large d B , it is of course impossible for d B − r + 1 to divide ( d A − / ( r − S ≤ ( d B − r + 1) + ( d A − r ). Discussion: applications and open questions.
Wehave determined the exact maximum dimension of sub-spaces of Schmidt rank ≥ r in any bipartite quantumsystem. The upper bound on the dimension is a gener-alisation of Parthasarathy’s argument [8] for a subspaceavoiding the manifold of product states, to the avoidingof a determinantal variety. Our constructive lower boundseems to differ from Parthasarathy’s (in the case r = 2),which is based on unextendible product vector systems.Comparing these results, using the Schmidt measure,with [17], where the entropy measure of entanglementis used, we have much tighter control on the entangle-ment in subspaces. For example, in the cited paper,the random subspaces that are constructed are necessar-ily highly entangled, simply because that is the genericbehaviour of random states. In contrast, here we findthe largest subspaces of bounded Schmidt rank over the whole range of the entanglement measure, including val-ues far away from typical. This is most clearly demon-strated by considering subspaces with Schmidt rankwithin a constant fraction of the maximum: r ≥ kd A . For k ≥ − d A / ( d B ln 2) , using the results of [17, Theorem IV.1]gives nothing better than the trivial one-dimensional sub-space, yet the exact result is asymptotically of order(1 − k ) d A d B , i.e. within a constant fraction of the entirespace!Our results can be used, in the spirit of [17], to con-struct highly mixed states of very large Schmidt mea-sure [7]: let ρ be the normalised projector onto a maxi-mum dimensional subspace S of Schmidt rank ≥ r . Then,since every pure state decomposition of ρ can only consistof state vectors from S , any entanglement measure builtfrom the Schmidt ranks of the constituent pure states hasto be at least r . For example, in arbitrarily large d × d –systems, we thus find for any p states of rank ≥ p d (i.e.entropy 2 log d +2 log p ) and Schmidt measure ≥ (1 − p ) d .The ideas and results described in this paper also haveapplications to the study of degradable quantum chan-nels [18].The present paper raises a number of questions: a firstis about multiparty generalisations, e.g. looking at sub-spaces with constraints on the Schmidt rank across all bi-partite cuts. The case of completely entangled subspaceswas solved in [8]. Note that, were random subspaces tosaturate the bound in Proposition 10, a random subspacesaturating the tightest of the bipartite constraints wouldautomatically satisfy all the other constraints. The mul-tipartite case would therefore reduce to the bipartite case.However, this would contradict the known result for com-pletely entangled subspaces, once again underlining thefact that progress in this type of problem requires goingbeyond the typical case. We could also attempt to make statements about moreoperationally motivated entanglement measures, espe-cially those based on von Neumann or R´enyi entropies,as for example in [17]. However, the algebraic techniquesused here do not seem to give any insight into these prob-lems. Acknowledgments.
The authors acknowledge sup-port by the European Commission, project “QAP”, andthe U.K. EPSRC through postgraduate scholarships, the“QIP IRC” and an Advanced Research Fellowship.We thank Aram Harrow, Richard Jozsa, Richard Lowand Will Matthews for various spirited discussions aboutthe content of this paper; especially the latter three fortheir solution of the case r = d A = d B , which was thestarting point of this work. We would also like to thankWee Kang Chua for drawing our attention to the questionof subspaces with upper-bounded Schmidt rank. [1] C. H. Bennett, G. Brassard, et al., Phys. Rev. Lett. ,1895 (1993).[2] C. H. Bennett and G. Brassard, in Proc. IEEE Int. Conf.Computers, Systems and Signal Processing, Bangalore,India (1984), pp. 175–179.[3] A. Ekert, Phys. Rev. Lett. , 661 (1991).[4] P. W. Shor, in Proc. 35 th Annual Symp. Foundations ofComputer Science , edited by S. Goldwasser (Los Alami-tos, CA, 1994), pp. 124–134.[5] R. Horodecki, P. Horodecki, et al., quant-ph/0702225 (2007).[6] M. Christandl, Ph.D. thesis, Cambridge (2006).[7] B. M. Terhal and P. Horodecki, quant-ph/9911117 ;Phys. Rev. A , R040301 (2000).[8] K. R. Parthasarathy, quant-ph/0307182 ; Proc. IndianAcad. Sci. (Math. Sci.) , 365 (2004).[9] P. Griffiths and J. Harris, Principles of Algebraic Geom-etry (John Wiley and Sons, New York, 1978).[10] J. Harris,
Algebraic Geometry: A First Course (Springer-Verlag, 1995).[11] I. Shafarevich,
Basic Algebraic Geometry 1: Varieties inProjective Space (Springer-Verlag, 1994), 2nd ed.[12] D. Cox, J. Little, and D. O’Shea,
Ideals, Varieties, andAlgorithms (Springer-Verlag, 2007), 3rd ed.[13] R. A. Horn and C. R. Johnson,
Matrix Analysis (Cam-bridge University Press, 1985).[14] S. M. Fallat, Amer. Math. Monthly , 697 (2001).[15] H. Flanders, J. London Math. Soc. , 10 (1962).[16] R. Westwick, Linear and Multilinear Algebra , 171(1987).[17] P. Hayden, D. Leung, and A. Winter, quant-ph/0407049 ;Comm. Math. Phys. , 95 (2006).[18] T. S. Cubitt, M.-B. Ruskai, and G. Smith, in preparation(2007).[19] H. Chen, J. Math. Phys.47