Exploiting Matrix Symmetries and Physical Symmetries in Matrix Product States and Tensor Trains

Transcription

1 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States and Tensor Trains Thoas K Huckle a and Konrad Waldherr a and Thoas Schulte-Herbrüggen b a Technische Universität München, Boltzannstr 3, Garching, Gerany; b Technische Universität München, Lichtenbergstr 4, Garching, Gerany We focus on syetries related to atrices and vectors aearing in the siulation of quantu any-body systes Sin Hailtonians have secial atrix-syetry roerties such as ersyetry Furtherore, the systes ay exhibit hysical syetries translating into syetry roerties of the eigenvectors of interest Both tyes of syetry can be exloited in sarse reresentation forats such as Matrix Product States s for the desired eigenvectors This aer suarizes syetries of Hailtonians for tyical hysical systes such as the Ising odel and lists resulting roerties of the related eigenvectors Based on an overview of Matrix Product States Tensor Trains or Tensor Chains and their canonical noral fors we show how syetry roerties of the vector translate into relations between the s atrices and, in turn, which syetry roerties result fro relations within the MPS atrices In this context we analyze different kinds of syetries and derive aroriate noral fors for MPS reresenting these syetries Exloiting such syetries by using these noral fors will lead to a reduction in the nuber of degrees of freedo in the MPS atrices This aer rovides a unifor latfor for both well-known and new results which are resented fro the ulti-linear algebra oint of view Keywords: Syetric ersyetric atrices; Quantu any-body systes; Sin Hailtonian; Matrix Product States; Tensor Trains; Tensor Chains AMS Subect Classification: 15A69 15B57; 81-08; 15A18 1 Introduction In the siulation of quantu any-body systes such as 1D sin chains one is faced with robles growing exonentially in the syste size Fro a linear algebra oint of view, the hysical syste can be described by a Heritian atrix H, the so-called Hailtonian The real eigenvalues of H corresond to the ossible energy levels of the syste, the related eigenvectors describe the corresonding states The ground state is of iortant relevance because it is related to the state of inial energy which naturally arises To overcoe the exonential growth of the state sace with syste size soeties referred to as curse of diensionality one uses sarse reresentation forats that scale only olynoially in the nuber of articles In quantu hysics concets like Matrix Product States have been develoed, see, eg, [16] These concets strongly relate to the Tensor-Train concet, which was introduced by Oseledets in [15] as an alternative to the canonical decoosition [5, 11] and the Tucker forat [3] In the s foralis vector coonents are reresented by the trace of a roduct of atrices, which are often of oderate size As will turn out, syetries and further relations in these atrices result in secial roerties of the vectors to Corresonding author Eail: waldherr@intude

2 Huckle, Waldherr, Schulte-Herbrüggen be reresented and, vice versa, that secial syetry roerties of vectors can be exressed by certain relations of the s atrices We will analyze different syetries such as the bit-shift syetry, the reverse syetry, and the bit-fli syetry, and we resent noral fors of s for these syetries, which will lead to a reduction of the degrees of freedo in the decoosition schees Organization of the Paer The aer is organized as follows: First, we list ertinent atrix syetries translating into syetry roerties of their eigenvectors Then we consider hysical odel systes and suarize the related syetries translating into syetries of the eigenvectors After a fixing notation of Matrix Product States, we resent noral fors of s and analyze how relations between the s atrices and syetries of the reresented vectors are interconnected Finally, we show the aount of data reduction by exloiting syetry-adated noral fors Matrix Syetries In this section we recall soe classes of structured atrices and list soe iortant roerties A atrix A is called syetric, if A A T ie a i, a,i and skewsyetric, if A T A A real-valued syetric atrix has real eigenvalues and a set of orthogonal eigenvectors If A is syetric about the northeastto-southwest diagonal, ie a i, a n +1,n i+1, it is called ersyetric Let J R n n, J i, : δ i,n+1, be the exchange atrix Then ersyetry can also be exressed by JAJ A T A atrix is syetric ersyetric, if it is syetric about both diagonals, ie or coonent-wise JAJ A T A a i, a,i a n+1 i,n+1 Note that a atrix with the roerty JAJ A is called centrosyetric Therefore, syetric ersyetric or syetric centrosyetric are the sae The set of all syetric ersyetric n n atrices is closed under addition and under scalar ultilication A atrix A is called syetric skew-ersyetric if JAJ A T A, or coonent-wise a i, a,i a n+1 i,n+1 The set of these atrices is again closed under addition and scalar ultilication Any syetric n n atrix A can be exressed as a su of a ersyetric and a skew-ersyetric atrix: A 1 A + JAJ + 1 A JAJ By J one ay likewise characterize vector syetries: a vector v R n is syetric if Jv v and skew-syetric if Jv v

3 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 3 As all the atrices of subsequent interest are built by linear cobinations of Kronecker roducts of saller atrices the following lea will be useful Lea 1: The Kronecker roduct of two syetric ersyetric atrices B and C is again syetric ersyetric Proof : Let J B and J C denote the exchange atrices which corresond to the size of B and C resectively Then the exchange atrix J of B C is given by J J B J C Therefore J B J C B CJ B J C J B BJ B J C CJ C B T C T B C Reark 1 : Each ower A k of a syetric ersyetric A is again syetric ersyetric Reark : For a syetric skew-ersyetric A, A is syetric ersyetric, and also the Kronecker roduct of two syetric skew-ersyetric atrices is syetric ersyetric Reark 3 : If atrix A is skew-syetric, then A is syetric Furtherore, the Kronecker roduct of two skew-syetric atrices is syetric Due to [3] we can state various roerties for syetric ersyetric atrices and the related eigenvectors As all the atrices of our interest have as size a ower of, we focus on the stateents related to even atrix sizes here The following lea oints out the ain results adated fro [3] Both the roof and siilar results for the odd case can be found in the original aer Lea [3]: Let A R n n be any syetric ersyetric atrix of even size n, the following roerties hold a The atrix A can be written as B C T A C JBJ with block atrices B and C of size, where B is syetric and C is ersyetric, ie C T JCJ b The atrix A can be orthogonally transfored to a block diagonal atrix with blocks of half size : 1 I J B C T I I I J C JBJ J J 1 B + JC + C T J + B B + JC C T J B B JC + C T J B B JC C T J + B B + JC B JC c The atrix A has skew-syetric orthonoral eigenvectors of the for 1 ui, Ju i

4 4 Huckle, Waldherr, Schulte-Herbrüggen where u i are the orthonoral eigenvectors of B JC A also has syetric orthonoral eigenvectors 1 vi Jv i where the v i are the orthonoral eigenvectors of B + JC The discussed transforation 1 to block diagonal atrices of saller size is quite chea and can be exloited to save coutational costs, see, eg, [] Reark 4 : In general, the transforation of syetric ersyetric atrices to block diagonal for 1 cannot be continued recursively because the atrix B ± JC is syetric but usually no longer ersyetric Altogether, any syetric ersyetric atrix has eigenvectors which are either syetric or skew-syetric, ie Jv v or Jv v However, one has to be careful with these stateents in the case of degenerate eigenvalues If the two blocks share an eigenvalue, A has as eigenvectors linear cobinations of syetric and skew-syetric vectors, so the eigenvectors theselves are in general neither syetric nor skew-syetric A atrix is called Toelitz atrix, if it is of the for, T r 0 r 1 r n 1 r 1 r 0 r1 r n+1 r 1 r 0 Toelitz atrices obviously belong to the larger class of ersyetric atrices Therefore, real syetric Toelitz atrices are syetric ersyetric An iortant class of Toelitz atrices are the circulant atrices taking the for r 0 r 1 r n 1 r C n 1 r 0 r1 r 1 r n 1 r 0 Any circulant atrix C with entries r : r 0, r 1,, r n 1 T can be diagonalized by the Fourier atrix F n f,k ; f,k 1 n e πik/n [10] via C F 1 n diagf n rf n F n diagf n rf n Analogously, a skew-circulant atrix looks like r 0 r 1 r n 1 r C s n 1 r 0 r1 r 1 r n 1 r 0

5 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 5 In general, an ω-circulant atrix with ω e iφ is defined by r 0 r 1 r n 1 ωr C ω n 1 r 0 r1 ωr 1 ωr n 1 r 0 These atrices can be transfored into a circulant atrix by the unitary diagonal atrix Ω n;ω diagω /n 0,,n 1 : r 0 r 1 r n 1 Ω H r n;ωc ω Ω n;ω Ω n;ω C ω Ω n;ω n 1 r 0 r1, 3 r 1 r n 1 r 0 where r k : ω k/n r k Multilevel circulant atrices are defined by the roerty that the eigenvector atrix is given by a tensor roduct of Fourier atrices F n1 F nk Block-Toelitz-Toelitz-Block atrices, also called -level Toelitz atrices, have a Toelitz block structure where each block itself is Toelitz More general, a ultilevel Toelitz atrix has a hierarchy of blocks with Toelitz structure 1 Reresentations of Sin Hailtonians For sin- 1 articles such as electrons or rotons, the sin angular oentu oerator describing their internal degree of freedo ie sin-u and sin-down is usually exressed in ters of the Pauli atrices P x 0 1, P 1 0 y 0 i i i and P z For further details, a reader wishing to aroach quantu hysics fro linear and ultilinear algebra ay refer to [9] Being traceless and Heritian, {P x, P y, P z } fors a basis of the Lie algebra su, while by aending the identity atrix I one obtains a basis of the Lie algebra u This fact can be generalized in the following way: for any integer a basis of the Lie algebra u is given by { Q1 Q Q ; Q i {P x, P y, P z, I} } To get a basis for su we have to consider only traceless atrices and therefore we have to exclude the identity, which results in the basis { Q1 Q Q ; Q i {P x, P y, P z, I} } \ {I I} Now, sin Hailtonians are built by suing M ters, each of the reresenting a hysical interaction These ters are theselves tensor roducts of Pauli atrices or identities H M k α k Q 1 Q k Q k }{{} k1 :H k M H k, 4 k1

6 6 Huckle, Waldherr, Schulte-Herbrüggen where the coefficients α k are real and the atrices Q k can be P x, P y, P z or I In each suand H k ost of the Q k are I: local ters have ust one nontrivial tensor factor, while air interactions have two of the Higher -body interactions with > usually do not occur as hysical riitives, but could be reresented likewise by Pauli atrices in the tensor roduct reresenting the -order interaction ter For defining sin Hailtonians we will need tensor owers of the identity I: I k : I I }{{} k For instance, in the Ising ZZ odel see eg [18] for the 1D chain with sins and oen boundary conditions, the sin Hailtonian takes the for 1 H I k 1 P z k P z k+1 I k 1 k1 + λ I k 1 P x k I k, k1 5 where the index k denotes the osition in the sin chain and the real nuber λ describes the ratio of the strengths of the agnetic field and the air interactions Using µ, ν {x, y, z}, one ay define H ν : I k 1 P ν k I k, 6 k1 1 H µµ : I k 1 P µ k P µ k+1 I k 1 7 k1 The ters 7 corresond to the so-called oen boundary case In the eriodic boundary case there are also connections between sites 1 and, which reads H µµ H µµ + P µ 1 I P µ 8 Note that in the literature often the identity atrices and the tensor roducts are ignored giving the equivalent notation H µµ : P µ k P µ k+1 od 9 k1 In analogy to the Ising odel 5, it is custoary to define various tyes of Heisenberg odels [1, 14] in ters of either vanishing or degenerate real constants x, y and z Table 1 gives a list of ossible 1D odels where, in addition, one ay have either oen or eriodic boundary conditions The oerators with the additional ter λh x are soeties called generalized Heisenberg odels The XX, res XXX odels are called isotroic

7 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 7 Table 1 List of different 1D odels Interaction Ising-ZZ Heisenberg-XX Heisenberg-XY Heisenberg-XZ Heisenberg-XXX Heisenberg-XXZ Heisenberg-XYZ Hailtonian zh zz + λh x xh xx + xh yy + λh x xh xx + yh yy + λh x xh xx + zh zz + λh x xh xx + xh yy + xh zz + λh x xh xx + xh yy + zh zz + λh x xh xx + yh yy + zh zz + λh x For sin-1 odels, the oerators take the for S x , S y 1 0 i 0 i 0 i, S z i The AKLT odel is defined as [1, 14] H k S k S k S ks k+1 11 where S k S k+1 : S x k S x k+1 + S y k S y k+1 + S z k S z k+1 More generally, the bilinear biquadratic odel has Hailtonian H k cosθs k S k+1 + sinθs k S k+1 1 These 1D odels can also be extended to and higher diensions Then the neighbor relations cannot be reresented linearly but they aear in each direction For exale, Eqn 7 would read H µµ P µ P µ k, <,k> where <, k > denotes an interaction between articles and k Being a su 4 of Kronecker roducts of structured atrices, any Hailtonians have secial roerties, eg, they can be ultilevel-circulant [4, 6] or skew-circulant, diagonal or ersyetric [3], which can be exloited to derive roerties of the resective eigenvalues and eigenvectors Syetry Proerties of the Hailtonians To begin, we list soe roerties of the Pauli atrices Proerties of the Pauli Matrices P x is syetric ersyetric and circulant Following Eqn, P x can be diagonalized via the Fourier atrix F : F P x F P z 13

8 8 Huckle, Waldherr, Schulte-Herbrüggen The atrix P y /i is skew-syetric ersyetric P y is skew-circulant and by using Eqn 3, it can be transfored into a circulant and even real atrix: Ω ; 1 P y Ω ; i 0 i i i 0 1 P 1 0 x, 14 which is due to 13 orthogonally siilar to P z P z is diagonal and syetric skew-ersyetric The identity atrix I is of course circulant, syetric ersyetric and diagonal Now we list syetry roerties of the atrices defined in Eqn 6 and 8 As the atrices are built by Kronecker roducts of -atrices it will be useful to exloit the fact that the exchange atrix can also be exressed as Kronecker roduct of -atrices: J J J P x P x Due to Lea 1 alied on this factorization the atrix H x as a su of Kronecker roducts of syetric ersyetric atrices is again syetric ersyetric Moreover, H x is ultilevel-circulant as it can be diagonalized by the Kronecker roduct of the Fourier atrix F : F F I k 1 P x k I k F F k1 F IF }{{ } k 1 F P x F k F }{{} IF }{{ } I 13 I P z k I k 1 k 6 P z k I H z k1 k1 k 15 Therefore the eigenvalues of H x are all ossible cobinations ±1 ± 1 ± ± 1 Trivially, the atrix H y /i is skew-syetric ersyetric and thus H y is Heritian It can be transfored to H x via the Kronecker roduct of the diagonal transfors considered in Eqn 14 + H an y, where each su- Even for the generalized anisotroic case H an H an x and k in both sus ay have an individual coefficient a k and b k, resectively, one can find an aroriate transfor To this end, consider H an a k I k 1 P x k I k + k1 b k I k 1 P y k I k k1 I k 1 a k P x k + b k P y k I k k1 k1 I k 1 0 a k ib k a k + ib k 0 I k

9 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 9 k1 Each tensor factor I k 1 0 rk e iφk I k r k e iφk 0 0 rk e C k : iφk 0 r k e iφk r k e iφk 0 e iφk r k e iφk 0 is ω-circulant ω k e iφk Following 3, C k can be transfored to a real atrix using the diagonal transfor D k Ω ;ωk : Therefore, the overall Hailtonian H an x H x ter: 0 rk D k C k D k r r k 0 k P x +H an y can be transfored to an anisotroic D 1 D I k 1 C k I k D 1 D k1 I k 1 D k C k D k I k k1 k1 r k I k 1 P x k I k H an x Analogously to H x see Eqn 15, the resulting atrix H an x can be diagonalized by the Kronecker roduct F F Therefore, the eigenvalues of H an x + H an y are given by all cobinations ±r 1 ± r ± ± r Let us return to analyzing the roerties of Hailtonians The atrix H z is obviously diagonal and skew-ersyetric The atrix H xx is again syetric ersyetric see Lea 1 Siilar to H x, H xx is again ultilevel-circulant as it can be diagonalized by the Kronecker roduct of the Fourier atrix F A coutation siilar to Eqn 15 results in F F H xx F F H zz The atrix H yy is real syetric ersyetric as becoes obvious fro P y P y 0 i i 0 0 i i being real and syetric ersyetric, which by Lea 1 translates into a real syetric ersyetric atrix H yy

10 10 Huckle, Waldherr, Schulte-Herbrüggen The atrix H zz is diagonal as it is constructed by a su of Kronecker roducts of diagonal atrices Moreover H zz is syetric ersyetric via P z P z according to Reark Obviously, the sin-1 oerators 10 have siilar syetry roerties as their counterarts: the atrix S x is real syetric ersyetric and has Toelitz forat, S y /i is a real and skew-syetric ersyetric Toelitz atrix, and S z is syetric skew-ersyetric and diagonal The Kronecker roduct S y S y reads S y S y , a real syetric ersyetric atrix coare Reark 3 Following Reark, the Kronecker roduct S z S z is syetric ersyetric Therefore, according to Reark 1 and Lea 1, Both the AKLT odel 11 and the generalized bilinear biquadratic odel 1 result in real syetric ersytric atrices Altogether all reviously introduced hysical odels such as the 1D odels listed in Table 1 define real and syetric ersyetric atrices Due to Lea, the related eigenvectors such as the ground state which corresonds to the lowestlying eigenvalue are either syetric or skew-syetric 3 Alication to Matrix Product States For efficiently siulating quantu any-body systes, one has to find a sarse aroxiate reresentation, because otherwise the state sace would grow exonentially with the nuber of articles Here efficiently eans using resources and hence reresentations growing only olynoially in the syste size In the quantu inforation QI society, Matrix Product States are in use to treat 1D robles 31 Matrix Product States: Foralis and Noral Fors This aragrah suarizes soe well-known basics about s We rovide both the s foralis and noral fors for s, which are well-known in the QI society, fro a ulti-linear algebra oint of view Afterwards we resent own findings to construct noral fors and discuss the benefit of such fors 311 Foralis For 1D sin systes, consider Matrix Product States, where every hysical site is associated with a air of atrices A 0, A 1 C D D+1, reresenting one of the two ossibilities sin-u or sin-down Let i 1, i,, i denote the binary reresentation of the integer index i Then

11 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 11 the ith vector coonent takes the for x i x i1,,i tr A i1 1 A i A i Hence, the overall vector x can be exressed as x x i e i i1 i 1,,i tr i 1,,i 1,, x i1,,i e i1 e i i 1,i,,i A i1 1 A i A i e i1 e i 16 A i1 1; 1, A i ;, 3 A i ;, 1 e i1 e i 1,, i 1 A i1 1; 1, e i1 i A i ;, 1 e i 1,,, a 1;1, a ;, 3 a ;, 1 with vectors a ;, +1 od of length These vectors are airs of entries at osition, +1 od fro the atrix air A i, i 0, 1 We distinguish between oen boundary conditions, where D 1 D +1 1 and eriodic boundary conditions, where the first and last articles are also connected: D 1 D +1 > 1 The first case corresonds to the Tensor Train forat [15], the latter to the Tensor Chain forat[13] Considerations on s fro a atheatical oint of view can be found in [1] 31 Noral Fors The s ansatz does not lead to unique reresentations, because we can always introduce factors of the for M M 1 between A i and A i+1 +1 In order to reduce this abiguity in the oen boundary case one can use the SVD to relace the atrix air A 0, A 1 by arts of unitary atrices see, eg [] To this end, one ay start fro the left right, carry out an SVD, relace the current air of s atrices by arts of unitary atrices, shift the reaining SVD art to the right left neighbor, and roceed recursively with the neighboring site Starting fro the left one obtains a left-noralized s reresentation fulfilling the gauge condition 0 HA 0 A + A 1 HA 1 I 17 Analogously, if we start the rocedure fro the right, we end u with a rightnoralized s reresentation fulfilling A 0 A 0 H + A 1 A 1 H I 18 In the eriodic boundary case these gauge conditions can only be achieved for all u to one site Still soe abiguity reains because we can insert W W H with any unitary W in the s reresentation 16 between the two ters at osition and + 1

12 1 Huckle, Waldherr, Schulte-Herbrüggen without any effect to the gauge conditions 17 or 18 To overcoe this abiguity a stronger noralization can be derived see, eg [7] It is based on different atricizations of the vector to be reresented and can be written in the for x i1i Γ i1 1 Λ1 Γ i Λ Γ i3 3 Λ 1 Γ i i A 1 1 Ai Ai3 3 A i 19 with diagonal atrices Λ containing the singular values of secial atricizations of the vector x The following lea states the existence of such an s reresentation Lea 31 [4]: Any vector x C of nor 1 can be reresented by an s reresentation fulfilling the left conditions or the right conditions A 0 Λ 0 HA 0 A + A 1 HA 1 A 0 A 0 H + A 1 A 0 Λ H + A 1 0 HΛ A 1 A 0 + A 1 A 1 1 A HΛ 1 A 1 I 0a H Λ 1 0b H I 1a Λ, 1b where the D +1 D +1 diagonal atrices Λ contain the non-zero singular values of the atricization of x relative to index artitioning i 1,, i, i +1,, i, diagonal entries ordered in descending order The roof of this lea is constructive and rovides s factors A i again as arts of unitary atrices, but satisfying two noralization conditions These conditions are well-known in the QI society, see, eg, [8, 4] The following roof is adated fro [7], but we reforulate it in atheatical atrix notation Proof : Let us rove reresentation 0 for a given vector x by orthogonalization fro the left We start with considering the SVD of the first atricization relative to i 1, X i1,i,,i U 1 Λ 1 W A 0 1 Λ 1W Γ 0 A 1 1 Λ 1 Λ 1W 1W Γ 1 1 Λ 1W with the notation A 1 U 1 Γ 1 and Λ 1 containing all ositive singular values Therefore, the coluns of U 1 are airwise orthonoral satisfying I A H 1 A 1 A 0 HA A 1 HA Now, the second atricization gives the SVD U 0 X i1,i,i 3,,i U Λ W 3 U 1 Λ W 3 3 Note that because both atricizations and 3 reresent the sae vector X, each colun of U 0 can be reresented as Γ 1 Λ 1 Γ 0 for soe Γ 0 This follows

13 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 13 by U 0 Λ W 3 Γ 1 Λ 1 W Picking a full-rank subatrix C of Λ W 3 and alying the inverse fro the right yields U 0 Γ 1 Λ 1 Ŵ The sae holds for U 1 with soe Γ 1 With these atrices Γ0 and Γ 1 we can write U 0 U Γ U 1 1 Λ 1 Γ 0 Γ Γ 1 Λ 1 Γ 1 1 A 0 Γ 1 A 1 with A 0 Λ A 1 : 1 Γ 0 Γ Λ 1 Γ 1 H 1 Γ 1Λ 1 Γ 0 Γ Γ H 1 Γ 1Λ 1 Γ 1 H 1 U0 Γ H 1 U1 4 In view of the SVD reresentation 3 of X i1,i,i 3,,i one finds I U H U A 0 HΓ H 1 Γ 1 A 0 + A 1 HΓ H 1 Γ 1 A 1 A 0 HA 0 + A 1 HA 1, which corresonds to the first noralization condition 0a Now we can rewrite the second atricization 3 as U 0 X i1,i,i 3,,i U 1 Λ W 3 Γ 1 Λ 1 Γ 0 Λ W 3 Γ 1 Λ 1 Γ 1 Λ W 3 Coaring this for of the vector X with the first atricization gives and therefore W Γ 0 Λ W 3 Γ 1 Λ W 3 I W W H Γ 0 Λ W 3 Γ 1 Λ W3 H W Λ 0 H Γ 3 W3 HΛ 1 H Γ Γ 0 Λ 0 H 1 Γ + Γ Λ 1 H Γ Multilying fro both sides with Λ 1 is ust the second condition 0b: Λ 1 Λ 1 Γ 0 Λ A 0 Λ Γ 0 A 0 HΛ1 + Λ 1 Γ 1 Λ 1 Γ H 1 + A Λ 1 H A HΛ1

14 14 Huckle, Waldherr, Schulte-Herbrüggen In the sae way we can use the two atricizations X i1,i,i 3,,i and X i1,i,i 3,i 4,,i to derive A 3, based on Λ, Γ, U 3, W 3, W 4 and Λ 3, satisfying the noralization conditions 0 Then A 4,, A follow siilarly Starting fro the right and using a siilar rocedure gives the reresentation satisfying the noralization conditions 1 Reark 1 : 1 The resulting s reresentation is unique u to unitary diagonal atrices as long as the singular values in each diagonal atrix Λ are in descending order and have no degeneracy are all different, coare [16] One ay consider the constructive roof as a ossible introduction of s [, 4] Then the conditions 0 or 1 aear naturally 3 The roof shows that, in general, an exact reresentation coes at the cost of exonentially growing atrix diensions D For keeing the atrix diensions liited one would have to introduce SVD-based truncations 4 The Vidal noralization [4] uses Γ i and Λ in 19 instead of A i 5 Starting fro a given s A-reresentation 16 it is ossible [] to build an equivalent ΛΓ-reresentation 19 without considering the atricizations exlicitly The construction starts fro a right-noralized s reresentation 18 and then iteratively coutes SVDs of odified decoositions related to two neighboring sites The conversion fro the ΛΓ-for to the A-for is siler: Fro 4 it becoes obvious to set Λ 1 Γ i in the right noralized case 1 we would define A i A i Λ 0 : 1 in the left-noralized case 0 Analogously, Γ i Λ, where Λ : 1 6 The ΛΓ-reresentation 19 corresonds to the Schidt decoosition, which is well-known in QI The Schidt coefficients are ust the diagonal entries of Λ [] 7 The diagonal atrices Λ contain the singular values of secial atricizations of the vector to be reresented Hence, local atrices A reflect global inforation on the tensor via the noralization conditions and the diagonal atrices Λ That is one of the reasons why s has roer aroxiation roerties [8] 313 Further Noral Fors Finally we roose own findings of concets to introduce ossible noral fors for s As an alternative to construct the gauge conditions 17 or 18 we roose coare [1] to consider two neighboring airs coare two-site drg [] A 0 A 1 U 0 U 1 A 0 +1 A 1 A 0 A 0 +1 A 0 A Λ U 0 +1 U 1 +1 A 1 A 0 +1 A 1 A 1 +1 U 0 U 1 U 0 U 1 SVD Λ U 0 +1 Λ U 1 +1 Λ Λ U 0 +1 U In this way all atrix airs A 0, A 1 u to one in the eriodic boundary case can be assued as art of a unitary atrix giving the noralization conditions

15 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States in the left-noralized case 5 or 18 in the right-noralized case 6 To circuvent the fact that the gauge conditions 17 or 18 still introduce soe abiguity we roose the following way to derive a stronger noralization Suose that the s atrices are already in the left-noralized for A 0 HA 0 + A 1 HA 1 I for 1,, The roosed noral for is now based on the SVD of the uer atrices A 0 U Σ V with unitary U, V V 0 : 1 and diagonal non-negative Σ, diagonal entries ordered relative to absolute value Then, every air A 0, A 1 is relaced by Ã0, à 1 V 1 U Σ, V 1 A 1 leading to the stronger noralization conditions Ã0 Hà 0 + à 1 Hà 1 V H 7 Σ H Σ + H I 8 with diagonal atrices Σ and Fro 8 we can read that this noral for rovides s atrices with orthogonal coluns For the uer atrices à 0 this fact is caused by construction, but it then autoatically follows also for the à 1 atrices Esecially for the left-ost site 1, the noralization condition 8 leads to à 0 1 1, 0 and à 1 1 0, 1 We ay of course also start the roosed noralization rocedure with a right-noralized for, resulting in a reresentation where the s atrices have orthogonal rows 314 Coarison of the Noral Fors All of the resented noral fors introduce soe kind of uniqueness to the s foralis, which initially is not unique Therefore, these noral fors hel to revent redundancy in the reresentations As a consequence we ay exect less eory deands as well as better roerties of nuerical algoriths such as faster convergence, better aroxiation, and iroved stability The noral for 0 is advantageous as it connects local and global inforation However, the construction involves the inverse of the diagonal SVD atrices which ay cause nuerical robles Our noral for 8 can be built without division by singular values, but the inforation is ore local 3 Syetries in s The results fro Section show that the atrices which describe the hysical odel systes have secial syetry roerties which result in syetry roerties of the related eigenvectors: the eigenvector of a syetric ersyetric Hailtonian has to be syetric or skew-syetric, ie Jv ±v One ight also think about other syetries which could be of the for a a v, v, or ore general Pv ±v a a with a general erutation P Furtherore, we can have vectors satisfying k different indeendent syetry roerties, eg P v ±v for erutations P,

16 16 Huckle, Waldherr, Schulte-Herbrüggen 1,, k At this oint the question arises how these syetry roerties can be exressed in ters of s, and, vice versa, how secial roerties such as certain relations between the s atrices eerge in the reresented vector Syetries in s already aear in different QI ublications: theoretical considerations on syetries in s can be found in [17, 1], syetries in ti s reresentations are exloited in [19], and the alication of involutions has been analyzed in [0] The ain goal of this aragrah is to resent an overview of different tyes of syetries in a unifying way and to give results concerning the uniqueness of such syetry-adated reresentation aroaches by roosing ossible noral fors Our results are intended for a theoretical urose siilar to [17, 1] but are also interesting for nuerical alications siilar to [19, 0] After soe technical considerations we discuss which roerties of the atrices that define an s vector x are related to certain syetry roerties of x Deriving noral fors for different syetries of s vectors will also be of interest A i 31 Technical Rearks In view of the trace taken in the s foralis 16, recall the following trivial yet useful roerties tr AB tr BA, 9 tr AB tr AB T tr B T A T for tr AB R, 30 tr AB tr AB H tr B H A H for tr AB C 31 in order to arrive at relations of the for tr A i1 1 A i A i 9 tr A ir+1 r+1 Ai 31 tr A irh r A ir 1H r 1 1 A ir r A i1 A i1h 1 A ih A ir+1h r+1 For the roof of the ain theores we will need the following three leata Lea 3: Let A, B K n, where K {R, C} If the equality tr AX tr BX holds for all atrices X K n, then A B Proof : The relation tr AX tr BX is equivalent to tr A BX 0 for all atrices X For the secial choice X A B H we obtain tr A BA B H A B F 0, which shows A B Lea 33: Assue that for U K n n and V K it holds X VXU 3

17 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 17 for all atrices X K n Then U ci n, V I /c with soe c 0 Proof : Obviously, V and U have to be non-zero and, oreover, they are regular Otherwise, if eg Va 0 for a 0, we can define X ab H with soe b 0 leading to a contradiction Choosing X ab H as rank-one atrix for any vectors a and b, it follows V 1 ab H ab H U Therefore, V 1 a and a have to be collinear V 1 a λa with soe λ K, and b H U and b H also have to be collinear b H U µb H Hence, U and V 1 and therefore also V have all vectors of aroriate size as eigenvectors, and therefore they are nonzero ultiles of the identity atrix, U c 1 I n and V c I Condition 3 finally shows c c 1 1/c Siilarly, we can derive the following result: Lea 34: Assue that for U K n n and V K it holds XU VX for all atrices X K n Then U ci n, V ci with a scalar c K Proof : First we rove that, if at least one of the two atrices U or V is singular, both of the have to be zero Obviously, if one of the two atrices is zero, the other one has to be zero as well If we now suose V to be singular and U to be nonzero, we can find vectors a 0 and b, such that Va 0 and b H U 0 H The choice X ab H 0 leads to a contradiction The sae arguent counts if we change the roles of U and V Otherwise, if both atrices are regular, the stateent of the lea is a direct consequence fro Lea 33 3 Bit-Shift Syetry and Translational Invariance To begin, consider the case where all atrix airs are equal, ie A 0 A 0 A 1 A 1 33 for all 1,, Then the s is site-indeendent and describes a translational invariant TI state on a sin syste with eriodic boundary conditions [16] The following theore states that the result of such a relation is a bit-shift syetry, ie x i1,i,,i x i,i 3,,i,i 1 x i,i 1,i,,i 1 Theore 35 [16]: If the s atrices are site-indeendent and thus fulfill Eqn 33 the reresented vector has the bit-shift syetry and in turn every vector with the bit-shift syetry can be reresented by a site-indeendent s

18 18 Huckle, Waldherr, Schulte-Herbrüggen Proof : To see that a ti s 33 leads to a bit-shift syetry, consider x i1,i,,i tr A i1 1 Ai A i 9 tr A i A i3 A i A i1 x i,i 3,,i,i 1 9 tr A i3 A i4 A i A i1 A i x i3,,i,i 1,i 9 tr A i A i1 A i 1 x i,i 1,i,,i 1 Let us now suose that the vector x has the bit-shift syetry and let x i1,i,,i tr B i1 1 Bi B i be any s reresentation for x Then the construction A i 1 0 B i B i 0 0 B i 1 B i 0 34 leads to a site-indeendent reresentation of x Reark : The construction 34 introduces an augentation of the atrix size by the factor The bit-shift syetry can also be generalized to block-shift syetry Assue that a block of r s atrix airs is reeated, ie A i1 1 Ai A ir r A ir+1 1 A ir+ A ir r A i r+1 1 A i r+ A i r to obtain syetries of the for x i1,,i r;i r+1,,i r;;i r+1,,i x ir+1,,i r;;i r+1,,i ;i 1,,i r Noral For for the Bit-Shift Syetry In the above eriodic TI ansatz 33 we can relace each A by MAM 1 with a nonsingular M resulting in the sae vector x Using the Schur noral for A 0 Q H R 0 Q or the Jordan canonical for A 0 S 1 J 0 A S we roose to noralize the s for by relacing the atrix air A 0, A 1 by Ã0, Ã 1 R 0, QA 1 Q H or Ã0, Ã 1 J 0 A, SA1 S 1 resulting in a ore coact reresentation of x with less free araeters For Heritian A 0 and A 1 the eigenvalue decoosition of A 0 Q H D 0 Q with

19 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 19 diagonal atrix D 0 can be used in the sae way leading to the noral for à 0, à 1 D 0, QA 1 Q H with à 0 as real diagonal atrix and à 1 as Heritian atrix 33 Reverse Syetry In this subsection we consider the reverse syetry x i1,,i x i,,i 1 35 The following theore shows a direct connection between the reverse syetry and an s reresentation with the secial syetry relations i A H S 1 Ai +1 S +1 for all 1,, 36 with regular atrices S of aroriate size, which additionally fulfill the consistency conditions S 0 S and S H S for 1,, 37 Theore 36 : If the s atrices fulfill the syetry relations 36, the vector to be reresented has the reverse syetry roerty 35 Vice versa, for any vector x fulfilling the reverse syetry, we ay state an s reresentation for x fulfilling the relations 36 Proof : For the vector x to be reresented, the relations 36 lead to x i1,,i tr A i1 1 Ai A i tr A i1 1 Ai A i tr A ih A i 1H 1 H A ih A i1h 1 S 1 tr A i 1 S 1 S 1 1 Ai 1 S S 1 1 Ai1 S tr A i 1 Ai 1 A i1 x i,,i 1, a reverse syetric vector So far we have seen that the relations 36 lead to the reresentation of a vector having the reverse syetry Contrariwise, it is ossible to indicate an s reresentation fulfilling the relations 36 for any reverse syetric vector To see this we consider any s for x: x i1,i,,i tr B i1 1 Bi B i with atrices B i of size D D +1 Such an s reresentation always exists, coare Lea 31

20 0 Huckle, Waldherr, Schulte-Herbrüggen Let us start with the case where this s reresentation is in PBC for The reverse syetry x i1,i,,i x i,i 1,,i 1 leads to x i1,i,,i xi1,i,,i + x i,i 1,,i 1 tr B i1 1 Bi B i i + tr B 1 Bi 1 B i1 tr B i1 1 Bi B i 1 tr B i1 1 Bi B i 1 tr B i B i1h + tr B i 1H 0 B ih 0 B i1h B i 0 0 B ih 1 B ih 1 BiH 1 1 BiH 1 B i 0 0 B ih 1 38 We ay now define A i : 1 B i 0 0 B ih and obtain A ih 1 B ih 0 0 B i I B i I I 0 0 B ih I 0 A i 0 I +1 I 0 0 I I 0 with I being identities of aroriate size Hence, the choice S 0 ID+1 I D+1 0 for all 1,, 40 gives A ih S 1 Ai +1 S +1, the desired atrix relations 36 In the OBC case we can roceed in a siilar way, but at both ends 1 and soething secial haens: as we want to reserve the OBC character of the s reresentation, the atrices A i1 1 and A i have to be vectors as well Therefore we define A i1 1 1 B i1 1 B i1h and A i 1 B i B ih 1 The choice S 1 leads to the desired relation A i1 H 1 S 1 1 Ai1 S Reark 3 : 1 The roof shows that the reverse syetry can occur in the eriodic boundary case, but also for the oen boundary case where A i1 1 and A i secialize to vectors Then, S 0 S S H 0 are sily even real 37 scalars

21 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 1 In the roof, the atrices S can be chosen to be unitary, coare 40 Thus, they can be diagonalized by a unitary transfor V giving S V V H with diagonal and unitary Because of S 1 SH 37 S the relations 36 read i A H S A i +1 S +1 V V H i A +1 V+1 +1 V+1 H The last equation can be rewritten to V H i A HV+1 V H Ai +1 V Defining à i : V H +1 A i V, the relations 36 take the for Ãi H à i with unitary diagonal atrices fulfilling H If the atrices S are unitary and also Heritian, the diagonal atrices have values ±1 on the ain diagonal 3 In the PBC case with s atrices B i of equal size D D, the S atrices 40 can be chosen to be site-indeendent, Heritian, and unitary In this case the relations 41 are fulfilled by diagi D, I D Noral For for the Reverse Syetry In the following theore we roose a noral for for s reresentations of reverse syetric vectors Theore 37 : Let x C be a vector with the reverse syetry If is even, x can be reresented by an s of the for x i1,i,,i U i tr 1 1 U i Σ U i +1H U ih 1 Λ 4 and if + 1 is odd, the reresentation reads x i1,i,,i U i tr 1 1 U i i U Σ U i+h U ih 1 Λ 43 with unitary atrices U i and real and diagonal atrices Σ and Λ Proof : We start with an s of the for 36 to reresent the given vector x, coare Theore 36 In the case of being even, we obtain x i1,,i tr A i1 1 Ai 36 tr tr A i A i1 1 Ai A i A i 1 1 Ai A i A i+1 +1 Ai S H A i+1h S H 1 S H 1 A ih 1 S H S H i A +1H S H A ih 1 44

22 Huckle, Waldherr, Schulte-Herbrüggen Following 37 we obtain S H S and thus we ay factorize S H WΛW H Using 9, Eqn 44 reads x i1,,i W tr H A i1 i 1 A A i S H A i+1h A i 1H W H A i HΛ 1 We use the SVD of W H A 1, W H A 0 1 U 0 W H A U 1 Λ 1 V 1, 1 to relace W H A 1 at both ends We then obtain tr U i1 1 Λ1 V 1 A i A i S H A i +1H Ai 1H V1 H Λ 1 U ih 1 Λ We roceed with the SVD for Λ 1 V 1 A i, ie Λ 1V 1 A i U i Λ V, to obtain tr U i1 1 Ui Λ V A i3 i 3 A S H A i+1h Proceeding in an iterative way finally gives U i tr 1 1 Ui U i Λ V S H VΛ H i U +1H }{{} :C Ai3H 3 V H i Λ U 1H U ih 1 Λ U i 1H U ih 1 Λ For, Eqn 37 yields S H S and thus C is also Heritian leading to Λ V S H V H Λ C C H XΣX H with unitary X and real diagonal Σ Altogether we obtain the s reresentation tr U i1 1 Ui U i X Σ X H U i+1h i U 1H U ih 1 Λ Relacing U i by the unitary atrix U i X gives the desired noral for 4 For the odd case + 1, we ay roceed in a siilar way and relace all factors u to the interior one related to + 1 by unitary atrices to obtain U i tr 1 1 Ui U i Λ V A i+1 +1 SH VΛ H }{{} :C i +1 For site + 1 the suosed atrix relations lead to A i+1 H +1 SH S A i+1 +1 H 36 U i +H S S -1 A i+1 +1 S U ih 1 Λ A i+1 +1 SH and thus the atrices C i+1 are both Heritian Using the SVD gives C i+1 U i+1 +1 ΣX XH ΣU i+1h +1 45

23 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 3 Hence, for the overall reresentation we obtain tr U i tr 1 1 U i i U +1 U i1 1 U i +1 ΣX U i+h X H XU i+1 i +1 Σ U + U ih 1 Λ X H H U i 1 HΛ Relacing U i by U i X H and U i for the odd case +1 by XUi +1 leads to the noral for 43 Reark 4 : In the odd case we ay also use the right-side SVD factorization C i+1 X H ΣU i+1h +1 in Eqn 45 leading to the noral for U i tr 1 1 U i i ΣU +1H +1 U i +H U ih 1 Λ This abiguity is reasonable as the interior factor in the odd case only has itself as counter art: A 1 A, A A 1,, A A +, A +1 A +1 Reverse Syetry in TI Reresentations Let us finally consider the reverse syetry in TI reresentations This additional roerty allows us to use site-indeendent atrices S S, which are Heritian, coare 37 Then the relations 36 take the for A i H S 1 A i S A i S H A i S Thus, we can reresent the vector with Heritian atrices à i : A i S In the QI society one can find considerations on TI systes using real syetric atrices, coare [19] 34 Bit-Fli Syetry Here we focus on the reresentation of syetric and skew-syetric vectors aearing, eg, as eigenvectors of syetric ersyetric atrices see Lea We will use the bit-fli oerator ī : 1 i for i {0, 1} First we show that the syetry condition Jx x corresonds to the bit-fli syetry To see this we consider Jx J J i 1,,i x i1,i,,i i 1,,i x i1,i,,i i 1,,i x ī1,ī,,ī x i1,i,,i x ī1,ī,,ī i 1,,i x i1,i,,i J e i1 J e i ei1 e ī ei1 e i ei1 e i

24 4 Huckle, Waldherr, Schulte-Herbrüggen Hence we obtain Jx x x i1,i,,i x ī1,ī,,ī for all i 1,, i 0, 1 Analogously, for a skew-syetric vector x one gets x i1,i,,i x ī1,ī,,ī In order to exress these relations in the s foralis consider A 1 U A 0 U +1 od for 1,, 46 with U being involutions, ie U vice versa to give I [0] Then Eqn 46 can also be exressed A i U A ī U +1 od 47 The following lea shows the corresondence between these relations and the bit-fli syetry Theore 38 : If the atrix airs A 0, A 1 are connected via involutions as in 46 the reresented vector has the bit-fli syetry and is hence syetric Contrariwise any syetric vector can be reresented by an s fulfilling condition 46 Proof : The atrix relations 46 translate into the syetry of the reresented vector x i1,i,,i tr A i1 1 A i A i 47 tr tr U 1 A ī1 1 U A ī1 1 Aī A ī x ī1,ī,,ī U A ī U 3 U A ī U 1 Let us now consider the construction of an s reresentation 46 for a syetric vector x fulfilling the bit-fli syetry x i1,i,,i x ī1,ī,,ī To this end we start with any s reresentation x i1,i,,i tr B i1 1 Bi B i with D D +1 atrices B i Starting fro the identity x i1,i,,i 1 x i1,i,,i + x ī1,ī,,ī we ay roceed in a siilar way as in 38 for the reverse syetry to obtain x i1,i,,i 1 tr B i1 1 0 B i 0 B ī B ī B i 0 0 B ī

25 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 5 This equation otivates the definition A i : i B 0 0 B ī In the OBC case the first and last atrices have to secialize to vectors: A i1 1 B i1 1 B ī 1 1 and A i B i B ī Using the involutions U : 0 ID I D 0 for 1,, 48 gives the desired relations 46 In the OBC case we have to define U 1 1 Reark 5 : If we want to reresent a skew-syetric vector x Jx, we ay also use the relations 47 at all sites u to one, say site 1, where we would have to add a negative sign: A i1 1 U 1 A ī1 1 U However, in the secial TI case, where all atrix airs have to be identical, this is not ossible: the relations 47 would read A 0 A 1 A UAV 49 at every site with site-indeendent involutions U and V Therefore, in the eriodic ti s ansatz 49 alied to syetric-ersyetric Hailtonians, only syetric eigenvectors can occur Noral For for the Bit-Fli Syetry As every involution, U ay only have eigenvalues { 1, 1} and thus U S 1 D ;±1 S, 50 where D ;±1 is a diagonal atrix with entries ±1: the Jordan canonical for ilies U S 1 J U S Moreover, the Jordan blocks in J U have to be involutions as well, so J U I and therefore J U D has to be diagonal with entries ±1 Consider A i 47 U A ī 50 U +1 S 1 D ;±1 S A ī S 1 +1 D +1;±1S +1, which results in S A i S 1 +1 D ;±1 S A ī S 1 +1 D +1;±1 51 }{{}}{{} à i à ī showing that the s atrices can be chosen such that the involutions in Eqn 47 can be exressed by diagonal atrices D ;±1 yielding A i D ;±1 A ī D +1;±1

26 6 Huckle, Waldherr, Schulte-Herbrüggen Often the distribution of ±1 in D ay be unknown So the exchange atrix J S 1 D J S is an involution where as diagonal entries in D J, +1 and 1 aear sizej/ If we double the allowed size D for the s atrices we can exect that J has at least as any +1 and 1 eigenvalues as all the aearing diagonal atrices D ;±1 Therefore, we ay heuristically relace each D ;±1 by J with larger atrix size D leading to an ansatz requiring no a-riori inforation Bit-Fli Syetry in TI Reresentations If the s atrices fulfill the bit-fli syetry relations 47 and are additionally site-indeendent, one has A ī UA i U with site-indeendent involutions U 50 S 1 D ±1 S The transforation 51 then reads SA } i {{ S 1 } D ±1 Ã i SA ī S 1 D ±1 }{{} Ã ī Thus, the vector can be reresented by a ti s fulfilling A ī D ±1 A i D ±1 with the sae involution D ±1 everywhere Siilar results can be found in [0] The D D involution 48 fro the roof has as eigenvalues as any +1 as 1 and thus the related diagonal atrix D ±1 can be written as diagi D, I D Therefore, instead of D ±1 we ay also use the D D exchange atrix J as ansatz for an involution Uniqueness Results for the Bit-Fli Syetry The technical rearks Leata 3, 33 and 34 ay be ut to good use in the following theore It deicts certain necessary relations for the s atrices to reresent syetric vectors Theore 39 : Let > 1 Assue that the s atrices over K are related by A 1 1 U A 0 1 V 1 and A 1 U 1 A 0 V for,, with square atrices V and U of aroriate size 1,, If any choice of atrices A 0 results in the syetry of the reresented vector x, Jx x then it holds U and V are u to a scalar factor involutions for all : U u I, V v I Furtherore, U c V, 1,, with constants c Proof : First, note that all U and V have to be nonsingular Otherwise, we could use a vector a 0, eg with U k 1 a 0, such that A 0 k ab H and A 1 k U k 1 A 0 k V k 0, giving x 1,1,,1 0, but with aroriate choice of the other A 0 we can easily achieve x 0,0,,0 0

27 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 7 Now, for all ossible choices of A 0, 1,,, it holds x 1,1,,1 tr A 1 1 A1 x 1,1,,1 x 0,0,,0 tr tr A 0 1 A0 U A 0 1 V 1 U 1 A 0 With the notation W V U, 1,,, we have V and 5 A 0 tr 1 A0 0 1 A for all A 0 Therefore, Lea 3 leads to W tr A 0 1 W 1 A 0 1 W 1 A 0 1 A0 A0 1 W A 0 1 W 1 W A 0 1 W 1 0 A and thus A 0 tr 1 A0 0 A 1 tr tr W A 0 1 W 1 W A 0 1 W 1 W 1 W A 0 1 W 1 W A 0 1 If we roceed in the sae way we iteratively reach A 0 1 A0 tr A 0 1 A0 A0 A0 W +1 W A 0 1 W 1A 0 W 1A 0 W and 53 tr W W +1 W A 0 1 W 1 W 1 A 0, 54 for 1,, 1 Thus, we finally obtain the identities tr A 0 A W W 3 W A 0 1 W 1 and 55 tr W 1 W W A 0 1, 56 which hold for all A 0 1 Lea 3 alied to 56 states I W 1 W W or W 1 1 W W Inserting this in 55 gives A 0 1 W1 1 A0 1 W 1 for all A 0 1, so due to Lea 33, W 1 w 1 I for soe constant w 1 0 If we ake use of this relation, Eqn 53 case leads to A 0 1 A0 w 1 W 3 W A 0 1 A0 W for all A 0 1, A0 Thus, Lea 33 states W w I By induction, Eqn 53 reads A 0 1 A0 A0 w 1 w w 1 W +1 W A 0 1 A0 A0 W

28 8 Huckle, Waldherr, Schulte-Herbrüggen for all A 0 1, A0, leading to W w I or U w V 1 for all 1,, 57 Considering instead x 1,0,,0 x 0,1,,1 gives tr U A 0 1 V 1A 0 A0 tr A 0 1 U 1A 0 V U 1 A 0 V Relacing A 0 1 by U 1 A 0 1 V 1 1 results in the above situation 5 Analogously W 1 V1 1 U 1, W V U 1 one gets U 1 c 1 V 1 and U c V Reeating this technique at all ositions for syetries of the for x 0,,0,1,0,,0 x 1,,1,0,1,,1 gives the identities x i1,,i 1,i,i +1,,i c V U 57 w V 1 for all 1,, Therefore, all U and V are involutions u to a factor, U w c I and V w c I Define u : w c and v : w c to finalize the roof Reark 6 : If we only allow unitary atrices U and V eg U V J as otivated above, the factors c and w and thus also u and v have absolute value 1 35 Full-Bit Syetry Now cobine the revious syetries and assue the following roerties of the s atrices A 0 A A H for all and A 1 JAJ for all 58 This ansatz results in reverse, bit-fli and bit-shift syetry Neglecting the ersyetry 58 for the oent and only assuing A 0 A 0 A 0 H and A 1 A 1 A 1 H, one ay diagonalize A 0 H A 0 U H ΛU and set B UA 1 U H Hence, we roose to define a noral for of the tye à 0 à 0 Λ and à 1 à 1 B B H 36 Reduction in the Degrees of Freedo The syetries discussed in the revious aragrahs lead to a reduction of the nuber of free araeters First let us discuss the reduction in the nuber of entries in the full vector x The bit-shift syetry x i1,i,,i x i,,i,i 1 reduces the nuber of different entries aroxiately to 1 Both bit-fli and reverse syetry lead to a reduction factor 1/ in each case Note that not

29 Exloiting Matrix Syetries and Physical Syetries in Matrix Product States 9 Table Listing index sets related to equal vector coonents for different syetries This table shows that the bit-shift syetry, the bit-fli syetry and the reverse syetry are rincially indeendent Bit-shift syetry , , , , , , , Bit-fli syetry , Reverse syetry , all of these syetries are indeendent, eg, the syetry x i1,i x i,i 1 is a consequence of either the bit-shift or the reverse syetry On the other hand the three syetries are indeed indeendent in general To see this we consider the following exale with 9 binary digits: i 1, i, i 3, i 4, i 5, i 6, i 7, i 8, i Table lists for all of the three classes of syetries all index sets which are related to equal vector coonents In the s ansatz we have siilar reductions The bit-shift syetry uses one atrix air instead of, giving a reduction factor The bit-fli syetry has a reduction factor if we ignore different choices for D ;±1, and in the reverse syetry only half of the atrices can be chosen Note, that this will not only lead to savings in eory but also to faster convergence and better aroxiation in the alied eigenvalue ethods because the reresentation of the vectors has less degrees of freedo and allows a better aroxiation of the anifold that contains the eigenvector we are looking for 37 Further Syetries In this aragrah we analyze further syetries such as ±b 1 b b x, 59a x, 59b x b b b ±b The following lea states results for the syetry 59a b 1 59c Lea 310: If the first atrix air is of the tye 0 A 1, A 1 1 B, B, 60 the reresented vector takes the for 59a and, vice versa, any vector of the for 59a can be exressed by an s fulfilling 60 Proof : The given s relation 60 ilies x 0,i,i 3,,i x 1,i,i 3,,i for all i,, i Hence, the reresented vector x is of the for 59a In order to secify an s reresentation for a vector x fulfilling 59a we consider any s reresentation see, eg, Lea 31 for the vector b, b i,,i tr B i Bi3 3 B i e i,i 3,,i

30 30 Huckle, Waldherr, Schulte-Herbrüggen The definition B 0 1 B 1 1 I D results in the desired relations x 0,i,,i x 1,i,,i tr B i1 1 Bi B i tr B i B i b i,,i Reark 7 : 1 The roof works for PBC and OBC In the latter case B i1 1 secializes to a scalar The second syetry 59b corresonds to the relation A 1 1 A 0 1 Adating the roof to this case would give B 0 1 I D and B 1 1 I D 3 The syetry tye 59c is related to A 1 ±A 0 The construction would analogously read B 0 I D and B 1 ±I D 4 Siilarly, we can iose conditions on the s reresentation that certain local atrix roducts are equal resulting in syetry roerties of x So the condition A 0 1 A0 A 1 1 A1 leads to x 0,0,i3,,i x 1,1,i3,,i Iosing the conditions A 0 1 A0 A 1 1 A1 A 0 1 A1 A 1 1 A0 leads to the syetry x 0,0,i3,,i x 1,1,i3,,i x 0,1,i3,,i x 1,0,i3,,i In the following theore we state certain necessary relations for the s reresentation of syetries, which are of the for 59a Theore 311 : Assue that the s atrices over K are related via A 1 1 VA 0 1 U with atrices V and U If any choice of atrices A 0, 1,, for fixed A 1 > 1, results in a vector x of the for b x b, then U ci V 1 and so A 1 1 A 0 1 Proof : The assution leads to the equation A 0 tr 1 VA0 1 U A i A i 0 for all choices of atrices A i, > Fro Lea 3 we obtain A 0 1 VA 0 1 U for all choices of A 0 1 Hence, due to Lea 33, U ci and V I/c for a nonzero c This gives U ci V 1 Reark 8 : The result of Theore 311 can be easily adated to the case 59b Moreover, it can be generalized to syetries such as 59c, which are of the for x i1,,i r,0,i r+,,i x i1,,i r,1,i r+,,i 38 Closing rearks on syetries Let us conclude this aragrah on syetries with soe rearks on alications So far, we have seen that there are different syetries which can be reresented by convenient relations between the s atrices Furtherore we roosed convenient noral fors and attested related uniqueness results It is ore difficult