The density matrix recursion method: genuine multisite entanglement distinguishes odd from

even quantum spin ladder states

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The density matrix recursion method: genuinemultisite entanglement distinguishes odd from evenquantum spin ladder states

Himadri Shekhar Dhar1, Aditi Sen(De)2 and Ujjwal Sen2,3

1 School of Physical Sciences, Jawaharlal Nehru University,New Delhi 110 067, India2 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi,Allahabad 211 019, IndiaE-mail: ujjwal@hri.res.in

New Journal of Physics 15 (2013) 013043 (13pp)Received 5 October 2012Published 18 January 2013Online at http://www.njp.org/doi:10.1088/1367-2630/15/1/013043

Abstract. We introduce an analytical iterative method, the density matrixrecursion method, to generate arbitrary reduced density matrices ofsuperpositions of short-range dimer coverings on periodic or non-periodicquantum spin-1/2 ladder lattices, with an arbitrary number of legs. The methodcan be used to calculate bipartite as well as multipartite physical properties,including bipartite and multi-partite entanglement. We apply this technique todistinguish between even- and odd-legged ladders. Specifically, we show thatwhile genuine multi-partite entanglement decreases with increasing system sizefor the even-legged ladder states, it does the opposite for odd-legged ones.

3 Author to whom any correspondence should be addressed.

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New Journal of Physics 15 (2013) 0130431367-2630/13/013043+13$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

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Contents

1. Introduction 22. The model state 33. Genuine multi-partite entanglement and the generalized geometric measure 44. The density matrix recursion method: the even ladder 55. The density matrix recursion method: the odd ladder 96. Even versus odd 97. Conclusion 11Acknowledgments 12References 12

1. Introduction

The quantum spin ladder is an interesting platform to investigate quantum many-body systemsin the intermediate sector between one-dimensional (1D) and two-dimensional (2D) latticestructures [1]. The possibility of relating doped even-legged quantum spin ladders to high-temperature superconductivity [2–4] makes such quantum systems extremely important. Thecharacteristic pseudo 2D structure of ladders has generated considerable interest in severalother areas of condensed matter physics [5, 6] and quantum information [7]. On the other hand,concepts such as entanglement [8] and other quantum correlations [9] have been applied tounderstand quantum critical phenomena in spin systems [10], including in spin ladders [7].The properties of such systems have also been tested experimentally in several systems andproposals thereof have been presented, including in compounds and cold gases [11]. Ladderstates formed by the superposition of short-range dimer coverings, also known as short-rangeresonating valence bond (RVB) states, can be simulated by using atoms in optical lattices [12]and, as shown recently, by using interacting photons [13]. These short-range RVB ladder statesare believed to be possible ground states of certain undoped Heisenberg spin ladders, supportedby various methods that include mean field theory [14], quantum Monte Carlo [4, 15] andLanczos [5, 16].

A striking feature of the quantum spin ladder is that the interpolation from the 1D spinchain to the 2D square lattice by gradually increasing the number of legs is not straightforward.For example, the quantum characteristics of the Heisenberg ladder ensure that the odd- and theeven-legged ladder ground states have different correlation properties. ‘Even ladders’ have afinite-gapped ground state excitation and exponential decay of two-site correlations, whereas‘odd ladders’ are gapless and have power-law decay [6, 17]. Such effects also occur in quantumspin liquids, such as discussed in [18]. The results of Heisenberg ladders cannot, therefore,be directly extrapolated to the 2D regime. This difference in quantum characteristic of odd-and even-legged ladders may lead to interesting features in the entanglement properties ofthe system. However, calculating the bipartite or multi-partite entanglement in large-sizedmulti-legged quantum spin ladder states formed by the superposition of short-range dimercoverings is numerically challenging. Within a dimer-covering approach [6], it is possible to

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derive energy density and spin-correlation functions for two-legged ladders by using generatingfunctions [19] or state iterations [20]. Approximate solutions may also be obtained for the four-legged ladder [21]. For further work in this direction, including density matrix renormalizationgroup calculations in multi-legged Heisenberg ladders, see [22].

In this paper, we introduce an analytical iterative technique, the ‘density matrix recursionmethod’ (DMRM), to obtain the reduced density matrix of an arbitrary number of sites of astate on a quantum spin-1/2 ladder with an arbitrary number of legs and with both open andperiodic boundary conditions. We consider the spin-1/2 ladder state to be a superposition ofshort-range dimer coverings. Specifically, within a dimer covering approach for the state of thequantum spin ladders, we find separate iterative formalisms for even and odd ladders. Thesepartial density matrices can be used to calculate and study the scaling and behavior of single-,two- and multi-site physical properties of the whole spin system, including two-site correlationsand bipartite as well as multi-partite entanglement. The iterative method introduced here canalso be a potential tool for studying the properties of reduced density matrices of generallarge superposed multipartite quantum states and hence can be used as an efficient method forinvestigating quantum correlations of multipartite quantum states.

We then apply our method to obtain the nature of genuine multi-partite entanglement,quantified by the generalized geometric measure (GGM) [23]. An understanding of the multi-partite entanglement content is potentially advantageous, in assessing the importance of thecorresponding state for future applications, over that of bipartite measures, as the former takesinto account the distribution of information between the multi-site sub-regions of the entiresystem and the hidden correlations are not ignored due to tracing out [24]. We compare the odd-and even-legged ladder states by using genuine multisite entanglement, and find that the GGMof odd-legged ladder states increases with system size while it decreases in the even ones. Wealso find that the convergence of the GGM, for a given odd- or even-legged ladder, occurs forrelatively small ladder lengths of the corresponding ladder.

This paper is organized as follows. We introduce the model state in section 2. The genuinemultipartite entanglement measure is defined in section 3. The main results are presented insections 4–6. In particular, the DMRMs for even and odd ladders, respectively, are introducedin sections 4 and 5. We present the conclusion in section 7.

2. The model state

Within a short-range dimer-covering approach, the state of the quantum spin-1/2 ladder consistsof an equal superposition of nearest neighbor (NN) directed dimer pairs on the spin-1/2 lattice,also called the RVB state. The ladder lattice under consideration can be divided into sublattices(see figure 1), A and B, in such a way that all NN sites of sublattice A belong to sublattice B,and vice versa. Such a lattice is called a bipartite lattice. Each bipartite lattice site is occupiedby a spin-1/2 qubit. The multi-legged quantum spin-1/2 ladder consists of coupled parallel spinchains withN sites on each chain labeled from 1 toN . The number of chains, referred to as legs,of the ladder are labeled from 1 to M . The total number of spin-1/2 sites in the entire bipartitelattice is s (s = MN ). The vertical chains, referred to as rungs, are labeled from 1 to n, similarto the sites on the chain, and each rung contains M spins (see figure 1). The periodic ladderis conditionally defined by allowing dimer states to be formed between rungs 1 and n [19].

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M

1 2 3

|2 12

|(ai ,i

j

4 n5…

|1 n

bj)

Figure 1. The M-legged, N -runged quantum spin-1/2 ladder in the form of abipartite lattice. The solid circles are of sublattice A, whereas the hollow onesare of B. The red arrows (solid) show the NN dimer states (|(ai , b j)〉) from a sitein sublattice A to another in B. The arrows (dashed) at the boundary indicate thatthe lattice has a periodic boundary condition. The dashed lines indicate that thenumber of legs and rungs can be extended beyond the illustrated number of sites.The figure also shows the two-site ladder state, |2〉, and the one-site rung state,|1〉, which will be profusely used in the iteration method presented.

Of course as numbers, n =N . The (unnormalized) quantum state under study is therefore [25]

|N 〉 =

∑k

∏i, j

|(ai , b j)〉

k

,

where |(ai , b j)〉 refers to a dimer between sites ai and b j on the bipartite lattice. (i, j) areNN sites with i 6= j . A product of all such dimers on the spin-1/2 lattice constitutes an RVBcovering. The summation refers to the superposition of all such dimer product states that giveus the superposed short-range dimer covering states. It is believed that these short-range dimercovered states are possible solutions of the 2D antiferromagnetic Heisenberg systems based onthe understanding of the RVB theory and possible explanation of the results obtained by usingnumerical methods such as the density matrix renormalization group [20, 26].

3. Genuine multi-partite entanglement and the generalized geometric measure

Quantification of multi-partite entanglement plays a crucial role in quantum informationprocessing. In the case of bipartite pure states, entanglement can be uniquely quantified as thevon Neumann entropy of local density matrices. In comparison, the quantification of multi-partite entanglement involves characterizations and criteria of multipartite pure quantum statesand cannot be uniquely defined [8]. For pure states, a genuinely multi-partite entangled statecan be defined as a multipartite pure quantum state that cannot be expressed as a productacross any bipartition. The ‘Greenberger–Horne–Zeilinger’ state of [27] and the ‘W state’of [28] are common examples of genuine multi-partite entangled states. Genuine multi-partite entanglement is an important resource from the perspective of many-party quantum

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communication [29] and is a potential resource for large-scale quantum computation [30].Cluster states are multi-partite entangled states that form promising candidates in building aone-way quantum computer [31]. Multi-partite entanglement also has a fundamental applicationin understanding the role of entanglement in many-body physics [10].

As mentioned above, a genuine multisite entangled pure quantum state is one which isentangled across every partition, into two subsets, of the set of the observers involved. A naturaldefinition of the amount of genuine multi-partite entanglement of a pure multipartite quantumstate is therefore the minimum distance of that state from the set of all states that are notgenuinely multipartite entangled. A widely used measure of distance is the fidelity subtractedfrom unity [32]. The GGM of an N -party pure quantum state |φN 〉 is defined as the minimum ofthis ‘fidelity-based distance’ of |φN 〉 from the set of all states that are not genuinely multipartiteentangled [23, 33]. In other words, the GGM of |φN 〉, denoted by E(|φN 〉), is given by

E(|φN 〉) = 1 − 32max(|φN 〉), (1)

where 3max(|φN 〉) = max |〈χ |φN 〉|, with the maximization being over all N -party quantumstates |χ〉 that are not genuinely multipartite entangled.

It was shown in [23, 33] that the GGM of a multipartite pure state can be effectivelycalculated by using

E(|φN 〉) = 1 − max{λ2A:B|A ∪ B = {1, 2, . . . , N }, A ∩ B = ∅}, (2)

where λA:B is the maximal Schmidt coefficient in the bipartite split A : B of |φN 〉. The GGMcan, therefore, be calculated once the reduced density matrices of the pure quantum state areobtained. We use the GGM to quantify the amount of genuine multisite entanglement present inmulti-legged ladder states of different sizes.

4. The density matrix recursion method: the even ladder

We will now describe the iterative methods for generating arbitrary local density matrices forthe even and odd ladders. We begin with the even ladders. The recursion relation for the non-periodic (i.e. with open boundary conditions) even-legged ladder can be written as

|N + 2〉 = |N + 1〉|1〉n+2 + |N 〉|2〉n+1,n+2

= |N 〉|2〉n+1,n+2 + |N − 1〉|2〉n,n+1|1〉n+2, (3)

where |N 〉 is an even-legged ladder containing N sites in each chain and corresponds to therungs numbered from 1 to n. On the other hand, e.g. |N 〉2,n+1 represents an even-legged laddercontaining N sites in each chain, and corresponds to the rungs numbered from 2 to n + 1. |1〉

is a vertical chain containing M spins representing a single rung of the ladder with the dimercoverings between adjacent sites. |2〉n+1,n+2 is a two-rung even-legged ladder of rungs n + 1 andn + 2. Also

|2〉n+1,n+2 = |2〉n+1,n+2 − |1〉n+1|1〉n+2.

The term |2〉n+1,n+2 contains all the coverings of a two-rung even-leg ladder |2〉n+1,n+2, apart fromthe coverings formed by the product of the two rungs |1〉n+1|1〉n+2. The subtraction removespossible repetition of states between the two terms in the recursion for |N + 2〉 in relation (3).

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We further note that if no site numbers are mentioned in the subscript of |N 〉, it is to beconsidered from 1 to n =N .

The periodic boundary condition entails that the ladder also forms dimer states between therungs 1 and n. Accounting for the additional states in the ladder system, due to the periodicity,the recursion relation can be modified into

|N + 2〉P = |N + 2〉1,n+2 + |N 〉2,n+1|2〉n+2,1, (4)

where the subscript P stands for a periodic ladder state. Throughout the paper, a state without asubscript P will imply a non-periodic state. The density matrix for the periodic ladder state canbe calculated by using the recursion relation in equation (4):

ρ(N+2)

P = |N + 2〉〈N + 2|P

= |N + 2〉〈N + 2| + |N + 2〉〈N |2,n+1〈2|n+2,1 + |N 〉2,n+1|2〉1,n+2〈N + 2|

+ |N 〉〈N |(2,n+1)|2〉〈2|(1,n+2). (5)

If, for example, we trace out all but two rungs (say n + 1 and n + 2) from the state, we willobtain a two-rung (mixed state) ladder containing 2M spins, where M is the number of legsof the ladder. Note that M is even here. These reduced states can then be used to calculate themultisite entanglement properties of the ladder system. In particular,

ρ(2)

P(n+1,n+2) = tr1,...,n(|N + 2〉〈N + 2|P) (6)

is the two-rung state of the periodic ladder. Now, using relation (3), we can obtain the trace forthe non-periodic part to have

tr1,...,n(|N + 2〉〈N + 2|) = tr1,...,n[|N 〉〈N ‖ 2〉〈2|(n+1,n+2) + |N − 1〉〈N − 1‖2〉〈2|(n,n+1)|1〉〈1|(n+2)

+|N 〉|2〉n+1,n+2〈N − 1|〈2|n,n+1〈1|n+2 + |N − 1〉|2〉n,n+1|1〉n+2〈N |〈2|n+1,n+2]. (7)

Tracing over the rungs 1 to n, we obtain

ρ(2)

(n+1,n+2) = ZN |2〉〈2|(n+1,n+2) +ZN−1ρn+1 ⊗ |1〉〈1|(n+2) + (|2〉n+1,n+2〈1|n+2〈ξN |n+1 + h.c.), (8)

where ZN = 〈N |N 〉, ρn+1 = trn[|2〉〈2|(n,n+1)] and

〈ξN |n+1 = 〈2|n,n+1〈N − 1|N 〉. (9)

Extending the trace to the periodic ladder state, we obtain

ρ(2)

P(n+1,n+2) = ρ(2)

n+1,n+2 + tr1..n[|N 〉〈N |(2,n+1)|2〉〈2|(1,n+2)

+(|N 〉2,n+1|2〉 + (|N 〉2,n+1|2〉1,n+2〈N + 2| + h.c.)]

= ρ(2)

n+1,n+2 + β21(n+1,n+2) + (β2

2(n+1,n+2) + h.c.). (10)

Here,

β21(n+1,n+2) = ZN−1|1〉〈1|(n+1) ⊗ ρn+2 +ZN−2ρn+1 ⊗ ρn+2 + (|1〉〈ξN−1|(n+1) ⊗ ρn+2 + h.c.) (11)

with

〈ξN−1|n+1 = 〈2|n,n+1〈N − 2|2,...,n−1N − 1〉2,...,n. (12)

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Further,

β22(n+1,n+2) = |2〉n+1,n+2〈1|n+1〈ξN |n+2 + |2〉n+1,n+2〈φN |n+1,n+2 + ρn+1 ⊗ |1〉n+2〈ξN−1|n+2

+1

A|ξ1〉n+1|1〉n+2〈1|n+1〈ηN−1|n+2, (13)

where

〈φN |n+1,n+2 = 〈2|1,n+2〈2|n,n+1〈N − 2|2,n−1N 〉, (14)

〈ηN−1|n+2 = 〈2|1,n+2〈N − 1|2,nN − 1〉1,n−1|1〉n (15)

and A= 〈1|1〉 = 〈1|1〉.The local density matrices can be calculated by using the iterations, derived below, of the

complete and partial inner products defined above. For M < 6, using (3), the normalization ofthe non-periodic ladder is given by

ZN = 〈N |N 〉

=AZN−1 +BZN−2 + 2CY1N−2 + 2DY2

N−1, (16)

where A= 〈1|1〉 and B = 〈2|2〉. C, D, C and D are given by 〈1|2〉 = C|1〉+D|1〉 and 〈1|2〉 =

C|1〉+D|1〉, where |1〉 is a vertical rung with a periodic dimer covering between the topmost andlowermost sites of that rung. The other terms in the normalization can be calculated as follows:

Y1N = 〈N |N − 1〉|1〉 =AZN−1 + CY1

N−1 +DY2N−1,

Y2N = 〈N |N − 1〉|1〉 = AZN−1 + CY1

N−1 + DY2N−1.

The other term in the expression for ρ(2)

n+1,n+2 that is to be calculated using iterations is

〈ξN | = 〈2|n,n+1〈N − 1|N 〉

= 〈1|(CA1N + CA2

N ) + 〈1|(DA1N + DA2

N ), (17)

where the iteration variables are given by

A1N =

N∑i=1

ZN−i gi−1,

A2N =

N∑i=2

ZN−i hi−1

(18)

withgi+1 = Cgi + Chi ,

hi+1 = Dgi + Dhi .(19)

Here, g0 = 1 and h0 = 0.The extra iterative terms in the periodic two-rung density matrix, ρ

(2)

P(n+1,n+2), can beexpressed as

〈φN |n+1,n+2 = 〈2|1,n+2〈2|n,n+1〈N − 2|2,n−1N 〉

= 〈2|1,n+2〈2|n,n+1(XN1 |1〉1|1〉n +XN2 |1〉1|1〉n +XN3 |1〉1|1〉n +XN4 |1〉1|1〉n

+ 〈2|n−2,n−1 . . . 〈223|2〉12 . . . |2〉n−1,n). (20)

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Here, the iterative variables, X Ni , are defined by using A1

N and A2N as

XN1 =

∑g2i(A1

N−1−2i + CA1N−2−2i) + g2iDA1

N−2−2i ,

XN2 =

∑g2i(A2

N−1−2i + CA2N−2−2i) + g2iDA2

N−2−2i ,

XN3 =

∑h2i(A1

N−1−2i + CA1N−2−2i) + h2iDA1

N−2−2i ,

XN4 =

∑h2i(A2

N−1−2i + CA2N−2−2i) + h2iDA2

N−2−2i ,

(21)

where the summation is from i = 0 to N . The variables gi and hi mimic the same relations asgi , hi , with initial conditions g0 = 0, h0 = 1. Similarly, we can derive

〈ηN−1|n+2 = 〈2|1,n+2〈N − 1|2,nN − 1〉1,n−1|1〉n

= (AA1N−1 + AA2

N−1)(C〈1| +D〈1|) + 〈2|n+2,1(XN−11 〈1|2〈1|n +XN−1

2 〈1|2〈1|n

+XN−13 〈1|2〈1|n +XN−1

4 〈1|2〈1|n)|2〉1,2|1〉n, (22)

where the iterative variables can be derived using (21). These iterations can be used to obtainthe partial density matrices of a periodic even-legged ladder, and hence its bipartite as well asgenuine multi-partite entanglement and other single- and multi-site physical quantities, providedthat the values of A, A, B, C, C, D, D, Z1, Y1

1 and Y21 are exactly calculated. The size of the

reduced density matrices depends on the type of ladder considered. For example, a two-rungreduced density matrix for an M-legged ladder is a τ = 2M-spin matrix. Other reduced densitymatrices of spins smaller than τ , required, e.g., for calculating the GGM, can be obtained fromthe τ -spin matrix by partial tracings.

For M > 6, the iterations involve algebra that is slightly more complicated. 〈1|2〉 =∑ki=1 α1

i |αi〉, where |α1〉 = |1〉 and |α2〉 = |1〉. |αk〉 (k 6=1, 2) are other singlet combinations ofan M-legged single vertical rung. For M = 4 (in the previous derivation), α1

k (k 6= 1, 2) = 0.α1

1 and α12 are C and D, respectively. Hence, for M > 6, we have the following relation:

n〈αk|2〉n,n+1 = (−1)n−1∑

j αkj |α j〉. The normalization terms then work out to be

ZN = 〈N |N 〉n

=AZN−1 +BZN−2 + 2(−1)n−1∑

j

α1jY

jN−1, (23)

Y jN = n〈α j |(1,n−1〈N − 1|N 〉1,n)

=A j1ZN−1 + (−1)n−1∑

k

αjkY

kN−1, (24)

where Ai j = 〈αi |α j〉. The rest of the iterations can be derived using these terms in a similarfashion to the derivation done for M < 6. The terms α

ji (i, j = 1 to k), A, B and Ai j need to be

exactly calculated.One of the main motivations for this work was that spin ladders of the type discussed in

this paper can be implemented in the laboratory. It is, therefore, important to find out whetherthe obtained effects are resilient to noise effects, such as small admixtures of higher multiplets

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or of singlets that are not of NNs. Since the genuine multipartite entanglement that we use inthe paper is a continuous function of the state parameters, such small admixtures of noise willnot substantially alter the amount of the measure present in the multipartite state. This featureis valid independent of whether we are considering even- or odd-legged ladders.

5. The density matrix recursion method: the odd ladder

The periodic recursion for the odd ladder is rather different from that of the even ladder. Thisis due to the fact that there exists no analogous state for |1〉 in the odd ladder. The non-periodic|N 〉 can be written as a series of N = 2 odd-legged ladders. |N 〉 = |2〉1,2|2〉3,4 . . . |2〉n−1,n. Theperiodic recursion for the odd M-legged N -spin-per-chain ladder is then given by

|N + 2〉P = |N 〉1,n|2〉n+1,n+2 + |N 〉2,n+1|2〉n+2,1, (25)

where |2〉n+1,n+2 is a two-rung odd-legged ladder at rungs n + 1, n + 2. We observe that there isno repetition of terms in |N+2〉P and hence no subtraction of states is required, as is needed forthe even ladder. The repetition is avoided by using the periodic condition in the initial recursion.This is possible due to the absence of |1〉 state in the odd ladder.

The density matrix can now be calculated as before, using (25): ρN+2P = |N + 2〉〈N + 2|P .

The reduced density matrices can be obtained by tracing out the requisite number of spins. Inparticular, for obtaining a two-rung reduced density matrix, we trace out the rungs ranging from1 to n:

ρ(2)

P(n+1,n+2) = tr1,...,n[|N + 2〉〈N + 2|P]

= tr1,...,n[|N 〉〈N |1,n|2〉〈2|n+1,n+2 + |N 〉〈N |2,n+1|2〉〈2|n+2,1

+(|N 〉1,n|2〉n+1,n+2〈N |2,n+1〈2|1,n+2 + h.c.)]. (26)

After simplification, the above equation reads

ρ(2)

P(n+1,n+2) = ZN |2〉〈2|(n+1,n+2) +ZN−2ρn+1 ⊗ ρn+2 + (|2〉n+1,n+2〈�N |n+1,n+2 + h.c.), (27)

where

〈�N |n+1,n+2 = 〈2|1,n+2〈N |2,n+1|N 〉1,n (28)

and the normalization is ZN = 〈N |N 〉 = ZN/22 . Now, for an M-legged ladder, Z2

corresponds to a two-legged M-site-per-chain ladder, and can be calculated fromthe previous section. The recursion for 〈�N | is 〈�N |n+1,n+2 = 〈2|1,n+2〈N |2,n+1|N 〉1,n =

〈2|1,n+2〈2|2,3 · · · 〈2|n,n+1|2〉1,2|2〉3,4 · · · |2〉n−1,n. Hence the computation involves writing analgorithm to iterate the step 〈2|i,i+3〈2|i+1,i+2|2〉i,i+1. Once the reduced density matrices areobtained by the iterative method, we can again use them to obtain the different single- andmulti-site physical quantities of the system.

6. Even versus odd

To illustrate the effectiveness of the DMRM, we apply it to obtain a multisite entanglement ofmulti-legged ladders with both even and odd numbers of legs. In particular, we consider two-and four-legged ladders among even ladders, and three- and five-legged ladders among odd

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2 4 6 8 10 12 14 16 18n

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

GG

M

2-leg4-leg

Figure 2. Genuine multisite entanglement decreases with system size for evenladders. We perform the iterations for M = 2 and 4. The iterations are carriedout up to 18 rungs, i.e. 36 and 72 sites, respectively, for M = 2 and 4. However,in both cases, the GGM converges much before those sizes. The vertical axisrepresents the GGM, while the horizontal one represents the number of rungs.Both axes are dimensionless.

ones. The iterative variables can be evaluated by using their explicitly calculated initial set ofvalues. The iterations provide the reduced density matrices of the system, which are thereafterutilized to obtain the GGM.

Specifically, for M = 2, the relevant initial parameters are

Z0 = 1, Z1 = 2,

A= 2, A= 2,

C = 1, D = 0, C = 0, D = 0,

Y11 = 2, Y2

1 = 0.

(29)

For M = 4, the initial parameters are

Z0 = 1, Z1 = 4,

A= 4, A= 2,

C = 5, D = 1, C = 2, D = 3,

Y11 = 4, Y2

1 = 2.

(30)

A similar analysis can also be done for higher even-legged ladders.In the case of odd ladders, the initial parameter required in the recursion for M = 3 is

Z2 = 44, and for M = 5 is Z2 = 804.We are now ready to compare the multisite entanglements for the even- and odd-

legged ladders. The GGMs obtained by the iterative methods clearly capture the characteristiccomplementary nature of even and odd ladders. We show that the GGM decreases with anincrease of system size in the case of even ladders (figure 2). The opposite is true for oddladders—the GGM increases with system size (figure 3).

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2 4 6 8 10 12 14 16 18n

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

GG

M

3-leg5-leg

Figure 3. Genuine multisite entanglement increases with system size for oddladders. We perform the iterations for M = 3 and 5. The iterations are up to18 rungs, i.e. 54 and 90 sites, respectively, for M = 3 and 5. Again in bothcases, the GGM converges much before 18 rungs. The vertical axis representsthe GGM, while the horizontal one represents the number of rungs. Both theaxes are dimensionless.

Although the comparison is made by taking M = 3 and 5 among odd ladders, we haveactually performed the computations also for M = 1, which again shows the characteristicincreasing GGM of odd ladders. In all cases, the GGM is calculated by considering reduceddensity matrices up to 2M spins: numerical exact diagonalizations corroborate that consideringup to 4 spins is already enough.

We have performed the iterative algorithms up to 18 rungs in all cases (which, e.g., isequivalent to 72 spins for the four-legged and 90 spins for the five-legged ladder). As seen infigures 2 and 3, the GGM has already converged much before the maximum number of rungsthat we have considered.

The entanglement properties of the quantum spin ladder states can be generalized andscaled to study other important aspects of quantum spin systems. The two-site reduceddensity matrix of the superposed dimer-covered state is a Werner state [34] and the bipartiteentanglement of the state is a monotone of the Werner parameter. The Werner parameter canalso be used to study the ground state energy and spin correlation length (see, e.g., [19–21]and references therein). There is also a correspondence between the entropic properties of spinladder states and entanglement [35].

7. Conclusion

We have introduced an analytical iterative technique, the DMRM, which can be efficiently usedto obtain arbitrary reduced density matrices of the states of spin-1/2 quantum spin ladders withan arbitrary number of legs, formed by the superposition of dimer coverings. This techniqueimmediately allows us to obtain single- and multi-site physical properties of the system. Inparticular, we use the method to obtain the scaling of genuine multisite entanglement in thestates of both odd- and even-legged ladders. We find that the genuine multisite entanglement

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can capture the disparity between the even and odd ladder states. These entanglement propertiesmay prove insightful to researchers interested in studying other physical aspects of spin systems.It would be interesting to find the extrapolation of the results obtained to the case of broad rungs.

The behavior of multi-partite entanglement of such systems has attracted a lot of interestowing to recent experimental developments. Simulating large qubit systems, using opticalsuperlattices [12] and interacting photon states [13], to generate dimer-covered superposedstates in the laboratory points to the future applications of multi-partite entanglement propertiesof these states. These multipartite entangled states can potentially find applications in buildingcluster states for large-scale quantum computation [31] and in quantum metrology [36].

Our analytical method enables investigation of the bipartite and multi-partite entanglementproperties in these superposed dimer-covered systems with relative control and ease even forsystems containing a considerable number of quantum spins. The results obtained may proveuseful in predicting the prospects of the entanglement properties of experimentally generatedstates for future applications.

Acknowledgments

HSD is supported by the University Grants Commission (UGC), India under the UGC-SeniorResearch Fellowship scheme. HSD thanks the Harish-Chandra Research Institute for hospitalityand support during visits. Computations were performed at the cluster computing facility at HRIand at the UGC-DSA computing facility at Jawaharlal Nehru University.

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