Density Matrix Recursion Method: Genuine Multisite Entanglement Distinguishes Odd from Even Quantum Heisenberg Ladders

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


INTRODUCTION
The quantum spin ladder is an interesting platform to investigate quantum many body systems in the intermediate sector between the one-and two-dimensional lattice structures [1]. The possibility of relating doped even-legged quantum spin ladders to high-temperature superconductivity [2][3][4] make such quantum systems extremely important. The characteristic pseudo 2-D structure of ladders evokes considerable interest in several other areas of condensed matter physics [5,6] and quantum information [7]. On the other hand, concepts like entanglement [8] and other quantum correlations [9] have been applied to understand quantum critical phenomena in spin systems [10], including in spin ladders [7]. Properties of such systems have also been tested experimentally in several systems or proposals thereof have been presented, including in compounds and cold gas [11]. Ladder states formed by the superposition of short-range dimer coverings, also known as short-range resonating 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 states are believed to be possible ground states of certain undoped Heisenberg spin ladders, supported by various methods which include mean field theory [14], quantum Monte Carlo [4,15], and Lanczos [5,16].
A striking feature of the quantum spin ladder is that the interpolation from the 1-D spin chain to the 2-D square lattice by gradually increasing the number of legs is not straighforward. For example, the quantum characteristics of the Heisenberg ladder ensures that the odd-and the even-legged ladder ground states have different correlation properties. "Even ladders" have a finite-gapped ground state excitation and exponential decay of two-site correlations, while "odd ladders" are gapless and have power-law decay [6,17]. Such effects also occur in quantum spin liquids, such as discussed in [18]. The results of Heisenberg ladders cannot therefore be directly extrapolated to the 2-D regime. This difference in quantum characteristic of odd-and even-legged ladders may lead to interesting features in the entanglement properties of the system. However, calculating the bipartite or multipartite entanglement in large-sized multi-legged quantum spin ladder states formed by the superposition of short-range dimer coverings is numerically challenging. Within a dimer-covering approach [6], it is possible to derive energy density and spin-correlation functions for two-legged ladders by using generating functions [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 renormalization group calculations in multi-legged Heisenberg ladders, see [22].
In this paper, we introduce an analytical iterative technique, the "density matrix recursion method" (DMRM), to obtain the reduced density matrix of an arbitrary number of sites of a state on a quantum spin-1/2 ladder with an arbitrary number of legs and with both open and periodic boundary conditions. We consider the spin-1/2 ladder state to be a superposition of short-ranged dimer coverings. Specifically, within a dimer covering approach for the state of the quantum spin ladders, we find separate iterative formalisms for even and odd ladders. These partial 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 correlations and bipartite as well as multipartite entanglement. The iterative method introduced here can also be a potential tool for studying properties of reduced density matrices of general large superposed multipartite quantum states and hence can be used as an efficient method for investigating quantum correlations of multipartite quantum states.
We then apply our method to obtain the nature of genuine multipartite entanglement, quantified by the generalized geometric measure (GGM) [23]. An understanding of the multipartite entanglement content is potentially advantageous, in assessing the importance of the corresponding state for future applications, over that of bipartite measures, as the former takes into account the distribution of information between the multisite sub-regions of the entire system 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 GGM of odd-legged ladder states increases with system-size while it decreases in the even ones. We also find that the convergence of the GGM, for a given odd-or even-legged ladder, occurs for relatively small ladder lengths of the corresponding ladder.
The paper is organized as follows. We introduce the model state in Sec. 2. The genuine multiparty entanglement measure is defined in Sec. 3. The main results are presented in sections 4, 5, and 6. In particular, the density matrix recursion methods for even and odd ladders are respectively introduced in sections 4 and 5. We present a conclusion in Sec. 7.

Model State
Within a short-range dimer-covering approach, the state of the quantum spin-1/2 ladder consists of an equal superposition of nearest neighbor directed dimer pairs on the spin-1/2 lattice, also called the resonating valence bond (RVB) state. The ladder lattice under consideration can be divided into sublattices (see Fig. 1), A and B, in such a way that all nearest neighbor (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 occupied by a spin-1/2 qubit. The multi-legged quantum spin-1/2 ladder consists of coupled parallel spin chains with N sites on each chain labelled from 1 to N . The number of chains, referred as legs, of the ladder are labelled as 1 to M . The total number of spin-1/2 sites in the entire bipartite lattice is s (s = M N ). The vertical chains, referred as rungs, are labelled from 1 to n, similar to the sites on the chain, and each rung contains M spins. See Fig. 1. The periodic ladder is conditionally defined by allowing dimer states to be formed between rungs 1 and n [19]. Of course as numbers, n = N . The (unnormalized) quantum state under study is therefore [25] where |(a i , b j ) refers to a dimer between sites a i and b j on the bipartite lattice.
(i, j) are NN sites with i = j. A product of all such dimers on the spin-1/2 lattice constitutes an RVB covering. The summation refers to the superposition of all such dimer product states that give us the superposed short-range dimer covering states. It is believed that these short-range dimer covered states are possible solutions of the two-dimensional antiferromagnetic Heisenberg systems based on the understanding of the RVB theory and possible explanation of the results obtained by using numerical methods like the density matrix renormalization group [20,26].

Genuine Multipartite Entanglement and Generalized Geometric Measure
Quantification of multipartite entanglement plays a crucial role in quantum information processing. In the case of bipartite pure states, entanglement can be uniquely quantified as the von Neumann entropy of local density matrices. In comparison, the quantification of multipartite entanglement involves characterizations and criteria of multipartite pure quantum states and cannot be uniquely defined [8].
For pure states, a genuinely multipartite entangled state can be defined as a multiparty pure quantum state that cannot be expressed as a product across any bipartition. The Greenberger-Horne-Zeilinger [27] and the W [28] states are the common examples of genuine multipartite entangled states. Genuine multipartite entanglement is an important resource from the perspective of many-party quantum communication [29] and is a potential resource for large-scale quantum computation [30]. Cluster states are multipartite entangled states that form promising candidates in building a one-way quantum computer [31]. Multipartite entanglement also has fundamental applications in understanding the role of entanglement in many-body physics [10]. As mentioned above, a genuine multisite entangled pure quantum state is one which is entangled across every partition, into two subsets, of the set of the observers involved. A natural definition of the amount of genuine multipartite entanglement of a pure multiparty quantum state is therefore the minimum distance of that state from the set of all states that are not genuinely multipartite entangled. A widely used measure of distance is the fidelity subtracted from unity [32]. The generalized geometric measure (GGM) of an N -party pure quantum state |φ N is defined as the minimum of this "fidelity-based distance" of |φ N from the set of all states that are not genuinely multiparty entangled [23,33]. In other words, the GGM of |φ N , denoted by E(|φ N ), is given by where Λ max (|φ N ) = max | χ|φ N |, with the maximization being over all N -party quantum states |χ that are not genuinely multipartite entangled. It was shown in Ref. [23,33] that the GGM of a multiparty pure state can be effectively calculated by using where λ A:B is the maximal Schmidt coefficient in the bipartite split A : B of |φ N . The GGM can therefore be calculated once the reduced density matrices of the pure quantum state is obtained. We use the GGM to quantify the amount of genuine multisite entanglement present in multi-legged ladder states of different sizes.

DMRM: The even ladder
We will now describe the iterative methods for generating arbitrary local density matrices for the even and the odd ladders. We begin with the even ladders. The recursion relation for the non-periodic (i.e. with open boundary conditions) evenlegged ladder can be written as where |N is an even-legged ladder containing N sites in each chain and corresponds to the rungs numbered from 1 to n. On the other hand, e.g. |N 2,n+1 represents an even-legged ladder containing 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 dimer coverings between adjacent sites. |2 n+1,n+2 is a two-rung even-legged ladder of rungs n + 1 and n + 2. Also, The term |2 n+1,n+2 contains all the coverings of a two-rung even-leg ladder |2 n+1,n+2 , apart from the coverings formed by the product of the two rungs |1 n+1 |1 n+2 . The subtraction removes possible repetition of states between the two terms in the recursion for |N + 2 in relation (3). We further note that if no site numbers are mentioned in subscript of |N , it is to be considered to be from 1 to n = N . The periodic boundary condition entails that the ladder also form dimer states between the rungs 1 and n. Accounting for the additional states in the ladder system, due to the periodicity, the recursion relation can be modified into where the subscript P stands for a periodic ladder state. Throughout the paper, a state without a subscript P will imply a non-periodic state. The density matrix for the periodic ladder state can be calculated by using the recusrsion relation in Eq. (4): If, for example, we trace out all but two rungs (say n + 1 and n + 2) from the state, we will obtain a two-rung (mixed state) ladder containing 2M spins, where M is the number of legs of the ladder. Note that M is even here. These reduced states can then be used to calculate the multisite entanglement properties of the ladder system. In particular, ρ P (n+1,n+2) = tr 1..n (|N + 2 N + 2| P ) is the two-rung state of the periodic ladder. Now, using relation (3), we can obtain the trace for the non-periodic part to have Tracing over the rungs 1 to n, we get where Z N = N |N ,ρ n+1 = tr n [|2 2 | (n,n+1) ], and Extending the trace to the periodic ladder state, we get Here, with Further, where 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 of the non-periodic ladder is given by whereĀ = 1|1 and B = 2 |2 . C, D,C, andD 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 and lowermost sites of that rung. The other terms in the normalization can be calculated as follows: The other term in the expression for ρ (2) n+1,n+2 that is to be calculated using iterations is where the iteration variables are given by with Here, g 0 = 1, and h 0 = 0. The extra iterative terms in the periodic two-rung density matrix, ρ P (n+1,n+2) , can be expressed as Here, the iterative variables, X N i , are defined by using A 1 N and A 2 N as where the summation is from i = 0 to N . The variablesḡ i andh i mimic the same relations as g i , h i , with initial conditionsḡ 0 = 0,h 0 = 1. Similarly, we can derive where the iterative variables can be derived using (21). These iterations can be used to obtain the partial density matrices of a periodic even-legged ladder, and hence its bipartite as well as genuine multipartite entanglement and other single and multi-site physical quantities, provided the values of A,Ā, B, C,C, D,D, Z 1 , Y 1 1 , and Y 2 1 are exactly calculated. The size of the reduced density matrices depend on the type of the ladder considered. For example, a two-rung reduced density matrix for an M -legged ladder is a τ = 2M -spin matrix. Other reduced density matrices of spins smaller than τ , required e.g. for calculating the GGM, can be obtained from the τ -spin matrix by partial tracings.
For M ≥ 6, the iterations involve algebra that is slightly more complicated. 1|2 = k i=1 α 1 i |α i where |α 1 = |1 and |α 2 = |1 . |α k (k = 1, 2) are other singlet combinations of an M legged single vertical rung. For M = 4 (in the previous derivation), α 1 k (k = 1, 2) = 0. α 1 1 and α 1 2 are C and D respectively. Hence, for M ≥ 6, we have the following relation; n α k |2 n,n+1 = (−1) n−1 j α k j |α j . The normalization terms then work out to be, where A ij = α i |α j . The rest of the iterations can be derived using these terms in a similar fashion to the derivation done for M < 6. The terms α j i (i, j = 1 to k), A, B and A ij need to be exactly calculated.
One of the main motivations for the present work is that spin ladders of the type discussed in this paper can be implemented in the laboratories. It is therefore important to find out whether the obtained effects are resilient to noise effects, like small admixtures of higher multiplets or of singlets that are not of nearest neighbors. Since the genuine multiparty entanglement that we use in the paper is a continuous function of the state parameters, such small admixtures of noise will not substantially alter the amount of the measure present in the multiparty state. This feature is valid independent of whether we are considering even-or odd-legged ladders.

DMRM: The Odd Ladder
The periodic recursion for the odd ladder is rather different from that of the even ladder. This is 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 . The periodic recursion for the odd M -legged N -spin-per-chain ladder is then given by where |2 n+1,n+2 is a two-rung odd-legged ladder at rungs n + 1, n + 2. We observe that there is no repetition of terms in |N + 2 P and hence no subtraction of states is required, as is needed for the 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): ρ N +2 P = |N + 2 N + 2| P . The reduced density matrices can be obtained by tracing out the requisite number of spins. In particular, for obtaining a two-rung reduced density matrix, we trace out the rungs ranging from 1 to n: After simplification, the above equation reads where and the normalization is . Now, for a M -legged ladder, Z 2 corresponds to a two-legged M -site-per-chain ladder, and can be calculated from the previous section. The recursion for Ω N | is Ω 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 an algorithm to iterate the step 2| i,i+3 2| i+1,i+2 |2 i,i+1 . Once the reduced density matrices are obtained by the iterative method, we can again use them to obtain the different single-and multi-site physical quantities of the system.

Even versus Odd
To illustrate the effectiveness of the DMRM, we apply it to obtain a multisite entanglement of multi-legged ladders with both even and odd number of legs. In particular, we consider two-and four-legged ladders among even ladders, and threeand five-legged ladders among odd ones. The iterative variables can be evaluated by using their explicitly calculated initial set of values. The iterations provide the reduced density matrices of the system, which are thereafter utilized to obtain the GGM.
Specifically, for M = 2, the relevant initial parameters are For M = 4, the initial parameters are 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 Z 2 = 44, and for M = 5 is Z 2 = 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 characteristic complementary nature of even and odd ladders. We show that the GGM decreases with the increase of system-size in the case of even ladders (Fig. 2). The opposite is true for odd ladders -the GGM increases with system-size (Fig. 3).  Although the comparison is made by taking M = 3 and 5 among odd ladders, we have actually performed the computations also for M = 1, which again shows the characteristic increasing GGM of odd ladders. In all the cases, the GGM is calculated by considering reduced density matrices upto 2M spins: numerical exact diagonalizations corroborate that considering upto 4 spins is already enough.
We have performed the iterative algorithms upto 18 rungs in all the cases (which e.g. is equivalent to 72 spins for the four-legged and 90 spins for the five-legged ladder). As seen in the Figs. 2 and 3, the GGM has already converged much before the maximum number of rungs that we have considered.
The entanglement properties of the quantum spin ladder states can be generalized and scaled to study other important aspects of quantum spin systems. The twosite reduced density matrix of superposed dimer-covered state is a Werner state [34] and the bipartite entanglement of the state is a monotone of the Werner parameter. The Werner parameter can also be used to study the ground state energy and spin correlation length (see e.g. [19][20][21] and references therein). There is also a correspondence between the entropic properties of spin ladder states with entanglement [35].

Conclusion
We have introduced an analytical iterative technique, the density matrix recursion method, which can be efficiently used to obtain arbitrary reduced density matrices of the states of spin-1/2 quantum spin ladders with an arbitrary number of legs, formed by the superposition of dimer coverings. This technique immediately allows us to obtain single-and multi-site physical properties of the system. In particular, we use the method to obtain the scaling of genuine multisite entanglement in the states of both odd-and even-legged ladders. We find that the genuine multisite entanglement can capture the disparity between the even and odd ladder states. These entanglement properties may 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 multipartite entanglement of such systems has created a lot of interest due to recent experimental developments. Simulating large qubit systems, using optical superlattices [12] and interacting photon states [13], to generate dimercovered superposed states in the laboratory points to the future applications of multipartite entanglement properties of these states. These multipartite entangled states can potentially find applications in building cluster states for large scale quantum computation [31] and in quantum metrology [36].
Our analytical method enables the investigation of the bipartite and multipartite entanglement properties in these superposed dimer-covered systems with relative control and ease even for systems containing a considerable number of quantum spins. The results obtained may prove useful in predicting the prospects of the entanglement properties of experimentally generated states for future applications.