Efficient description of many-body systems with Matrix Product Density Operators

Matrix Product States form the basis of powerful simulation methods for ground state problems in one dimension. Their power stems from the fact that they faithfully approximate states with a low amount of entanglement, the"area law". In this work, we establish the mixed state analogue of this result: We show that one-dimensional mixed states with a low amount of entanglement, quantified by the entanglement of purification, can be efficiently approximated by Matrix Product Density Operators (MPDOs). In combination with results establishing area laws for thermal states, this helps to put the use of MPDOs in the simulation of thermal states on a formal footing.

(here for a chain of N d-level spins), where the A [k] i k are D × D matrices, except for A [1] i1 and A [N ] iN which are 1 × D and D × 1 matrices, respectively.On the one hand, any MPS obeys an area law by construction.On the other hand, it has been shown that any state which obeys a suitable area law for the entanglement can be faithfully approximated by an MPS, that is, with a D which only grows polynomially in the system size N and the desired accuracy (in contrast to the Hilbert space dimension d N ) [12,13].This clarifies why MPS are well suited to describe ground states and excitations of quantum manybody systems, allowing for efficient simulations.
For thermal states, or more generally mixed states, the situation is much less clear.Again, we can ask the same two key questions: First, what is the structure of entanglement, or more generally correlations, in thermal states of physical systems -in particular, is there some analogue to the area law?And if yes, second: Given a mixed state ρ which obeys an area law, what is the structure of ρ, and in particular, can it be well approximated by a Matrix Product Density Operator (MPDO) (where the sum runs over i k , j k = 1, . . ., d, and the The problem is further impeded by the fact that for mixed states there exists a whole zoo of different entanglement measures which are often not related in a simple fashion.As it turns out, the first of those questions has been addressed previously: It has been shown that thermal states of local Hamiltonians obey an area law for the mutual information, which quantifies both quantum and classical correlations in the system [14].The second question, however -relating entanglement scaling and approximability by MPDOs -is yet open, and this is what the present work deals with. In this paper, we show that also for mixed states, a suitable family of entanglement area laws -even with a logarithmic correction -implies that the state can be efficiently approximated by MPDOs.Specifically, given a mixed state ρ on a chain of N d-level spins, we prove that if there exist constants c > 0 and 0 < λ < 1 such that the α-Rényi entanglement of purification E p,α (defined below) [15] satisfies then ρ can be efficiently approximated by an MPDO σ D : As long as the bond dimension D scales polynomially, D = N κ for any κ > 2c 1−λ , the error ε := ρ − σ D 1 in trace norm goes to zero super-polynomially in N (i.e., faster than any inverse polynomial) [16].By using the trace norm -which exactly bounds the error in expectation values of arbitrary bounded observables (with largest eigenvalue 1) -we obtain a bound on the error incurred in arbitrary simulations of physical processes.This establishes that MPDOs are precisely the framework needed to faithfully describe mixed states which obey an entanglement area law of the form above.
The proof will on the one side follow the approach in the pure state case [12].On the other side, we need to deviate from it at some key steps.The reason is that we require a good approximation in trace norm, which -unlike the 2-norm -does not induce a scalar product, which in turn is essential to build norm-preserving projections.At the same time, we cannot bound the 2-norm instead: The relative bound ρ 1 ≤ √ D ρ 2 , with D = d N the dimension of the total space, is tight (saturated by the maximally mixed state), that is, the trace norm can be exponentially larger in N than the 2-norm, breaking efficiency of the approximation.
We will use the conventional graphical calculus for MPS/MPDOs [4][5][6][7][8], where a (mixed) many-body state is denoted as a box with legs (double legs denote ket+bra), Fig. 1a, and an MPDO is expressed as a tensor network, where tensors are boxes, each leg denotes a tensor index, and connecting legs corresponds to contraction, Fig. 1b.
Let us briefly sketch the proof strategy: First, we show that for any bipartition, a bound on the entanglement implies that the target state ρ can be well approximated by a low-rank decomposition across that cut.An area law thus implies that ρ has low-rank approximations across every cut.The crucial step will then be to merge these approximations.To start, we will show how to merge two approximations in such a way that (i) we still obtain a good approximation and (ii) the internal structure of the two states is preserved (specifically, existing lower-rank approximations across other cuts), as this allows us to iterate the procedure.In a final step, we then show how to nest this merging procedure in such a way as to obtain a good MPDO approximation of the target state ρ.
We start by defining the α-Rényi entanglement of purification E p,α (ρ AB ) [15].For a bipartite state ρ AB , it is given by where the minimum is taken over all purifications |ψ AA ′ BB ′ of ρ AB , i.e. tr A ′ B ′ |ψ ψ| = ρ AB , and ), the α-Rényi entanglement entropy quantifying the pure state entanglement between AA ′ and BB ′ .For the remainder of this paper, we restrict to 0 ≤ α < 1.
A key result from the pure state case [12] is that a small E α (|ψ ) implies a rapid decay of the Schmidt coefficients, and thus, there exists a low-rank approximation to |ψ : Concretely, for any D p there exists a (5) (and thus D p scales as an inverse polynomial in the error η).This is equivalent to with • 1 the trace norm (i.e. the sum of the singular values) [17].By tracing A ′ B ′ , and using the fact that tracing (as a completely positive trace preserving map) is contractive under the trace norm, we arrive at for some with rank D = D 2 p (where Let us now turn towards a spin chain of length N whose state ρ obeys an area law, that is, there is an E α max such that E p,α (ρ AB ) ≤ E α max for any bipartition A = 1, . . ., L, B = L + 1, . . ., N .Combining Eqs. ( 7) and ( 5), we have that for each cut, there exists a rank D = D 2 p decomposition of the form (8) with trace norm error What remains to be seen is whether it is possible to merge these different low-rank approximations.However,
at this point we can no longer use the purifications to resort to the pure state result, since the optimal purifications (minimizing E p,α ) for different cuts need not be related [18].We thus require a different approach.
To start, consider a bipartite state ρ ≡ ρ AB (obtained by blocking sites), a truncated approximation and another approximation σ 2 , obtained e.g. by truncating across a different cut.Let us now try to connect those two approximations.To this end, consider a (not necessarily orthogonal) projection P 1 onto span{A i }, P 1 (A i ) = A i , which can be written as for some basis Âi = c ik A k of span{A i } and some (dual) matrices Â′ i satisfying tr[( Â′ i ) † Âj ] = δ ij .P 1 can be naturally embedded into the full space as Now consider see Fig. 2a.First, it also has rank D across the cut; second, the left part is spanned by A k , and thus inherits the structure of the left part of σ 1 ; and third, the right part is obtained from σ 2 by tracing its left part with (A ′ i ) † , and thus inherits the structure of the right part of σ 2 .In particular, if σ 1 and σ 2 have parts on the left and right, respectively, which are already in Matrix Product form, both of these are inherited by P 1 (σ 2 ), see Fig. 2b.We can then iterate this scheme, starting from truncations at individual cuts, to obtain an MPDO approximation.
What is the approximation error of the merged truncation P 1 (σ 2 )?Using P 1 (σ 1 ) = σ 1 from (10), we have with Starting from (12), a series of elementary inequalities [19] gives To keep P 1 (X) 1 small, we thus ideally want to choose Âi and Â′ i } indeed exist, a standard result in functional analysis [20]: Choose a so-called Auerbach basis of the normed space A = span{A i } with norm • 1 , that is, a basis { Âi } together with a set of linear functionals â′ j : A → C such that â′ j ( Âi ) = δ ij and Âi = â′ j = 1 (such a basis always exists) and extend the bounded functional â′ j to a bounded functional tr[( Â′ j ) † • ], Â′ j ∞ = 1, whose existence is guaranteed by the Hahn-Banach theorem.
By inserting these { Âi }, { Â′ i } in Eq. ( 15), we arrive at P 1 (X) 1 ≤ D X 1 , which together with (14) and That is, we have merged the two approximations σ 1 and σ 2 , with new error as above; if both δ 1 , δ 2 ≤ δ, the new error is at most (2D + 1)δ.At this point, we can start concatenating truncations using (16).However, we cannot do this sequentially as one would do for the 2-norm (where one can choose P 1 FIG. 3. Merging of cuts in a tree-like fashion.In each step, the number of cuts is doubled, and the error grows by a factor of (2D + 1).
the orthogonal projection for which P 1 (X) ≤ X ; note that this yields an alternative proof for the result of Ref. 12): The prefactor (2D + 1) would grow exponentially with the number of steps, rendering the bound useless.To overcome this issue, we choose a renormalizationlike procedure, where we concatenate the cuts in a treelike fashion, as illustrated in Fig. 3, using Eq. ( 16) in each step.One can readily check that each step doubles the number of cuts and multiplies the error with (2D + 1); if the number of cuts is not a power of 2, we can start some branches of the tree later, or we can pad the spin chain with trivial (uncorrelated) spins.For a chain of length N , this scheme thus requires K = ⌈log 2 (N −1)⌉ ≤ (log 2 N ) + 1 steps, and thus incurs a total error of We are now at the point where we can combine our results: Combining Eqs. ( 9) and (17) yields If we now -following Eq. ( 3) -choose with ∆ = 1 λ (κ(1 − λ) − 2c) > 0, which thus goes to zero super-polynomially as N → ∞.This completes the proof of our result.
In summary, in this work we have established when MPDOs can efficiently describe quantum many-body systems.We have derived conditions which a state ρ has to fulfill such that it can be approximated by an MPDO with a polynomial bond dimension.In particular, we have shown that for a sequence of density operators ρ on a spin chain of length N , an entanglement area law implies an efficient approximability of ρ by MPDOs.More concretely, we have considered a family of area law bounds for the Rényi entanglement of purification which limit the quantum correlations to grow at most logarithmically with the system size N ; in this setting, we have found that there exist MPDO approximations to ρ with a bond dimension which grows polynomially in the system size N , and for which the approximation error decreases faster than any inverse polynomial in N .This shows that MPDOs provide a faithful approximation to density operators which satisfy an area law, and are thus well suited for the numerical simulation as well as analytical study of such systems.
in α, we can also choose any smaller α in (3).
[17] This is readily checked by working in the two-dimensional space spanned by the two vectors; note that |χD need not be normalized.
[18] Different purifications are related by a unitary on the purifying system A ′ B ′ , which however mixes A ′ and B ′ and thus changes the entanglement properties.

FIG. 1 .
FIG. 1.(a) Tensor notation of density matrix ρ: Each pair of legs denotes the ket and bra index at one site.(b) Tensor network for MPDO, Eq. (2).Legs denote indices, connected lines contraction (summation) of indices, corresponding to the matrix products in Eq. (2).