Around the same time, other groups have developed similar approaches in which quantum information plays a predominant role as, for example, in DMRG implementations for periodic boundary conditions , and for studying mixed-state dynamics in one-dimensional quantum lattice systems. Those last approaches actually provide a formalism that is more general than the original TEBD approach, as it also allows to deal with evolutions with matrix product operators; this enables the simulation of nontrivial non-infinitesimal evolutions as opposed to the TEBD case, and is a crucial ingredient to deal with higher-dimensional analogues of matrix product states. Consider a chain of "N" qubits, described by the function formula_1  This form, known as a Matrix product state, simplifies the calculations greatly. To understand why, one can look at the Schmidt decomposition of a state, which uses singular value decomposition to express a state with limited entanglement more simply. Consider the state of a bipartite system formula_2. Every such state formula_3 can be represented in an appropriately chosen basis as:  where formula_5 are formed with vectors formula_6 that make an orthonormal basis in formula_7 and, correspondingly, vectors formula_8, which form an orthonormal basis in formula_9, with the coefficients formula_10 being real and positive, formula_11. This is called the Schmidt decomposition (SD) of a state. In general the summation goes up to formula_12. The Schmidt rank of a bipartite split is given by the number of non-zero Schmidt coefficients. If the Schmidt rank is one, the split is characterized by a product state. The vectors of the SD are determined up to a phase and the eigenvalues and the Schmidt rank are unique. For example, the two-qubit state:  has the following SD:  with  On the other hand, the state:  is a product state:  At this point we probably know enough to try to see how we explicitly build the decomposition (let's call it "D"). Consider the bipartite splitting formula_18. The SD has the coefficients formula_19. By expanding the formula_20's in the local basis, one can write:  The process can be decomposed in three steps, iterated for each bond (and, correspondingly, SD) in the chain:  Step 1: express the formula_22's in a local basis for qubit 2:  The vectors formula_24 are not necessarily normalized.