Quantum data compression, quantum information generation, and the density-matrix renormalization-group method

We have studied quantum data compression for finite quantum systems where the site density matrices are not independent, i.e., the density matrix cannot be given as direct product of site density matrices and the von Neumann entropy is not equal to the sum of site entropies. Using the density-matrix renormalization-group (DMRG) method for the one-dimensional Hubbard model, we have shown that a simple relationship exists between the entropy of the left or right block and dimension of the Hilbert space of that block as well as of the superblock for any fixed accuracy. The information loss during the RG procedure has been investigated and a more rigorous control of the relative error has been proposed based on Kholevo’s theory. Our results are also supported by the quantum chemistry version of DMRG applied to various molecules with system lengths up to 60 lattice sites. A sum rule that relates site entropies and the total information generated by the renormalization procedure has also been given, which serves as an alternative test of convergence of the DMRG method.

Figure 1

Schematic plot of the system and environment block of DMRG. $B_l$ and $B_r$ denote the left and right block of length $l$ and $r$, and of dimension $M_l$ and $M_r$, respectively, • stands for the intermediate sites ($s_l$ and $s_r$) with $q_l$ and $q_r$ degrees of freedom. The blocks $B_L=B_l•,B_R=•B_r$ have dimension $M_L$ and $M_R$, respectively.

Figure 2

Block entropy, $S$, and $\ln \left(\dim \Lambda \right)$, $\ln \left(\dim {\Lambda }_{SB}\right)$ as a function of iteration steps for the half-filled 1D Hubbard model with $N=80$ for $U=1$ with fixed number of block states ($M=512$) using PBC. For an iteration step less than 39 the infinite lattice method has been used.

Figure 3

Same as Fig. 2 but using DBSS approach for ${\mathrm{TRE}}_{\mathit{max}}={10}^{-4}$ and $M_{\mathit{min}}=16$.

Figure 4

Parameter $\beta$ as a function of rescaled iteration step using the DBSS approach for ${\mathrm{TRE}}_{\mathit{max}}={10}^{-4}$, $M_{\mathit{min}}=16$, $U=1$, $N=40,60,80$.

Figure 5

Schematic plot of the upper and lower bonds on accessible information for a binary channel.

Figure 6

The relative error as a function of the truncation error and the threshold value of Kholevo’s bound on accessible information obtained with the DBSS method and by Eq. (12), respectively.

Figure 7

The first panel shows the relative error as a function of the threshold value of Kholevo’s bound on accessible information obtained by Eq. (12). The second panel shows the error in the sum rule given by Eq. (19).