Experimental observation of classical analogy of topological entanglement entropy

Long-range entanglement is an important aspect of the topological orders, so efficient methods to characterize the long-range entanglement are often needed. In this regard, topological entanglement entropy (TEE) is often used for such a purpose but the experimental observation of TEE in a topological order remains a challenge. Here, we propose a scheme to observe TEE in the topological order by constructing specific minimum entropy states (MESs). We then experimentally construct the classical microwave analogs of the MESs and simulate the nontrivial topological order with the TEE in Kitaev toric code, which is in agreement with theoretical predictions. We also experimentally simulate the transition from Z2 topologically ordered state to topologically trivial state.

. After a measuring process through the PROJ part followed by several mixing and filtering processes for multiplex signals, the desired frequency components selected by 9 FFT-based digital filters { 1 9 , ,

Supplementary Tables
Supplementary Table 1. Distinct frequency components (MHz) selected by bandpass filters 1 F and 2 F in the first stage, and then 5 F , 6 F and 7 F in the second stage in Supplementary Figure 9 and their corresponding     16 A  } selected by the FFT digital filter at the bottom of Supplementary Figure 9 Figure 17 for the corresponding selected basis terms in Supplementary

Supplementary Note 1. Finding minimum entropy states (MESs) for square lattices with 8 and 12 spins
When the square lattice has 8 spins, the Hamiltonian for this system is x The schematic representation has been addressed in Supplementary Figure 1.
There also exists four degenerated ground states for this Hamiltonian. Due to lengthy expressions of these four states, we only provide two of them ( g, 1 8  and g,2 8  ) below.
For the system with 8 spins, we can also define string operators as those in the system with 4 spins. Here, we generate two operators associated with two closed loops, one is along x-direction, and the other is along y-direction, Given the expressions of minimum entropy state (MES) in Ref. [1,2], we take one MES for the system with 8 spins When the square lattice has 12 spins, the Hamiltonian for this system is x The schematic representation has been addressed in Supplementary Figure 3.
There also exists four degenerated ground states for this Hamiltonian. Due to lengthy expressions of these four states, we only provide two of them ( g, 1 12  and g, 2 12  ) below.
For the system with 12 spins, we can also define string operators as those in the system with 4 spins. Here, we generate two operators associated with two closed loops, one is along x-direction, and the other is along y-direction,     1 2 3 4 5 6 1 7 ,12 ,12 , . , , , .
Given the expressions of MES in Ref. [1,2], we take one MES for the system with 12 spins as   g,1 g,2 12 12 12 With 12  , when separating the system into two subsystems as shown in Supplementary Figure 4, we can obtain the von Neumann entropy for the subsystem containing spins 1, 2, 3, 4, 5 and 6 as 4ln2.

Supplementary Note 2. Finding MESs for toric code models with 8 and 12 spins
Here, we present the entanglement entropy for real toric code model, with the spin occupying at the bond of lattice.
which is the smallest toric code model exhibiting the nontrivial entanglement entropy. The schematic representation is addressed in Supplementary Figure 5.
When we have obtained the four degenerated ground states for this model, we look for the MES of such model.
We find that the MES has the similar form as in the equivalent square lattice. The von Neumann entropy for this MES is 2ln2.
When the system contains 12 spins, the Hamiltonian for this system is 12 . x The schematic representation for this model is addressed in Supplementary Figure 6.
When we have obtained the four degenerated ground states for this model, we look for the MES of such model.
is the corresponding down-converted angular frequency set as In the experiment, the classical analogy of the projective measurement can be realized by subsequent where m ( | ( , ) is the conjugate transpose for . Moreover, here we follow the proposed concept of cebit [4,5] to name the vector form of a signal pair as the classical counterpart of a single-qubit quantum state, and adopt the parentheses notation (parent (| and thesis |) ) to represent it as . The cebits constitute an inner product space where the inner product is given by parentheses (|) .
After the projection measurement process in the PROJ, the output signals 1 2 { ( ), ( )} S t S t are sent to a multiplier (×) and then a digital filter 12 F , where the resulting product signal is selected at the sum frequency 8.0MHz and the corresponding complex amplitude A  of this final filtered signal is expressed as 1  2  3  4  1  2  3  4   1  2  3  4  1  2  3  4   1 2 3 4   m  1  m  2  m  3  m  4  m  1  m  2  m  3  m  4   m  1  m  2  m  3  m  4  m  1  m  2  m  3 As shown in Supplementary Figure 9, these eight signals M , a bandpass filter 2 F (passband: 3.5-10.0MHz), and an adder. Then, these two summed signals 12 S and 34 S are sent into the second stage, processed by a multiplier 5 M for mixing and three filter 5 F (7.18MHz), 6 F (passband: 11.0-13.5MHz) and 7 F (17.53MHz) which together select eight desired sum-frequency components  Supplementary Table 1 denoted as 1 4 S  and the sum signal of eight components { 5 6 7 8 Supplementary Table 2 represented as 5 8 S  are processed by a multiplier 7 M in Supplementary Figure 9. Then the combined signal passes through a collection of FFT-based digital filters 16 A  } of the desired frequency components are selected respectively corresponding to those 16 superposed terms in Supplementary Equation 20. The filtered frequencies and the corresponding terms are listed in Supplementary Table  3. Finally which corresponds to the MES presented in Supplementary Figure 10.
We set the down-converted frequencies of the 24 signals in channels { 1 As shown in Supplementary Figure 10, these 24 signals go through the PROJ and then enter a stage-by-stage mixing and filtering process. Note the processes designed in the first and second stages are similar to those in Supplementary Figure 9. For simplicity, here we only list the parameters of these bandpass filters  .
Then, we focus on the signal processing in the third stage. The mixed signal resulting from the mixing of two former signals 1 4 S  and 5 8 S  by a multiplier 10 M is processed by a collection of filters 10 can be put into one group since they own the same joint part of 9

Supplementary Note 5. The compressed sensing method for state tomography
Compressed sensing concerns the problem of recovering structured signals (e.g., sparse signals and low rank matrices) from a small number of measurements [12][13][14]. When the desired density matrices are low-rank, it has been shown that one can stably reconstruct these matrices from highly incomplete Pauli measurements via some convex recovery procedures [15][16][17].
where tr  denotes the nuclear norm of a matrix and  represents a parameter indicating the estimating errors of the experimental data. For the 8-cebit analogy cases, we choose m=6000(

Supplementary Note 6. The construction of MESs for square lattices with symmetrically applied fields
In our study, when an external field is added on the system, the Hamiltonian of the square lattice changes to ) , The schematic representation is shown in Supplementary Figure 11. We express the states with 4 spins as 4 When the system belongs to topologically trivial phase ( 0.34 g  ), there are no nearly degenerated ground states and the topological properties can be revealed from its unique ground state. In order to describe the topological properties of system with external fields, we need to obtain the corresponding ground states for systems. Since there are no analytic forms of ground states for the system with external fields, we numerically obtain the ground states of system ( g 4  ) and list as below. When the system has 4 spins and 0.9 g  , we find that some coefficients in the ground state g 4  are same, that is Here, the degenerated ground state     a a a a a a a a ) When the system belongs to topologically trivial phase ( 0.34 g  ), there are no nearly degenerated ground states and the topological properties can be revealed from its unique ground state. In order to describe the topological properties of system with external fields, we need to obtain the corresponding ground states for systems. Since there are no analytic forms of ground states for the system with external fields, we numerically obtain the ground states of system ( g 8  ) and list as below. When the system has 8 spins and 0.9 g  , we find that some coefficients in the ground state g 8  are same, that is Here, the nearly degenerated ground state           with a passband as 3.0-5.0MHz to select four sum frequency terms: 1 2 | ) h h at 4.0MHz, 1 2 | ) h v at 3.6MHz, 1 2 | ) v h at 4.5MHz, and 1 2 | ) v v at 4.1MHz, which are then recombined into a mixed signal 12 S . Similarly, the original 3 F and 4 F are replaced by 34 F with a passband as 7.0-9.0MHz to select four sum frequency terms: 3 4 | ) h h at 7.9MHz, 3 4 | ) h v at 8.4MHz, 3 4 | ) v h at 7.5MHz, and 3 4 | ) v v at 8.0MHz, which are then recombined into a mixed signal 34 S .
Then, the two summed product signals 12 S and 34 S are mixed by a multiplier The parameters for the FFT filters and modulators with their corresponding terms are listed in Supplementary Table   11. The circuit designed in the DSP module for the cases with a large g (=10,5,2,1,0.9) is shown in Supplementary  Figure 17. The incoming signals