Suppose we have a system with a Lindblad operator $\hat{A}$. The question is what is the Lindblad operator in a composite system. Presumably, $\hat{A} ⊗ 𝟙$, but one could also think that with $𝟙 = ∑|k⟩⟨k|$, the composite system could have one Lindblad operator $\hat{A} ⊗ |k⟩⟨k|$ for each basis vector of the second system. That is, the composite system would have multiple Lindblad operators?
Are those two things potentially equivalent? They are not! The difference is in the term $\hat{A} \hat{\rho} \hat{A}^\dagger$ in the Liouville equation. With a single Lindblad operator $\hat{A} ⊗ 𝟙$, we find
$$(\hat{A} ⊗ 𝟙) \hat\rho (\hat{A} ⊗ 𝟙)^\dagger = \sum_{ijnm} \rho_{(ij),(nm)} \hat{A} |i⟩⟨n| \hat{A}^{\dagger} \otimes |j⟩⟨m| \tag{1}$$
where $|ij⟩⟨jm|$ is the double-indexed basis in which $\hat{\rho}$ is represented, and $\rho_{(ij),(nm)}$ is the matrix element for those indices.
On the other hand, with multiple Lindblad operators, we have
$$\sum_k (\hat{A} ⊗ |k⟩⟨k|) \hat\rho (\hat{A} ⊗ |k⟩⟨k|)^\dagger = \sum_{ink}\rho_{(ik),(nk)} \hat{A} |i⟩⟨n| \hat{A}^{\dagger} \otimes |k⟩⟨k| \tag{2}$$
That is, we are missing the cross-terms $|j⟩⟨m|$ in Eq. (1) for $j \ne m$.
These matrices are coherences in the second system that are lost in the dynamics. That hints at the choice of multiple Lindblad operators being unphysical: Simply tensoring a non-dissipative system to a dissipative system should not affect the non-dissipative system at all, the dissipative system should not be inducing dissipation into the non-dissipative system.
We can illustrate this for the example of a single qubit with spontaneous decay. We will simulate the dynamics of the qubit, and then the dynamics of the system tensored with a second, non-dissipative qubit, for the two possible choices of composite Lindblad operators.
Then, the population dynamics of the sub-systems, with the other system traced out:
Calculate the partial trace of the given density matrix ρ
ρ_red = partial_trace(ρ, (N_i, N, N_j))calculates the reduced density with the middle degree of freedom integrated out.
It is assumed that the density matrix ρ has the structure
$$ρ = \sum ρ_{(inj)(i^′n^′j^′)} |inj⟩⟨i^′n^′j^′|$$
with the triple-index $(inj)$ where $i$ is in the range $(1, N_i)$, $n$ is in the range $(1, N)$, and $j$ is in the range $(1, N_j)$
The reduced density matrix is defined as
$$ρ^{(red)} = \sum \underbrace{\sum_{n} ρ_{(inj)(i^′nj^′)}}_{= \rho^{(red)}_{(ij),(i^′j^′)}} |ij⟩⟨i^′j^′|$$
Note that assuming a structure of 3 tensored Hilbert spaces and always tracing out the middle Hilbert space allows handling an arbitrary number of actual Hilbert spaces, and tracing out arbitrary degrees of freedom: All Hilbert spaces to the left of the degree of freedom to be integrated out can be combined into a single Hilbert space of dimension $N_i$, and all the Hilbert spaces on the right can be combined into a space of dimension $N_j$. This includes the situation where we have two actual Hilbert spaces with dimensions N and M, respectively. If we wanted to trace out the first Hilbert space, we could use
ρ_red = partial_trace(ρ, (1, N, M))If we wanted to trace out the second Hilbert space, we would instead use
ρ_red = partial_trace(ρ, (N, M, 1))The given ρ can be a matrix or vectorized matrix. The reduced density matrix will be returned correspondingly.
And, the reduced density matrix of the second TLS (first TLS traced out):
We can now use
We'll consider a completely static TLS:
Now we tensor the system with a second TLS. This second TLS is non-dissipative.
and the purity
Note: the code for this notebook is available at https://github.com/goerz-research/2025-09-11_Composite_Dissipation
Note that the purity starting from a mixed state should be $1 - \frac{1}{2} \left(e^{-\gamma t} + e^{-2\gamma t}\right)$: The purity first drops due to the dissipation and then rises again because in the infinite time limit, all the population ends up in the (pure) ground state.
Uses two Lindblad operators, one for each of the levels in the second TLS.
When analyzing the dynamics, what we'll be looking for is that the second system remains unaffected by the dissipation in the first system. To this end, we'll have to calculate the reduced density matrices for the first and second TLS, using a partial trace:
The difference comes in via the purity (the coherences) in the sub-systems:
The population dynamics even for the sub-systems still looks correct:
tensors the original Lindblad operator with the identity for the second TLS.
Second,
mimetext/htmlrootassigneelast_run_timestampA1*|persist_js_state·has_pluto_hook_features§cell_id$f8d76484-05e6-4a72-98c8-3f33b06f4306depends_on_disabled_cells§runtime published_object_keysdepends_on_skipped_cells§errored$20eb5f97-ca69-434d-80ec-5c99b8f153c1queued¤logsrunning¦outputbodyprefixStringelements "|00⟩⟨00|"text/plain"|01⟩⟨00|"text/plain"|10⟩⟨00|"text/plain"|11⟩⟨00|"text/plain"|00⟩⟨01|"text/plain"|01⟩⟨01|"text/plain"|10⟩⟨01|"text/plain"|11⟩⟨01|"text/plain "|00⟩⟨10|"text/plain "|01⟩⟨10|"text/plain "|10⟩⟨10|"text/plain "|11⟩⟨10|"text/plain "|00⟩⟨11|"text/plain"|01⟩⟨11|"text/plain"|10⟩⟨11|"text/plain"|11⟩⟨11|"text/plaintypeArrayprefix_shortobjectidc813cf94dbba48dcmime!application/vnd.pluto.tree+objectrootassigneesuperop_labelslast_run_timestampA1Cpersist_js_state·has_pluto_hook_features§cell_id$20eb5f97-ca69-434d-80ec-5c99b8f153c1depends_on_disabled_cells§runtimë́published_object_keysdepends_on_skipped_cells§errored$0fabd75f-e9d1-46d5-8ba5-7e334101d1c2queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampA1upersist_js_state·has_pluto_hook_features§cell_id$0fabd75f-e9d1-46d5-8ba5-7e334101d1c2depends_on_disabled_cells§runtime 밵published_object_keysdepends_on_skipped_cells§errored$378380e4-7043-434d-8933-72b36e3a8981queued¤logsrunning¦outputbody4×4 SparseArrays.SparseMatrixCSC{ComplexF64, Int64} with 4 stored entries: ⋅ ⋅ ⋅ 0.0+0.1im ⋅ 0.0-0.05im ⋅ ⋅ ⋅ ⋅ 0.0-0.05im ⋅ ⋅ ⋅ ⋅ 0.0-0.1immimetext/plainrootassigneeLlast_run_timestampA1& persist_js_state·has_pluto_hook_features§cell_id$378380e4-7043-434d-8933-72b36e3a8981depends_on_disabled_cells§runtime !Xpublished_object_keysdepends_on_skipped_cells§errored$0eaf2661-c7cd-4c3d-872e-00d4eaef74e0queued¤logsrunning¦outputbody%pops (generic function with 1 method)mimetext/plainrootassigneelast_run_timestampA1 persist_js_state·has_pluto_hook_features§cell_id$0eaf2661-c7cd-4c3d-872e-00d4eaef74e0depends_on_disabled_cells§runtime ~published_object_keysdepends_on_skipped_cells§errored$337e927c-6c23-4afe-9a62-bf30f3a12c0aqueued¤logsrunning¦outputbodyeFirst, we look at the population dynamics of the combined system:
We have two candidates for the Lindbladian of the composite system. First,
Just to test our implementation of the partial trace a bit more for the general case, we can check that we can recover the components of a random separable state:
We can look at which matrix elements are missing.
L1 and L2 do not have the same number of non-zero entries!
We'll add to that spontaneous decay, with a decay rate of
This is clearly unphysical: without interactions between system 1 and system 2, the dissipation in system 1 should not be causing dissipation in system 2! System 2 should remain pure at all times.
Thus, we can conclude that a single Lindblad operator $\hat{A} \otimes 𝟙$ is the correct approach.
The Liouvillian can be constructed as a super-operator, via
We can check that the intial state decomposes as expected:
which we can analyze in terms of the populations
The result of the propgation is
The reduced density matrix of the first TLS (second TLS traced out):
to simulate the dynamics of a coherent superposition of |0⟩ and |1⟩ as an initial state, over a time grid
We repeat this analysis for the second candidate of a Liouvillian, with two Liouville operators.
The population dynamics in the combined system looks unsuspicious:
mimetext/htmlrootassigneelast_run_timestampA1Ԑ~persist_js_state·has_pluto_hook_features§cell_id$d45416d2-ba22-43d3-bc6a-6e7edf050be8depends_on_disabled_cells§runtime published_object_keysdepends_on_skipped_cells§errored$296fa066-fc76-4a1b-9d70-57c0fe080966queued¤logsrunning¦outputbody16×16 SparseArrays.SparseMatrixCSC{ComplexF64, Int64} with 16 stored entries: ⎡⠀⢄⠀⠀⠀⠑⠀⠀⎤ ⎢⠀⠀⠀⢄⠀⠀⠀⠑⎥ ⎢⠀⠀⠀⠀⠑⢄⠀⠀⎥ ⎣⠀⠀⠀⠀⠀⠀⠑⢄⎦mimetext/plainrootassigneeL1last_run_timestampA1bpersist_js_state·has_pluto_hook_features§cell_id$296fa066-fc76-4a1b-9d70-57c0fe080966depends_on_disabled_cells§runtime ^published_object_keysdepends_on_skipped_cells§errored$9e8682c2-0e3d-4515-bb1f-28a8d339402aqueued¤logsrunning¦outputbodyټAs we expect, the population deviates from the initial superposition only for the first system.
Lastly, we look at the purity in the sub-systems:
Again, we see the purity deviating from 1 only in the first system.
It might be interesting to further look into the interpretation of exactly how these matrix elements affect the dynamics. But in lieue of that, we'll simply simulate the dynamics.
As the initial state, we'll use a coherent superposition both in the first and in the second TLS:
Now, we can compare the dynamics under our two candidate Liouvillian L1 and L2:
But the dynamics are certainly not identical!