Entanglement spectrum of mixed states
Evert van Nieuwenburg Oded Zilberberg






Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
California Institute of Technology, Pasadena, CA 91125, USA

Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland


Entanglement plays an important role in our ability to understand, simulate, and harness 
quantum many-body phenomena.
In this work, we investigate the entanglement spectrum for open one-dimensional systems, 
and propose a natural quantifier for how much a 1D quantum state is entangled while being subject 
to decoherence. 
We demonstrate our method using a simple case of single-particle evolution and find 
that the open system entanglement spectrum is composed of generalized concurrence values, as well as 
quantifiers of the state's purity. 
Our proposed entanglement spectrum can be directly obtained using a correct scaling of 
a matrix product state decomposition of the system's density matrix.
Our method thus offers new observables that are easily acquired 
in the study of interacting 1D systems, and sheds light on the approximations employed in matrix product 
state simulations of open system dynamics.




Entanglement spectrum of mixed states
    Oded Zilberberg
    December 30, 2023
=====================================





§ INTRODUCTION

The wave nature of quantum-mechanical particles leads to fundamental physical implications such as interference and entanglement. 
The latter can be understood as a quantifier of how nonlocal a specific quantum state is, and has become a ubiquitous tool for understanding quantum many-body physics <cit.>. Indeed, the study of entanglement plays an important role in research fields such as quantum information and quantum computation, which push technological advances towards the utilization of quantum mechanics in real-life applications <cit.>. 

Entanglement of a quantum mechanical state |ψ⟩ can be considered with respect to a bipartition of the state into two parts A and B.
Typically, the bipartitioning is taken as a spatial cut that divides the system into two equal halves. 
If the state can be decomposed into a product of state |ψ_A⟩ in subsystem A and a state |ψ_B⟩ in subsystem B, the state |ψ⟩=|ψ_A⟩⊗|ψ_B⟩ is considered non-entangled.
Conversely, when the state can not be decomposed into a product with respect to the bipartitioning, it is entangled.
Note that these statements are made with respect to a given basis, i.e. a state can be non-entangled in one basis but entangled in another.
Additionally, a given state can be entangled for one bipartition but not for another. 
 
One method of quantifying the entanglement of a pure state with respect to a given bipartitioning, is using the entanglement entropy S. 
To compute it, one considers the density matrix ρ = |ψ⟩⟨ψ| corresponding to the state |ψ⟩. 
The bipartitioning into parts A and B is obtained through a partial trace s.t. ρ_A = Tr_B ρ. 
The reduced density matrix ρ_A contains the full information of the state within subsystem A. 
Additionally, it stores information on the amount of entanglement that exists between subsystems A and B. 
Namely, one can interpret the reduced density matrix as ρ_A = e^-H_e, where H_e is known as the entanglement Hamiltonian <cit.>. 
The eigenvalues ϵ_i of H_e are known as the entanglement spectrum (ES). 
Typically, one works with the eigenvalues λ_i=e^-ϵ_i of ρ_A directly, and then computes the entanglement entropy S≡ -Trρ_A logρ_A =-∑_i λ_i logλ_i as a measure of entanglement. If S=0 the state is non-entangled; non-zero S signals entanglement.



The reduced density matrix generally describes a mixed state <cit.>, i.e. its purity 𝒫_A≡ Tr{ρ_A^2} is less than unity. 
This procedure of tracing out a subsystem is a natural description of open quantum systems, in which the traced out part is the environment. In this work we consider entanglement between subsystems A_1 and A_2 of an open quantum system (cf. Figure <ref>), where the latter may initially be non-pure due to the tracing out of an environment B. Due to the linearity of the trace operation, this is in principle equivalent to considering ρ_A_1 = Tr_A_2 ∪ B ( ρ_A_1 ∪ (A_2 ∪ B) ), representing a different bipartitioning of the whole system. Nevertheless, considering the route of first tracing out B provides a physical interpretation of the resulting entanglement properties.

Here we investigate a natural definition for the entanglement spectrum of open systems that can serve as a direct measure for entanglement and correlations between two subsystems A_1 and A_2. 
We perform a bipartitioning of the mixed state of an open system by interpreting the corresponding density matrix as a vector (i.e. by vectorizing it), and analyze the obtained generalization of the entanglement spectrum for mixed states <cit.>, referred to as the operator space entanglement spectrum (OSES) <cit.>. 

We find here that the OSES directly encodes information about the purity of the mixed state as a whole, as well as of its bipartitioned subparts. 
Focusing on the case of a single-particle excitation we highlight that the OSES additionally encodes nonlocal bipartite entanglement information (in the form of generalized concurrence values), similar to the closed system entanglement spectrum. 
Importantly, the OSES can be readily obtained numerically using matrix product state (MPS) decomposition of the system's vectorized density matrix. 
As result, our work highlights what kind of information is neglected when using such algorithms with a finite bond dimension, and appropriate transformations before truncation may be used to preserve these quantities <cit.>.

Our procedure for defining the OSES of the mixed state operator ρ_A is identical to that used in the discussion of operator space entanglement entropy (OSEE) <cit.>. The procedure for numerically obtaining the OSES differs in an important way from the OSEE procedure by a re-scaling factor. The OSES has rarely been considered directly, but was previously shown to indicate symmetry protected topological states in open systems via its degeneracy structure <cit.>, by a direct extension of the known results for the closed system <cit.>.

Our paper is structured as follows: in Sec. <ref>, we review the formalism for obtaining the OSES. In Sec. <ref>, we obtain an analytic description for the case of single-excitation mixed states, followed by two demonstrations of these results for simple one-dimensional spin-chains on small chains in Sec. <ref>. We discuss the impact and scope of our results in Sec. <ref>.



§ OPERATOR SPACE ENTANGLEMENT SPECTRUM


Inspired by the process of obtaining the entanglement spectrum at a bipartitioning of a pure state <cit.>, we present here a method for obtaining a meaningful entanglement spectrum for a mixed state. 
We refer to Appendix <ref> for more details on the pure state entanglement spectrum. 

We consider the density matrix ρ of an open system A, e.g. ρ may be obtained by tracing out an environment. The density matrix can be written in a Liouville space as a vector |ρ⟩ <cit.> by a procedure referred to as vectorization.
Then, in analogy to the construction of density matrices from pure states, we can define a Hermitian matrix Q as 

    Q=|ρ⟩⟨ρ| .
	

We now consider a bipartition of Q into subparts A_1 and A_2, and trace out the latter to obtain

    Q_A_1 = Tr_A_2 Q    .
	
We can write Q_A_1 = e^-H̃_e, where H̃_e is interpreted as some Hamiltonian with a spectrum ϵ̃_i, which is the operator space entanglement spectrum. 
In the following, we will work with the eigenvalues Λ_i=e^-ϵ̃_i of Q_A_1 and refer to these exponential values as the OSES. 

The main aim of our work is to investigate the physical information encoded in the Λ_i.
Before presenting a more rigorous analysis of the OSES, we want to highlight one immediate consequence of this definition of Λ_i.
From the structure of Q it is apparent that the trace over the matrix Q is Tr Q = Trρ^2, i.e., it equals the purity of the system. 
Considering that (i) Λ_i are the eigenvalues of Q_A_1 and (ii) Tr Q= Tr_A_1 Q_A_1, we readily obtain that the sum of the OSES is the purity Trρ^2=∑_i Λ_i. 
This important property is sometimes overlooked in matrix product state simulations of open systems, because the Λ_i are typically scaled s.t. ∑_i Λ_i^2 = 1.



 §.§ Partial trace of Q

Let us now continue with the partial-tracing of Q. 
We begin by introducing a set of local basis matrices that span a local Hilbert space ℋ, and denote them Γ^[n]_i_n. The superscript n indicates the site n, and the subscript i_n = 1… d^2 with d = dimℋ. A general density matrix ρ on L sites can now be decomposed using a tensor product of these basis matrices as:


    ρ = ∑_i_1 … i_L P_i_1 … i_LΓ^[1]_i_1⊗⋯⊗Γ^[L]_i_L .

In doing so, we have assumed that every site has the same local Hilbert space of dimension d. 
We then denote the vectorized matrices |Γ^[n]_i_n⟩, and they are chosen to obey the following on-site normalization condition:


    ⟨Γ^[n]_i_n|Γ^[n]_i_m⟩ = 1/dTr Γ^[n]†_i_nΓ^[n]_i_m = δ_i_n,i_m,


where the first equality defines the inner product between vectorized matrices. In the following, we will assume that the local Hilbert space is that of a spin-1/2, i.e. the local basis can be composed of the identity matrix plus Pauli matrices, σ_i_n with i_n=0,1,2,3. In general for spins S, the generators of SU(2S+1) provide such a basis. We will introduce our choice for the basis matrices in the next section. For ease of notation, we will drop the superscript site labels wherever possible.

Using the vectorized density matrix in this basis, the matrix Q can be written as follows:


    .9!Q  = ∑_i_1… i_L,α_1…α_L P_i_1… i_N P_α_1…α_N|Γ_i_1⟩⋯|Γ_i_L⟩⟨Γ_α_1|⋯⟨Γ_α_L| ,


The process of tracing out the sites m + 1 through L then leads to

    .9!Tr_m+1… L Q = ∑_j_m+1… j_L⟨Γ_j_m+1|⋯⟨Γ_j_L| Q |Γ_j_m+1⟩⋯|Γ_j_L⟩
       .9!= ∑_i_1… i_m,α_1…α_m C_i_1… i_m;α_1…α_m|Γ_i_1⟩⋯|Γ_i_m⟩⟨Γ_α_1|⋯⟨Γ_α_m| ,

where we have defined the prefactors

    C_i_1… i_m;α_1…α_m
    =
    ( ∑_j_m+1… j_L P_i_1… i_m;j_m+1… j_L P^*_α_1…α_m;j_m+1… j_L) ,

which can be placed in a matrix C of size d^2m× d^2m. 
When considering a local basis composed of Pauli matrices for a spin-1/2 system, C is therefore of size 4^m× 4^m.

Having defined a procedure for obtaining the partial-trace of Q, the OSES Λ_i is obtained by diagonalizing the remaining matrix Q_A_1. 
This is equivalent to obtaining the eigenvalues of the matrix C. This is a challenging task in general, and we will focus on the case of a single excitation in the following. We note that efficient numerical methods for obtaining the OSES in the case of many-body systems exist, see Appendix. <ref>.



§ SINGLE EXCITATION


In this section, we detail a specific example for the type of information encoded in Λ_i.
We restrict ourselves to the case of a single excitation, and do not consider particle-loss processes here. Rather, we will include general decoherence processes that make the state non-pure.

As a set of basis matrices, we choose the set 

    Γ^[n]_i_n∈{[ 1 0; 0 0 ], [ 0 0; 0 1 ], [ 0 0; 1 0 ], [ 0 1; 0 0 ]},

and introduce the shorthand notation {σ_e, σ_f, σ_+, σ_-} for these matrices, respectively. The matrices σ_e,f describe an empty or full site, and σ_± describe the corresponding off-diagonal elements. Notice that with this definition of the basis matrices, the 1/d normalization factor in Eq. <ref> is not needed.

Using this notation, the basis states for the L-site system with a single excitation can be divided into two groups. 
The first group are basis states that are made out of products of σ_e,f, corresponding to the diagonals of the density matrix. In particular, for a single excitation this means only those states in which a single site n has σ_f as its local basis, whilst the others are empty with σ_e. The other group of basis states spans the off-diagonal elements (the coherences) of the density matrix and consists of states where a pair of sites are connected using σ_± and the others are empty. These are the only allowed basis states for a single excitation, and we use the shortcut notation for the coefficients in Eq. (<ref>) of the former type as P_j=f (corresponding to e.g. P_e… efe… e with the f at position j), and the latter as P_j=+,l=-.

Since we will be considering mixed states, it is useful to decompose the i,j-entry of the density matrix as ξ_ijρ_ij, where ξ_ij are complex numbers representing decoherence, i.e., ξ_ii≡ 1 on the diagonal of the density matrix and |ξ_ij| ≤ 1 for the off-diagonal elements. In particular, a pure state has ξ_ij = 1 for all i,j.

In the above notation the coefficients P_j=f correspond simply to ρ_jj, i.e. to the density n_j at site j. Additionally, it sets an important relation for the case of an initially pure state, namely,

    P_j=+,l=- P_j=-,l=+ = |ξ_jl|^2 P_j=fP_l=f .

This relation stems from the fact that the density matrix is a Hermitian operator that describes a pure state that can decohere. Since the density matrix is Hermitian, the left-hand side of Eq. (<ref>) can be further reduced to |P_j=+,l=-|^2.

We are now in a position to use these relations to obtain an exact expressions for the OSES. 
To do so, we construct the matrix C obtained by a bipartitioning of the system between sites m and m+1 [cf. Eq. (<ref>)] and obtain its eigenvalues Λ_i.
In the single excitation case, the matrix C decomposes into three subblocks corresponding to the cases where: (i) the particle is fully traced out (i.e. it was in subsystem A_2 and no coherences connected it to subsystem A_1), (ii) the particle is fully on the remaining side (subsystem A_1), and (iii) the particle has some coherent part that spans across the bipartition. The OSES will be composed of the eigenvalues of these subblocks, which we discuss in detail in the next sections.



 §.§ The particle is fully in subsystem A_2


In this case the subblock consists of just a single element. 
The indices of C correspond to empty chains due to the fact that the particle remained in the subsystem that was traced out, i.e., the indices of C are i_k = e and α_k = e for all 1≤ k ≤ m. Therefore, the OSES that originates from this block is directly the C-matrix element itself, i.e.,


    Λ_1    ≡    C_𝐞_m ; 𝐞_m
       =   ∑_j=m+1^L | P_j=f |^2 + 2 ∑_j=m+1^L∑_l=m+1,l≠ j^L | P_j=+,l=- |^2
       =   ∑_j,l=m+1^L |ξ_jk|^2 P_j=fP_l=f≡𝒫_A_2 ,


where we have used notation 𝐞_m≡e… e_m times. We have, additionally, inserted the shorthand single-particle basis notation into Eq. (<ref>), and have used Eq. (<ref>). Importantly, the last equality highlights that the expression (<ref>) that we have obtained encodes the purity of subsystem A_2, i.e., 𝒫_A_2≡ Trρ_A_2^2 with ρ_A_2≡ Tr_A_1ρ. One of the single excitation OSES' entries therefore corresponds to the purity of the traced-out subsystem.



 §.§ The particle is fully in subsystem A_1


In this case, the non-zero elements of C have the form

    C_i_1… i_m ; α_1 …α_m = P_i_1 … i_m; 𝐞_L-m P^*_α_1 …α_m; 𝐞_L-m ,

representing the case where the single particle and its basis decomposition in the A_1 subsystem solely appear in independent C elements. The size of this block is equal to the number of possible single particle configurations within the basis i_1 … i_m, i.e., all the density and coherences within subsystem A. There are m of the former, and m(m - 1) of the latter, and hence the block is of size m^2 × m^2.

The subblock of this case can be directly expressed as 𝐯·𝐯^†, where 𝐯 is a column vector with elements P_i_1 … i_m; 𝐞_L-m as its entries. Correspondingly, there is a single eigenvalue for this subblock given by

    Λ_2    ≡ Tr 𝐯·𝐯^† = 𝒫_A_1 .


In other words, the OSES has another entry corresponding to the purity of the subsystem A_1 that was not traced out.



 §.§ The particle's coherence extends across the bipartition


We are left with coherences between sites that are on opposite sides of the cut. This means that we deal here with matrix elements with indices i_1… i_m and i_m+1… i_L marking empty sites with a single position where s=± can appear. Correspondingly, this block is of size 2m× 2m. 

we denote G+_j as a string of e's with a single + at index j, and G-_j as the same string but with a - at index j. The most general form of such elements of C is

    C_Gs1_j ; Gs2_k = ∑_s3∑_l=m+1^L P_j=s1 ; l=s3 P^*_k=s2; l=s3 ,

where the indices s1,s2,s3=± and 1≤ j, k ≤ m. Taking into account that, in our single-particle case, s3=-s1 but also s3=-s2, we have nonvanishing elements solely when s1=s2=s. As a result, this block of C splits into two independent subblocks, each of size m× m.
		
Each of these m× m subblocks have two types of entries. On the diagonal of each subblock, j=k,
		
		
    C_Gs_j, Gs_j = ∑_l=m+1^L | P_j=s ; l=-s |^2 .

		
		The off-diagonal elements of each subblock, then, take the form
		
		
    C_Gs_j, Gs_k = ∑_l=m+1^L P_j=s ; l=-s P^*_k=s; l=-s ,

		
As in Sec. <ref>, we identify that each s-subblock is constructed as ∑_l=m+1^L𝐯_s,l·𝐯_s,l^† with v_s,l being a column vector with entries P_j=s, l=-s, where j=1… m. 
Because of the Hermiticity of the density matrix, these vectors are complex-conjugate to one another, i.e., 𝐯_s,l = 𝐯_-s,l^*.  Hence the two subblocks are identical and we expect to obtain m two-fold degenerate eigenvalues contributing to the OSES from this case.

As in the previous two cases, we wish to relate these eigenvalues to a more physical intuition. To better understand the information encoded in the eigenvalues arising from the coherence block (Sec. <ref>), we can write each complex coefficient in Eq. (<ref>) as P_j=s ; l=-s=|P_j=s ; l=-s|e^iϕ_j,l,s. Using Eq. (<ref>) we then write

    P_j=s ; l=-s=|ξ_jl|√(P_j=fP_l=f)e^iϕ_j,ls .

Plugging Eq. (<ref>) into Eq. (<ref>), we obtain


    .9!C_Gs_j, Gs_k = ∑_l=m+1^L|ξ_jl||ξ_kl|P_l=f√(P_j=fP_k=f)e^i(ϕ_j,l-ϕ_k,l)s.


Let us now consider the special case in which contributions to decoherence are such that ξ_jl factorizes into ξ_jξ_l. In this special case, C_Gs_j, Gs_k takes the form


    .9!C_Gs_j, Gs_k = (∑_l=m+1^L|ξ_l|^2 P_l=f)ξ_jξ_k√(P_j=fP_k=f)e^i(ϕ̃_j-ϕ̃_k)s,


where we have defined ϕ̃_j=ϕ_j,l-θ_j-θ_l-ϕ̃_l using the local decohorence contributions collected as ξ_j=|ξ_j|e^iθ_j. Note, that ϕ̃_j is the phase associated with the amplitude of the pure state at position j.

The decoupling offered by the local decoherence assumption is quite illuminating: the sum over l enters as a constant prefactor in all elements of the coherence block of C, cf. Sec. <ref>. As a result, the subblock can be described, in similitude to (ii), as a product of a a single vector 𝐮 with elements 𝒜ξ_j√(P_j=f)e^iϕ̃_j where we have defined 𝒜=√((∑_l=m+1^L|ξ_l|^2 P_l=f)). We, therefore, obtain in this special case that the coherence block contributes two degenerate OSES values solely. Their value is the squared norm of the vector, |𝐮|^2, and corresponds to a generalized cross-boundary concurrence, i.e., these values take the form

    Λ_3,4 = (∑_l=m+1^L|ξ_l|^2 P_l=f)(∑_j=1^m|ξ_j|^2 P_j=f) ,

which is identically the squared norm of a vector made of concurrences between all pairs of two sites across the boundary, √(P_j=fP_l=f), attenuated by local decoherence factors.
 
We conclude that under the local decoherence assumption the OSES of a single-particle will be composed of strictly four values regardless of the size of the system. 
It is important to note that the case of a pure state, ξ_ij≡ 1 fits under this limiting assumption. 
Indeed, for a single-particle pure state case, the ES has only two values (see Appendix <ref>) and for pure states the OSES is just the tensor product of the ES with itself (c.f. Appendix <ref>), in conjunction with our result here of having four OSES values. Deviations away from this simple case will generate additional values arising from the interplay of dissipation and the state's coherence in the block discussed in Sec. <ref>. Nevertheless, these deviations are usually small and we commonly observe four dominant values in the single-particle case.
We have rigorously analyzed the single-particle case of the OSES. In the following two subsections we provide two intuitive examples to illustrate our result.



§ DEMONSTRATIONS




 §.§ Example I: Charge-qubit Rabi oscillations

As we have shown above, the OSES values contain important bipartite entanglement information. In order to demonstrate the physical relevance of these values, we will start with a simple case study of a charge qubit, i.e., a single-excitation hopping between two sites. The Hamiltonian describing such a setup simply is

    H = -t c^†_1 c^†_2 + h.c. ,

where c_1 and c_2 are the single-particle annihilation operators on sites 1 and 2, respectively. We have assumed a vanishing on-site potential and use t as the hopping amplitude. 

The corresponding 4× 4 density matrix ρ(t) describing state at any time t only has a single 2× 2 non-vanishing subblock corresponding to the single particle sector:

    ρ^(t) = ( [   ρ_11(t)   ρ_12(t); ρ_12^*(t)   ρ_22(t) ]) .

Coupling the system to local baths that interact capacitively with the sites, we describe the system dynamics using a standard master equation in Lindblad form


    ∂_t ρ = -i [ H, ρ] + γ∑_i ( 2 n̂^†_i ρ n̂_i^† - {n̂_i^†n̂^†_i, ρ}) ,


where n̂_i is a local density operator at site i, and γ is the coupling strength to the (infinite temperature) bath.

For this simple case, one can easily identify the nonvanishing coefficients P_1=f=ρ_11(t), P_2=f=ρ_22(t), and P_1=+,2=-=ρ_12(t). There is only one possible spatial bipartition and the resulting C-matrix is already diagonal with the following OSES:

    Λ_1=   ρ_11^2(t)≡ n_1(t)^2 ,
    Λ_2=   ρ_22^2(t)≡ n_2(t)^2 ,
    Λ_3,4=   |ρ_12(t)|^2 = |ξ_12(t)|^2 n_1(t)n_2(t) ,

where n_i is the density at site i. 
Above, we have used Eq. (<ref>) to relate the off-diagonal density matrix elements to the diagonal ones using the decoherence prefactor ξ_12(t). 
Consequently, while Λ_1 and Λ_2 follow trivially the square of the densities at the two sites (which are indeed the purity of the subsytems), the values Λ_3,4 are degenerate and correspond to the square-root of the quantum state's concurrence.



In Fig. <ref>(a), we plot the Λ_i for a pure state evolution (γ=0). Whenever the particle is equally spread over the sites, the OSES values are degenerate, whereas mostly they have a “1,1,2”-degeneracy structure corresponding to squared-densities and a doubly-degenerate concurrence value. The lower panel shows the respective operator space entanglement entropy and total purity computed from the OSES.
In Fig. <ref>(b), we present the obtained OSES and respective measures for non-Hamiltonian evolution (c.f. (<ref>), with γ=0.25). Over time, the Rabi oscillations reduce alongside a decaying concurrence towards an equal weight statistical mixture. This can also be seen in a reduction with time t→∞ of the entanglement entropy saturates at log 2 and purity at 1/2, as expected.

It is clear that for such a simple case the four OSES values contain all the information on the single-particle density matrix on two-sites, since there are merely four elements in the density matrix in total. This simple example, however, points to a natural understanding of the bipartite information that the OSES contains, namely, we have two values that contain weights on how much of the particle is to either side of the bond and two degenerate values that correspond to the amount of cross-bond coherence.



 §.§ Example II: Four-site case

We now extend our demonstration to the case of a single-particle hopping across four sites. The Hamiltonian keeps its simple structure, i.e.

    H_ cq = -t ∑_i=1^3 c^†_i c^†_i+1 + h.c. ,

with a corresponding density matrix ρ(t) of size 2^4 × 2^4 having  an effective 4 × 4 non-vanishing single-particle subblock. 
Also in this case we couple a dephasing bath to the sites of the system, c.f. Eq. <ref>.
 
In this case, the excitation has a chance to develop a full coherent spread to either side of the central bond as well as cross-bond coherence. 
In other words, there are two sites to each side of the central bond, each allowing for a charge qubit. 
We focus here on the OSES of the central bond, and by the construction of section <ref> we have:

    Λ_1=   n_1^2 + 2|ξ_12|^2 n_1 n_2 + n_2^2 
    Λ_2=   n_3^2 + 2|ξ_34|^2 n_3 n_4 + n_4^2

which correspond to the purities of both subsystems. In this case, there are two sets of doubly-degenerate pairs of OSES entries, Λ_3,4 and Λ_5,6, corresponding to cross-bond coherence. If the coherence factors ξ_ij were to factorize to ξ_ij = ξ_i ξ_j as in section <ref>, only two degenerate cross-boundary generalized concurrence values would remain:


    Λ_3,4= ξ_1ξ_3 n_1 n_3 + ξ_1ξ_4 n_1 n_4 + ξ_2ξ_3 n_2 n_3 + ξ_2ξ_4 n_2 n_4 .


Due to our local coupling to the baths, we indeed find (c.f. Fig. <ref>) that Λ_5,6 are generically much smaller than the other entries and appear as a perturbative deviation.



In Fig. <ref>(a), we plot Λ_i for a pure state evolution (γ=0). We see the coherent quantum walk of the particle between the four sites reflected in the OSES. The bipartition yields an effective qubit decomposition with weights to each side of the bond: (i) Λ_1 and Λ_2 correspond in the pure case to the square of the total density to the left and right of the bond, (ii) Λ_3,4 are a generalized concurrence across the bond, and (iii) Λ_5,6≡ 0. In the lower part, the respective generalized entanglement entropy and total purity are computed from the OSES.
In Fig. <ref>(b), the γ=0.3 case is depicted. From the OSES, we see that the local purities to each side of the bond decay to
√(1/8), symbolizing a homogeneous delocalization of the particle across the system, i.e., n_i=1/4. The cross-bond coherence is dominated by two main values that decay over time. In the limit t→∞ the operator space entanglement entropy goes to (3/4)log 2 and purity saturates at 1/4, as expected.




§ CONCLUSION AND OUTLOOK


In this work we study the bipartitioning of a mixed state density matrix, and obtain the eigenspectrum of the resulting reduced operator. Our procedure is analogous to that used for pure states, where the obtained eigenspectrum is referred to as the entanglement spectrum <cit.>. We obtain an operator space entanglement spectrum (OSES) that can be related to physical properties of the system. In particular, the OSES contains information on (i) the total purity of the system, (ii) the purity of the two subsystems, as well as (iii) the cross-boundary coherence. 

For the case of a mixed state density matrix with a single excitation, the bipartitioning can be performed analytically. We consider an important limiting case of local dissipation. This local dissipation ansatz directly matches both the pure state case as well as the fully decohered density matrix scenario, where only four values are sufficient to describe the state. 
The obtained OSES in intermediate cases of partially decohered states can be thought of perturbative corrections away from this point.

Whereas in the single-particle case the C-matrix (<ref>) could be decomposed into three blocks, in the many-body case there will be many more blocks to be considered. Nonetheless, we expect similarities to persist. 
For example, the block corresponding to all particles on the left of the bipartitioning will always correspond to the purity of that subsystem. 
Hence, as matrix product decomposition allows for a straightforward numerical way of obtaining the OSES also in many-body cases, the knowledge of what the values encode (even without their exact analytical derivation), is then useful for simulating many-body mixed state dynamics. 
Unitary transformations of the basis before truncation can then help preserve these properties <cit.>.

Last, in our analysis, we rely on the fact that the particle-number of the system is conserved in order to find a meaningful block decomposition of the C-matrix in Eq. (<ref>). Hence, the OSES of systems where couplings between particle sectors occur are not covered in this work. 

We would like to thank C. Carish, R. Chitra, M. Ferguson, M. Fischer, S. Huber and C.D. White for useful discussions on this work. We acknowledge financial support from the Swiss National Science Foundation (SNSF).



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§ MPS IMPLEMENTATION


A practical way of obtaining the entanglement spectrum is through the use of a matrix product state (MPS) decomposition. In MPS simulations, the coefficients of the wavefunction in a given basis |i_1 … i_L⟩ are replaced by a product over local matrices (for extensive reviews on the topic of MPS, we point the reader to <cit.>):


    |ψ⟩_i_1… = Tr[ Γ^[1]_i_1Λ^[1]…Γ^[L]_i_LΛ^[1]] |i_1… i_L⟩.


In this representation, the diagonal matrix Λ^[m] contains the entanglement spectrum values for a bipartitioning between sites m and m+1. The dimension of the matrices (called the bond dimension) is what limits the number of independent coefficients that can be constructed, and hence controls the level of the approximation of the MPS <cit.>. 
We remark that such a representation of an arbitrary wavefunction can be exact as long as the size of the local matrices is chosen large enough <cit.>. 
In the current case of our single-particle analysis, the use of MPS methods is uncalled for. 
They provide, however, a straightforward extension to the multi-particle case.

Algorithms such as DMRG or TEBD are designed to find the set of Γ_i_n and Λ^[n] matrices for which the MPS best approximates the actual wavefunction of the system. 
Instead of considering pure states, we may construct an MPS of the vectorized density matrix |ρ⟩ of Eq. (<ref>) as in Ref. <cit.>. 
The corresponding Λ^[n] matrices then correspond to the Schmidt values of Q when bipartitioned at n, which themselves are equal to the square roots of the eigenvalues Λ_i of Q_A_1. 
These values can be directly interpreted as the OSES, with the additional remark that in standard TEBD algorithms they are typically normalized such that ∑_i Λ_i^2 = 1. 
In the above, we have taken out this normalization factor so that instead this sum equals the purity of the state. For the computation of any observables this normalization factor should of course be included.




§ SINGLE-PARTICLE ENTANGLEMENT SPECTRUM


In this Appendix we derive the expressions for the entanglement spectrum in 1D systems containing a single-particle. It serves as a step to the full proof in the case of the mixed state.

Consider a system of L sites, where the local basis states are |0⟩ and |1⟩. Let us denote the first L/2 sites as subsystem A and the last L/2 sites as subsystem B. We introduce the single particle basis states for each of the halves as


    |n⟩_A = ⊗_i=0^L/2-1 |δ_in⟩,   and   |n⟩_B = ⊗_i=L/2^L |δ_in⟩.


The full single particle basis states can then be spanned by the states 


    |n ⟩ = θ(n < L/2) |n⟩_A ⊗ |0⟩_B + θ(n ≥L/2) |0⟩_A ⊗ |n⟩_B,


where |0⟩ denotes a tensorproduct of only the |0⟩ basis states on the respective subsystems. Even though the expression in Eq. (<ref>) looks like a superposition, these basis states are all pure product states due to the Heaviside θ-function. 

The partial trace over subsystem B of the density matrix ρ = ∑_n,m |n⟩⟨ m| can now be performed as



    Tr_B ρ   = ∑_α  n  m_B⟨α | ρ | α⟩_B + _B⟨0 | ρ | 0⟩_B 
       = ∑_α  n  m c_n c_m^*  [ θ(n,m ≥L/2)   |0⟩_A ⟨0 |  ⟨α | n ⟩_B  ⟨ m | α⟩_B +  θ(n,m < L/2) |n⟩_A ⟨ m| ] 
       = ∑_α=L/2^L |c_α|^2 |0⟩_A ⟨0 | + ∑_i=0^L/2-1 |c_i|^2 |ψ⟩_A ⟨ψ |,



where in the last step, | ψ⟩ is a normalized wavefunction corresponding to ∑_i∈ A c_i |i⟩_A.
Hence the partial traced density matrix consists of two blocks, each with a single eigenvalue. These eigenvalues, λ_1 = ∑_α=L/2^L |c_α|^2 and λ_2 = ∑_i=0^L/2-1 |c_i|^2, correspond to the total density of subsystems B and A respectively. Notably, for the pure state case the partial-trace procedure generates two independent subblocks. For the mixed case, an extra subblock will appear that corresponds to possible coherences of the particle across the bipartioning.



 §.§ Relation to OSES


For a pure state, we compare the MPS coefficients Tr[ Γ^♯_i_1λ^[1]♯…Γ^♯_i_Lλ^[L]♯] of a density matrix |ρ⟩ to those obtained from |Ψ⟩⟨Ψ|:




    |Ψ⟩⟨Ψ|   = ∑_{i_1… i_L}∑_{j_1… j_L}Tr[ Γ_i_1λ^[1]…Γ_i_Lλ^[L]] Tr[ Γ^*_j_1λ^[1]…Γ^*_j_Lλ^[L]]| i_1… i_L⟩⟨ j_1… j_L| 
       = ∑_{i_1… i_L}Tr[ (∑_j_1Γ_i_1⊗Γ^*_j_1) λ^[1]⊗λ^[1]…(∑_j_LΓ_i_L⊗Γ^*_j_L) λ^[L]⊗λ^[L]] | i_1… i_L⟩⟨ j_1… j_L|



Via a basis transformation, we may relate |i_1 ⟩⟨ j_1 | to the σ matrices. This transformation will only redefine the Γ matrices however, whereas for the λ matrices we immediately find the relation λ^[n]♯ = λ^[n]⊗λ^[n]. This relationship also directly leads to the generalized entanglement entropy for a pure state being twice the entanglement entropy, since lnλ⊗λ = lnλ⊗ 1 + 1 ⊗lnλ, and hence


    Trλ⊗λlnλ⊗λ   = Tr (λlnλ) ⊗λ + λ⊗ (λlnλ) 
       = 2 Tr (λlnλ ) Trλ
       = 2 Trλlnλ .



