Channel-State duality with centers
Simon Langenscheidt Daniele Oriti Eugenia Colafranceschi









 

 ecolafra@uwo.ca

 Department of Physics and Astronomy, Western University, N6A 3K7, London ON, Canada

 Department of Physics, University of California, Santa Barbara, CA 93106, USA



 s.langenscheidt@physik.lmu.de

Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany 
 Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 München, Germany


doriti@ucm.es

  Depto. de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Plaza de las Ciencias 1, 28040 Madrid, Spain, EU 

 Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 München, Germany



    We study extensions of the mappings arising in usual Channel-State duality to the case of Hilbert spaces with a direct sum structure. This setting arises in representations of algebras with centers, which are commonly associated with constraints, and it has many physical applications from quantum many-body theory to holography and quantum gravity. 
    We establish that there is a general relationship between non-separability of the state and the isometric properties of the induced channel. 
    We also provide a generalisation of our approach to algebras of trace-class operators on infinite dimensional Hilbert spaces.


Channel-State duality with centers
    Daniele Oriti
    April 29, 2024
==================================



  

    


    





§ INTRODUCTION

The focus in this work is on mappings encoding information transport properties in quantum systems, specifically those which have an algebra of observables possessing nontrivial center, and, a representation of the same algebra on Hilbert spaces with a direct sum structure<cit.>.
These mappings can also be seen as way to characterize entanglement between subsets of quantum degrees of freedom of the same systems, in a context in which no unambiguous notion of subsystem is immediately available, due to the Hilbert space lacking a tensor product decomposition.

Let us give a simple example of the maps we consider, but in a much simpler context, i.e. 
the simplest bipartite quantum system made from 2 qubits. 
For Bell states of 2 qubits, the action of an operator on one of the qubits can be expressed through an operator on the other, more specifically its transpose:[For the |Ψ^±⟩ states, the corresponding operator is X σ^1, so multiplied by a Pauli matrix.]

    (X ⊗_B)|Φ^±⟩ = ( _A ⊗± X^t)|Φ^±⟩

This is possible due to the maximal correlation between the two qubits in this state, which implies that acting on or measuring any property on one of the two is equivalent to performing some related action or measurement on the other. 
In the following, we study generalisations of this property, but with focus on sets of operators, and on how states with this property are selected. Already in this simple case, using this state, we can define a correspondence between operators on the two qubits that we call a transport superoperator: a mapping 

    _Φ^±:(^2_A) →(^2_B)

which takes any (bounded) operator on a subsystem and turns it into an operator on its complementary subsystem. Such a mapping is state-dependent. How well subsystems are mapped onto each other by its action depends both on how similar the subsystems are (in particular, the dimensions of the subsystem or their tensor product substructure) and how strongly correlated they are in the given state.

Given any operator X ∈(^2_A), we can write it, using a notation that applies straightforwardly in the general case, as

    _Φ^±(X) = (^2_A)·_A[ (X⊗_B) |Φ^±⟩⟨Φ^±| ] ∈(^2_B).

One can verify easily in components that this does indeed give the transpose ± X^t of the operator X for |Φ^±⟩. Written like this, it becomes clear that this is a particular case of Channel-state duality<cit.>.

The duality connects sets of k-positive operators on a bipartite system = _A⊗_B, 

    ℒ_k() = { O∈(): ∀|ψ⟩
     of Schmidt rank  k  or less, ⟨ψ| O |ψ⟩≥ 0 }

with the set of k-positive linear maps from (_A) to (_B), 

    BL_k((_A),(_B)) = { T:(H_A)→(H_B): 
    _^k⊗ T  positive}

in the form

    BL_k((_A),(_B)) ≅ℒ_k()      ∀ k≤(_A).

The most commonly used example of this duality is for k=(_A), where the set of operators becomes that of (unnormalised) states on , and the set of maps is that of completely positive(CP) ones. Then, restricting to normalised states/trace-preserving(TP) maps, one recovers a correspondence between channels and states. (For example, between |Φ^±⟩ and _Φ^±).

Given a (1-)positive operator ρ∈(), we can get the channel side of the duality in the form

    _ρ (X) = _A [(X⊗_B)Σ(ρ)]

where either Σ(ρ) = ρ or Σ(ρ) = ρ^t_A with the partial transpose (-)^t_A. The two options here may be referred to, respectively, as the Jamiolkowski-Pillis and the Choi mapping. The difference between them is that, while both establish an isomorphism for k=1, only the Choi map extends this to arbitrary k.  

The converse requires a noncanonical choice. Given a k-positive linear map :(_A)→(_B), we must choose a reference maximally entangled state |ϕ⟩∈. With such a choice, we can define the associated k-positive operator ρ_∈()

    ρ_ := (𝕀_A⊗)|ϕ⟩⟨ϕ|

often referred to as the Choi matrix of the channel. The reference state must be chosen maximally entangled in order for the resulting operator to carry the same information as the channel, but split between the subsystems A,B. (If it were for example a product state, then the resulting operator would only capture one point in the image of .) Different reference states |ϕ⟩ generally lead to different →ρ_ correspondences, although there are some exceptions (see e.g. <cit.>). We will disregard these ambiguities in what follows, as we are primarily interested in the inverse of the correspondence, i.e. ρ→_ρ . Still, we will point out, when needed, which choice we make to establish the →ρ_ direction of the generalised channel-state duality. This is particularly relevant because in the case of direct-sum Hilbert spaces, the correct analogue of a maximally entangled state is not immediately obvious. We will later see that for a sensible notion of channel-state duality for the direct-sum setting, it will simply consist of a maximally entangled state per sector.

We can see that the property of Bellness, i.e. maximal quantum correlation, of |Φ^±⟩ does not come into play in making _Φ^± a channel. Rather, regardless of the quantum correlations present in the state, as long as ρ is normalised and completely positive, it induces a channel. In fact, we will demonstrate in the following that maximal correlation, translated in terms of maximal entanglement entropy, instead corresponds to an additional property of the channel: it is isometric, where isometry is defined in the Hilbert-Schmidt inner product on operators ⟨ X,Y⟩_HS = [X^† Y]:

    ⟨_Φ^± (X),_Φ^± (Y)⟩_HS,B = ⟨ X,Y⟩_HS,A .

The notion of (information) transport superoperators associated with states, and their properties with respect to this inner product, can characterise the underlying state in a useful operational fashion. 
 In particular, it is a way to characterize quantum correlations  beyond the case where the subsystems have identical Hilbert spaces, or when the total Hilbert space does not factorise at all in terms of subsystem Hilbert spaces.


Superoperators of this kind have found use in tensor network models of holography<cit.>, where the systems A and B are seen as, respectively, the bulk and boundary of a spatially compact system, and state or operator reconstruction on boundaries are used as hallmarks of holographic behaviour. Indeed, this is a case in which the relevant Hilbert spaces are not, in general, composed of factors of the same size when they factorize over A and B, or do not factorize at all <cit.>. Motivated by these models, we study  transport superoperators on bi- and tripartite systems subject to constraints which prevent the total Hilbert space to fully factorize, enforcing a structure of the form

    = ⊕_E _I,E⊗_O,E.

where labels I and O refer to the possible interpretation of the two subsystems as, respectively, "input" and "output", and where E is the value of the constraint charge.
In particular, we ask the questions:   
Given such a setup, what is a sensible notion of complementary subsystems, and of a transport operator  from a subsystem to a complementary one? Further, Is there an associated version of channel-state duality for this direct sum setting?


This setting is, as we stressed, quite generic. Consider two systems with a given series of eigenenergies E, each with eigenspace _E, and fix their total energy to ℰ. The corresponding Hilbert space is
    
    _ℰ = ⊕_E+E'=ℰ_E⊗_E'.

    So, we may generically see such direct sum spaces as indicating the imposition of a constraint in some larger (possibly factorized) Hilbert space.

    To give a couple more examples, we can first consider a spin system on a lattice Λ, described by a qubit on each site v ∈Λ. If the dynamics is such that the total angular z-momentum J = ∑_v σ^z_v is conserved, it makes sense to split the Hilbert space into its eigenspaces
    
    _free = ⊗_v∈Λ_v^2 ≅⊕_J _J

    In this case, the sectors are labeled by the total spin J (whose range depends on the size of the lattice) and do not have a direct tensor product factorisation due to the global constraint, unlike the total (free) Hilbert space. If, however, the lattice is split into complements Λ_L|R, we can equally write them as
    
    _J ≅⊕_J_L+J_R = J_Λ_L,J_L⊗_Λ_R,J_R

    in direct analogy to the energy case before. In this case, then, the sector label E is given by the total spin J of the subregions.
    Contrast this with a case of local constraints on each site, as in ℤ_2 lattice gauge theory, with qubit on each link e∈Λ. The system is subject to the Gauss constraints
    
    G_v = ∏_e∩ vσ^x_e !=_v

    which are the lattice counterpart of the continuum Gauss constraint of gauge field theory and which impose a ℤ_2 gauge invariance of states. The resulting gauge-invariant Hilbert space _Λ does not have immediate spatial factorisation properties in terms of lattice sites, like in the example before. However, given a split of the lattice into complements Λ_L|R along a set of links S, one can still give a direct-sum tensor split
    
    _Λ≅⊕_E_Λ_L,E⊗_Λ_R,E

    in which the sectors are labeled by the set of X_e-eigenvalues on S,
    E:={s_e∈ℤ_2| e∈ S}. Each of the tensor factors is simply the gauge invariant Hilbert space on the respective sublattice, subject to the boundary condition X_e = s_e  ∀  e∈ S. Let us stress that  the same absence of local factorization is true in the continuum case and holds for any (field) system with gauge invariance<cit.>.

    These examples show also that as long as an a-priori tensor factorisation is present before imposing the constraints, one can keep said structure on the constrained subspace sector-by-sector in a direct-sum decomposition over the conserved charges.
    

    The lack of tensor factorisation makes unusable several technical and conceptual tools. The operator algebra associated with the full system no longer factorises nicely, and a clear notion of subsystems defined in terms of Hilbert subspaces also disappears. Further, the standard notion of entanglement, based on the deviation from a product state ρ_A⊗ρ_B on _A⊗_B, is also no longer valid<cit.>, as no notion of a product state is available to compare to.

    We may of course choose to embed the direct sum in some bipartite system, with a factorized Hilbert space, but a choice of such embedding is highly nonunique and possibly unphysical. In fact, such a choice can be understood as equivalent to a choice of algebraic subsystems for A,B, in the sense of choosing subalgebras _A|B⊂=()<cit.>. We deal with all of these issues in the following sections. 

    The structure of this paper is as follows. In section <ref>, we review the problem and introduce relevant concepts (e.g. algebraic subsystems, extension and partial trace maps). In section <ref>, we study the case of transport operators on tripartite systems with no direct sum, and then extend the discussion to direct sums of tripartite systems. Section <ref> presents the generalization of the Channel-State duality to the direct sum case. Section <ref> discusses possible extensions of these notions to the infinite dimensional case.






  §.§.§ Intro v2


In this note, we wish to study a set of questions related to information transport and channel-state duality in systems whose Hilbert spaces do not admit a straightforward tensor product factorisation, but rather as direct sums of such factorised Hilbert spaces, of the form

    = ⊕_E _I,E⊗_O,E.

In particular, we would like to ask the questions:   
Given such a setup, what is a sensible notion of complementary subsystems, and of a transport operator  from a subsystem to a complementary one? Further, Is there an associated version of channel-state duality for this direct sum setting?

This setting is, in fact, quite generic: Consider having two systems with given series of eigenenergies E, each with eigenspace _E, and fix their total energy to ℰ. Then the corresponding Hilbert space is
    
    _ℰ = ⊕_E_E⊗_ℰ-E.

    So we may generically see such direct sum spaces as indicating the imposition of a constraint in some larger Hilbert space.

    The lack of tensor factorisation spoils several technical and conceptual tools: The operator algebra associated to the full system does not longer factorise nicely, and a clear notion of subsystems through Hilbert subspaces also disappears. Further, the standard notion of entanglement, defining entangled states through deviation from a product state ρ_I⊗ρ_O on _I⊗_O is also no longer valid, as no notion of a product state is available to compare to.

    We may of course choose to embed the direct sum in some bipartite system, but a choice of such embedding is highly nonunique and possibly unphysical. In fact, such a choice can be understood as equivalent to a choice of algebraic subsystems for I,O, in the sense of choosing subalgebras _I|O⊂=B(). 

    
    We also wish to elaborate on the question of channel-state duality in this case. The usual statement refers to the isomorphism between the sets of k-positive operators ρ on bipartite systems _I⊗_O, and k-positive linear maps _ρ : _I →_O (so quantum channels if k≥() and ρ is normalised), through the correspondence
    
    : ℒ_k()    ⟶ BL_k(BL(_I),BL(_O))
    
            ρ   ⟼_ρ,  _ρ(X) = _I[X ρ^t_I]

    with the partial transpose ρ^t_I on the 'input' system _I. We would like to extend this duality as far as possible into the direct sum setting introduced above. This is useful, for example, in tensor network toy models of holography.
    


  §.§.§ Intro v1

    
    The purpose of this note is to elaborate on a solution to a peculiar problem:
    Given some pure state |ψ⟩ of a system equipped with a bipartition ≅_I⊗_O into input(I) and output(O), we can create an antilinear mapping _I∋|I⟩↦⟨ I |ψ⟩∈_O turning an input into an output. Turning this into a mapping on operators, we can make a linear map by using a partial transpose:
    
    X_I ↦_I[X_I ρ^t_I] = (X_I)

    This kind of superoperator has found usage in tensor network models of holography, where ρ represents a tensor network state with open legs, which form the input and outputs. As a mapping on operator spaces, we may interpret it as acting on the full system with the operator X_I, which results in an effective change of the output system that is represented by (X_I). The way this connects to holography is that if the mapping  is suitably isometric, one will be able to relate general statements of the input to statements about the output. Choosing the input to be a 'bulk' and the output a 'boundary', this gives a simple, concrete and powerful version of holography.

    An issue arises in the case where no simple factorisation is available. This is, in fact, generic: Seeing for example the bulk and boundary as systems coupled by currents J and boundary conditions, the hilbert space of the combination has the form
    
    ≅⊕_J _I,J⊗_O,J.

    We can also arrive at such direct sum structures by applying constraints to a bipartite system that couple the two parts: Consider a system of two Qudits carrying a representation V_j of SU(2) each. It carries a joint generator J = J_1⊗_2+_1⊗ J_2, with Casimir J^2. Requiring the constraint J^2 = 0 will project the system into a Hilbert space that is no longer bipartite, but rather a direct sum of irreps of SU(2) again.

    
    Inv(V_j⊗ V_j) ≅⊕_m,n∈ℐ(j)  V_m⊗ V_n

    In fact, as an even simpler example, consider just having two systems with given series of eigenenergies E_n, and fix their total energy to ℰ. Then the corresponding Hilbert space is
    
    _ℰ = ⊕_n_n⊗_ℰ-n.

    So we may generically see such direct sum spaces as indicating the imposition of a constraint in some larger Hilbert space.

    This spoils several technical and conceptual tools: The operator algebra associated to the full system does not longer factorise nicely, and a clear notion of a single bulk or boundary algebra also disappears (by virtue of several possible boundary conditions). Further, the standard notion of entanglement also is no longer valid, as no notion of a product state is available to compare to.

    We may of course choose to embed the direct sum in some bipartite system, but a choice of such embedding is highly nonunique and possibly unphysical. In fact, such a choice can be understood as equivalent to a choice of algebraic subsystems for I,O, in the sense of choosing subalgebras _I|O⊂=B(). 

    We wish to address the following question: Given such a setup, what is a sensible notion of transport operator ? Further, Is there an associated version of channel-state duality for this direct sum setting?




    fsdfs




§ THE SETTING

    Let us start from the algebra 𝒜 = () of operators[We will for simplicity work in this section with bounded operators on finite dimensional Hilbert spaces.] acting on a Hilbert space  of our choice. Then, select as subsystems two subalgebras 𝒜_I,𝒜_O, to be later seen as inputs and outputs of the transport superoperator. These are understood as operations or observables of the subsystems in consideration. These two algebras are not necessarily a partition of 𝒜, where a (bi)partition would be rather specified as the output being the algebraic complement (the commutant) of the input,
    
    (𝒜_I)' = 𝒜_O      (𝒜_O)' = 𝒜_I,

    so that the operators commuting with inputs are precisely the output operators. This structure, when present, captures the general properties we may expect of any operational definition of splitting of the system into two parts. [ Notice that, in particular, this is a pendent of properties in local QFT, as formalised by Haag duality, where subsystems are identified and distinguished by their localization on the spacetime manifold.]
    We have here labeled the two parts of the bipartition 'input' and 'output' in analogy to a quantum channel, but the labels could refer to any form of separation of subsystems (it could equally well be 'Left/Right', 'Inside/Outside', 'System/Measurement apparatus' or 'Alice/Bob', etc). For more about this algebraic perspective on subsystems, we recommend the review<cit.>.

    In general, we have that 𝒜_I ∪𝒜_O ≠𝒜, and, more importantly, we may have a nontrivial center 𝒵 = 𝒜_I∩𝒜_O, which consists of operators which commute with all others. If the center is trivial, i.e. consisting only of multiples of the identity λ, we have that 𝒜_I ∪𝒜_O factorises into _I⊗_O. Its representations, then, also factorise into tensor products of Hilbert spaces. This is the simplest setting. In this special case, we can define subsystems as Hilbert subspaces in the tensor factorisation and extend subsystem operators uniquely: X_I ↦ X_I ⊗_O[We assume here that the algebras contain a unit, so an identity operator.]; moreover, the  entanglement for pure states is well-defined and can be quantified, e.g. through von Neumann entropy. These properties do not generalise to the case with nontrivial center.<cit.> 

    In the case of nontrivial center, we have a commutative subalgebra of 𝒜. In fact, because the input and output systems commute with this center, we may seperate representations of the algebra into sectors labeled by eigenvalues of operators in 𝒵. This gives rise to the characteristic structure of the Hilbert space
    
    = ⊕_E_E =  ⊕_E_I,E⊗_O,E

    in which the central operators have been diagonalised with eigenvalues given by E. A very concrete way to present this sort of algebra is through block matrices, where each block corresponds to a sector E in the above decomposition. Lacking 
    a clear notion of separable states, the simple notion of entanglement as non-product states does not hold up, but,  of course, this is not the end of the story, and for example entropies may still be calculated<cit.>. 

    Still, the particular structure of the algebras and Hilbert spaces considered above indicates a natural way forward. 
    
    We still have a notion of subsystem in each sector due to factorisation, given by operators on input and output Hilbert spaces, in each of them. That is, there are associated subalgebras
    
    ⊕_E(_I,E)⊗𝕀_O,E            ⊕_E𝕀_I,E⊗(_O,E)

    of operators on the individual subsystems for each sector. These are a consistent definition of complementary subsystems in the case of the Hilbert space structure above - they form subalgebras, have the correct commutant relation and the right center given by sums of identity operators in each sector, which represent the diagonalised operators from 𝒵. 

    They also have unique extensions from the image of naive partial traces to the full algebra, and are the largest set to have this property. This means that, if we take the naive partial trace of an operator X = ∑_E,F X_E,F∈() (splitting into blocks over the different sectors), which is given by
    
    _O [X] = ∑_E__O,E[X] = ∑_E__O,E[X_E,E]

    then we only keep the diagonal blocks in X, because there is no notion of a trace on the non-diagonal blocks. The reason for this is that, while there is a natural 'evaluation' or trace map on V^∗⊗ V for any vector space V, this is not true for 
    
    (_E,_F) ≅_E^∗⊗_F

    when E≠ F (or any vector spaces which are not equal). 
    Without providing such maps by hand (which amounts to a different choice of partial trace map), we can only arrive at operators of the above form by reducing to subsystems. 
    Similarly, if we want to extend some abstract subsystem operator, for example given by U = ∑_E,F U_E,F∈(⊕_E _I,E) on the input subsystem, to the full system, then we would naively do so by extending it with 'identity operators' _O;E,F
    
    i_I(U) := ∑_E,F U_E,F⊗_O;E,F,

    which however are only unambiguously defined, again, for E=F. This is simply the statement that the off-diagonal blocks will not have a clear notion of a diagonal, and certainly not of an 'identity', without prescribing it by hand. 
    In this sense, the choice of subsystem we indicate here is the only unambiguous one - for others, we would need to prescribe by hand extra data for defining any extension and restriction. 
To summarize, the natural way forward is to use the unique unambiguous definition of subsystems in each sector and extend it to the sum over sectors, to obtain information channels and a generalised channel/state duality for the whole system. What we do in the following is to show that this way forward can indeed be pursued, that it leads to a well-defined result, and that the resulting construction is, in the sense we clarified, the only natural one.
    
    


§ MAPPINGS ON ALGEBRAS WITH NONTRIVIAL CENTERS


        In order to define generalisations of the transport superoperator from before, we can follow the path indicated above, using a few ingredients:
    
        
  * Choices of input and output systems _I|O, e.g. _I|O = ⊕_E (_I|O,E),
        
  * Identifications/Injections i_I|O:_I|O↪, whose images we identify as the complementary subsystems _I|O,
        
  * Conjugate partial trace maps P_I|O:→_I|O that reduce an operator on the full system to a subsystem,
        
  * A mapping Σ: →, usually  related to a density matrix ρ, e.g. Σ(X) =  X ρ^t_I.
    
    
    We will now go into detail about this construction in the case of a trivial center at first, which corresponds to a system with simple tensor product factorisation in its Hilbert space. This will illustrate that the notion of transport operators is useful also in this simple case, and already shows the main behaviour of their properties, namely that there exists a 2-out-of-3 implication for purity of the state ρ, trace preservation and isometry of the mapping associated to it[This 2-out-of-3 property appears to be due to the relatively rigid way entanglement shows itself in pure states, as manifested through there being a (mostly unique) measure of entanglement for pure states, which is not the case for mixed states.].
    After that, we show the generalisation to the case with multiple blocks/sectors or nontrivial center, and find that the same thing holds, but the conditions split per sector.
    
            
        

 §.§ Trivial center

            For starters, consider the C*-algebra = () of bounded linear operators on a finite dimensional Hilbert space , equipped with the Hilbert-Schmidt inner product., together with a tripartition of ≅_I⊗_O⊗_B into input, output and background spaces. In concrete cases, this induces subsystems _I|O|B and associated extension 
            
    i_I|O|B: _I|O|B↪

            and partial trace maps
            
    P_I|O|B:𝒜→_I|O|B

            which are, respectively, injective and surjective and, up to normalisation, inverses of each other. As before, we identify _I|O|B = Im(i_I|O|B).

            We prescribe usually the subsystems as the obvious choice of subalgebras of ():
            
    _I = B(_I)       _O = B(_O)       _B = B(_B).

            In this concrete case, they are simply given by
            
    i_I(X) = X ⊗_BO     P_I(X) = _BO[X]

            and will from now on refer to  and  interchangeably where there is no risk of confusion.

            In fact, the partial trace is best defined in terms of an adjoint to an extension.
             For example, for the bipartite case, it is the defining property of the partial trace that
            
    X_I⊗_O,Y_HS = X_I,_O[Y]_HS,I
    ∀  X_I∈_I,   Y∈_O

            Using this relation, we can define more general partial trace and extension maps which share the same behaviour.
            
            These extension and partial trace maps may be used to create various kinds of transport maps from the input to the output system. In general, such a mapping will take the form
            
    _Σ :_I→_O
    
                        X ↦ P_O [ Σ( i_I(X) ) ]

            where Σ: → is some linear mapping that twists the trivial extension-restriction operation. For Σ=𝕀_, this gives a completely depolarising channel up to normalisation. We are interested in a twisting by multiplication with a density operator ρ of the full system. These have the interpretation of first preparing the system in the state given by a density matrix ρ, acting on a subsystem I with some operator and then looking at the results of that action in subsystem O. This gives an effective induced operator in O, and therefore provides a notion of 'operator transport' similar to the 2-qubit case that we discussed in the introduction. 
            
            The first concrete case we are interested in is the choice Σ(X) = K· X ρ with some positive constant K, which produces the Jamiolkowski-Pillis mapping
            
    _ρ (X)  = K   P_O [ i_I(X)ρ ].

            For the purpose of generality, we do not assume the density matrix has been trace-normalised and keep its appearance explicit in the following. This mapping is characterised by the relation in Hilbert-Schmidt inner products
            
    _ρ(X),Y_O    = K i_I(X)ρ,i_O(Y)
       = X,K  P_I[ρ^† i_O(Y)]_I.

            The middle form here gives a clear interpretation of the inner products: we extend both X from the input and Y from the output to the full system, then take their inner product with the density matrix in between. Due to the cyclicity of the full trace on , this is the same as the expression
            
    K_[i_O(Y)i_I(X)^†ρ ]    = K ρ, i_O(Y)i_I(X)^†
       = Ki_O(Y)i_I(X)^†_ρ

            which is just, up to scaling, the expectation value of the operator given by X^† on the input subsystem and by Y on the output subsystem, in the state ρ.

            
            
            We can, of course, change the twisting map Σ to a different operation, but there is no uniquely compelling alternative. The perhaps most obvious alternative comes from a seemingly innocuous difference:
            the Choi mapping 
            
    _ρ (X)  = K   P_O [ i_I(X)ρ^t_I ].

            uses the partial transpose of the state with respect to the subspace _I. 
            This may be defined canonically using a series of isomorphisms and the swap operator:
            
    vec:    𝐁() →⊗^∗
    ℛ:    ⊗^∗→⊗
    _I:    _I ⊗_O ⊗_I ⊗_O →_I ⊗_O ⊗_I ⊗_O    .

            Letting the first be the canonical identification of linear operators with vectors, and the second the Riesz isomorphism, we can then define 
            
    X^t_I := vec^-1( ℛ^-1(_Iℛ(vec(X))) )    .

            The Choi mapping has a number of more favorable properties compared to the Jamiolkowski-Pillis mapping. In particular, unlike the latter, the Choi mapping provides an isomorphism between the sets of CP maps _I→_O and of (unnormalised) states on _I⊗_O. Also, for bipartite systems, the Choi mapping for a pure state |ϕ⟩⟨ϕ| can always be written as
            
    _ρ (X) = Φ X Φ^†

            where the map Φ :_I→_O has components ⟨o|Φ|ι⟩ = ι,o|ϕ. This can be seen through 
            
    ⟨o|_ρ (X)|o⟩   = ∑_i,i⟨i|X|i⟩⟨i,o| (|ϕ⟩⟨ϕ|)^t_I|i,o⟩
    
                       = ∑_i,i⟨i|X|i⟩⟨o|Φ|i⟩⟨i|Φ^†|o⟩.

            

  §.§.§ Trace and isometry conditions

            We will ask two important questions about this mapping: first, when it is a channel, and second, when it is an isometry (in the Hilbert-Schmidt sense).

            The former is a standard question, but the latter has an interesting new aspect to it: if the mapping is isometric, we can see the system itself as providing an 'information funnel' from input to output (this, in turn, has been used as a proxy of holographic behaviour in the literature).

            First, if ρ≥ 0, it is clear that the mapping is completely positive. Trace preservation amounts to 
            
    K  P_I[ρ] = _I
     K = (_I)/[ρ],  P_I[ρ/[ρ]] = _I/(_I) .

            In other words, being a quantum channel fixes the overall normalisation of the mapping and also puts the requirement on the reduced input state that it must be flat.

            Isometry may be expressed easily as well, using swap operators[These simply take two factors in a tensor product and swap them, |a⟩⊗|b⟩ = |b⟩⊗|a⟩.] on the Hilbert spaces:
            
    (X),(X) _O 
       = _O[(X)^†,(Y)] 
       = _O^⊗ 2[((X)^†⊗(Y))_O]
    
                       =K^2 _^⊗ 2[(X^†⊗ Y)(ρ⊗ρ)_O]
    
                       = K^2 _I^⊗ 2[(X^†⊗ Y) _OB^⊗ 2[(ρ⊗ρ)_O] ]

            which translates into the requirement
            
    K^2 _O^⊗ 2[(ρ_IO⊗ρ_IO)_O] = _I.

            This in turn implies the two equalities (the second from multiplying the isometry condition by _I)
            
    K^2 [ρ]^2 e^-S_2(ρ_O) = D_I
    
                        K^2 [ρ]^2 e^-S_2(ρ_IO) = D_I^2

            expressed using the second Rényi entropy
            
    e^-S_2(ρ) = [ρ^2]/[ρ]^2   .

            Combining these two leads to the general, normalisation-independent requirement
            
    e^-S_2(ρ_IO)+S_2(ρ_O) = D_I    .

            The form of the exponent suggests looking for a subsystem inequality for Rényi entropies - however, it is known that such inequalities do not exist<cit.>[However, we might still use measured Rényi entropies and mutual information<cit.>. These have a known expression and satisfy nice properties as an analogue of the von Neumann mutual information.]. Still, in general these conditions fix K^2[ρ]^2 to be in the interval 
            
    [D_I,D_I D_O]∩ [D^2_I,D^3_I D_O] = [D_I^2,D_I D_O].

            The minimum value K = D_I/[ρ] is part of the trace condition above, while the maximum is K = D_I/[ρ]√(D_O/D_I) is incompatible with being trace-preserving, in general. However, we should not preemptively choose the former value. Indeed, if we do, then the above conditions turn into
            
    D^2_I e^-S_2(ρ_O) = D_I  S_2(ρ_O) = log(D_I) 
    
                        D^2_I e^-S_2(ρ_IO) = D_I^2  S_2(ρ_IO) = 0

            which means that the reduced state ρ_IO must be pure, and therefore the state must factorise ρ=ρ_IO⊗ρ_B, and the reduced input (and output) state must be maximally mixed S_2(ρ_O)=S_2(ρ_I) = log(D_I). 

            In other words, isometry (ISOM) and trace preservation (TP) imply purity of the state ρ_IO (PURE).
            
    ISOM∧TPPURE

            
            However, this setup is too restrictive. On the other hand, the maximal value implies
            
    e^-S_2(ρ_O) = 1/D_O S_2(ρ_O)= log(D_O)
    
                        e^-S_2(ρ_IO) = D_I/D_O S_2(ρ_IO) = log(D_O) - log(D_I),

            so once again the state reduced to the output system is maximally mixed. Now, however, the reduced state ρ_IO no longer needs to be pure. So quite intriguingly, the mapping we propose cannot be a quantum channel and an isometry in general, unless the state used factorises in a nice way.

            Additionally, we may ask when the mapping we defined is unital. This gives an input-output swapped version of the trace preservation condition:
            
    Kρ_O =_O      K = D_O/[ρ]  , ρ_O/[ρ] = _O/D_O

            and we can again check when this is compatible with the mapping being isometric: we need D^2_O ∈ [D_I^2,D_I D_O], but when D_I ≤ D_O this is only the case iff D_I = D_O. In that case, isometries are unitaries, and trace preservation and unitality are equivalent. Additionally, K is fixed uniquely to the value K = D_I/[ρ] = D_O/[ρ] and there is no other option than ρ_IO being pure. 

            We can frame this simple result as follows. If we fix a state ρ, then select manually input and output systems such that they are of equal size, then there is no way to have an isometry between the operator spaces from the Jamiolkowski-Pillis mapping if the state does not factorise into pure states.

            Even in this simple setting, operator transport has clear limitations in the multipartite case. The 'environment' or 'bath' B generically makes it impossible for the mappings above to be isometric.

            
            To specialise this discussion, let us assume that the state ρ_IO = |ϕ⟩⟨ϕ| is pure and we use either the Choi or Jamiolkowski-Pillis mapping (the requirements, for both of them, turn out to be the same). Then the isometry condition is
            
    |K|^2 _O^2[ρ_IO^⊗ 2_O] = |K|^2 _O[ρ_IO]^⊗ 2_I = _I    .

            This is simply the requirement of the reduced input state being flat:
            
    ρ_I = _I/|K| = _I/D_I.

            This is precisely the condition we found before for trace preservation. So for pure states ρ_IO, trace preservation and isometry are in fact equivalent:
            
    PURE∧TPISOM
    PURE∧ISOMTP   .

            This, together with the implication we found before, shows a 2-out-of-3 property of the Jamiolkowski-Pillis or Choi mappings. Phrased in terms of entanglement properties, we may say that for pure states, isometry holds precisely when the induced transport is trace preserving, or equivalently when the two subsystems are maximally entangled. 

            
            We can also study the opposite case and ask what happens when the state ρ_IO is seperable. In that scenario, we find (assuming a normalised ρ) that
            
    _ρ(X),_ρ(Y)_O/X_ρ_IY_ρ_I = C_seperable = K^2 e^-S_2(ρ_O)    ,

            which is, importantly, independent of X and Y. In simple terms, this is just the situation in which the inner product in O factorises between X and Y.  Of course, the same thing happens in the maximally mixed case - if ρ_IO = /D_I D_O in the standard setup, then the above formula holds for maximal entropy of ρ_O.
            This suggests that, while the isometry condition indeed seems to favour entangled states, it also disfavours mixed states, generally.

            We note that already  in <cit.>, it had been shown that even for mixed states one can link separability with properties of the induced transport superoperator. It would be interesting to extend these considerations to our more general setting, but we leave this for future work.

            

  §.§.§ An example

            We illustrate, for concreteness, the Choi mapping on the classic Werner states on 2 qubits
            
    ρ = p Ψ^- + (1-p) /4

            where the Bell state Ψ^- = |ψ^-⟩⟨ψ^-| is maximally entangled, and so the mapping is expected to give isometry. The Choi map (here for K=2) is linear in the state ρ, and, for the Bell state alone, induces a conjugation by the 2nd Pauli matrix:
            
    _Ψ^-(X) = σ_2 X σ_2

            Therefore, the Werner states induce the superoperator
            
    _ρ (X) = p σ_2 X σ_2 + (1-p)_O/2   .

            The isometry condition can therefore be checked directly:
            
    _ρ (X) , _ρ (Y)  = p^2 X,Y_I + 1-p^2/2_I[X]_I[Y]

            and we can see that isometry only holds for the pure case p=1; it is not a consequence of entanglement by itself, but rather of entanglement together with purity. 
            
            

  §.§.§ Isometry degree of the average state


            We can achieve a generic understanding of the tripartite case by employing Page-type averaging arguments<cit.>. We can in principle just consider a random pure state ρ= |ψ⟩⟨ψ| of the full system and compute quantities according to the unitary average -_U, where states are given as U|ψ_ref⟩. Then we can check the isometry condition in the average as well:
            
    |K|^2 _O^⊗ 2[(ρ_IO⊗ρ_IO)_O]_U
     
                    =
                    |K|^2 _OB^⊗ 2[ρ^⊗ 2_U _O]    ,

            and use the result (found via Schur's theorem for the permutation group on the two copies of the system):
            
    ρ^⊗ 2_U    = ∫_𝒰(D) dμ_Haar(U)  (U^†ρ U)^⊗ 2
       = _⊗+_⊗/D(D+1)   .

            This means that on average the left side of the isometry condition becomes
            
    |K|^2 D_O^2 D_B /D(D+1) (_I + D_B/D_O_⊗)

            which shows two conditions which must hold on average:
            
    r= D_B/D_O << 1
                        
                    |K|^2 = D(D+1)/ D_O^2 D_B ≈ r D_I^2 D_O    .

            So we can see already that only small environments allow for the average state to still give rise to isometries. This is unsurprising: in that scenario, a typical reduced state ρ_IO is close to being pure. We can again take traces of this expression with _I and _I to find 
            
    D_I !=1    .

            We interpret this as follows. In order to have a system whose average pure state gives rise to an isometric map, the system sizes must follow the above conditions. Of course, if we restrict the average to a smaller class of states, we might find more lenient conditions. For example, we may only work with states of the form
            
    ρ = Π^†|ψ⟩⟨ψ|Π

            with some projector Π:_B→ P⊂_B to a subspace of the environment, suitably extended to the full system. This essentially restricts the environment into a class of states. Then, the above calculation goes through as before, but replacing D_B by D_B = (P). Such a projection can then make the first condition superfluous by choosing P to be small enough. So better knowledge of the state of the environment makes the effective Choi map more isometric. If we also want to check for trace preservation in this setting, we get the condition
            
    K D_O D_B/D = 1 ⟷ K_TP = D_I 
     K_Isom = √(D_B) D_I    .

            So we have in fact that once again, choosing a small P makes trace preservation and isometry nearly equivalent. So by either making the environment small, or choosing its coupling to the system to be small, or by assuming strong knowledge of the system (for example assuming it to be in a pure state, making D_B = 1), we can find isometries in the tripartite case. 

            
            

 §.§ Nontrivial center

                Let us first discuss the bipartite case.
                Consider an algebra  with representation space , pre-selected subsystems _I|O, such that (_I)' = _O, but with nontrivial center[This assumes an extension map and associated partial trace operation have been chosen.] 𝒵=_I ∩_O. The case of interest to us is that of Hilbert spaces of the form
                
    = ⊕_E _I,E⊗_O,E

                with the full algebra = (), and subsystem algebras _I|O  =⊕_E (_I|O,E). In this sector-split Hilbert space setting, extension and partial trace operations are defined sector-wise.
                
    i_I(X) = ∑_E X_E ⊗_O_E     P_I[X] = ∑_E _O_E[X_E]    ,

                which are adjoints to each other under the Hilbert-Schmidt scalar products on the algebras. We also identify _I|O, the true subsystems, as the images of ℬ_I|O under the extension maps. In practical terms, any operator that may be reached by partial tracing needs to be in ℬ_I|O. Similarly, any operator that is obtained from extending one in ℬ_I|O must be in _I|O. 


                We may once again define a Jamiolkowski-Pillis (or Choi with partial transpose) mapping via the property <ref>
                which is also fulfilled in the case of trivial center. We allow ourselves to rescale this mapping again by a constant K:
                
    _ρ (X)     = K   P_O [ i_I(X)ρ ] 
       = ∑_E K c_E  _I_E [ (X_E,E⊗_O_E)ρ_E,E ]    ,

                where we decompose the state as
                
    ρ = ∑_E,E√(c_E c_E)ρ_E,E   ,

                with [ρ_E,E] = δ_E,E and c_E = _E[ρ] ≥ 0, ∑_E c_E = 1. 

                
                For the tripartite case, we can proceed analogously. We assume: 1) a Hilbert space structure
                
    = ∑_E_I,E⊗_O,E⊗_B,E   ;

                2) input/output algebras
                
    _I = ⊕_E B(_I,E)     _O = ⊕_E B(_O,E)    ;

                3) the mapping
                
    _ρ (X)     = K   P_O [ i_I(X)ρ ] 
       = ∑_E K c_E  _I_E B_E [ (X_E,E⊗_O_EB_E)ρ_E,E ]    ,

                and investigate about trace preservation and isometry.

                Trace preservation is just the property
                
    _B_EO_E[ρ_E,E] = _I_E/D_I_E             c_E = D_I_E/K   .

                

                Identifying isometry is made easier by the aforementioned relation <ref>, which entails that the adjoint to _ρ is (unsurprisingly) given by
                
    ^∗_ρ (X) = P_I(ρ^† i_O(X))    .

                Letting σ_E = _B_E[ρ] and rescaling our definitions by K, we obtain the isometry condition
                
    (^∗∘)(X) 
       = ∑_E |K|^2 _O_E[(_I_E⊗_I_E[(X_E⊗_O_E)σ_E])σ_E^†]
       = 
                        ∑_E |K|^2∑_a,b,c,db|X_E,E|a·
       ·|c⟩⟨d|·_O_E[
                        a|σ_E|bc|σ_E|d
                        ]
       != X = ∑_E
                        ∑_a,b,c,db|X_E,E|a·|c⟩⟨d|·δ_a,dδ_b,c   ,

                where we choose some orthonormal basis of I_E labeled by a,b,c,d in the last line. This leads directly to the condition
                
    |K|^2  _O^2_E[σ^⊗ 2_E _O_E] = _I_E,

                as before. Notice however now that we use the same prefactor for all sectors, meaning the requirement hinges more on the properties of the reduced states σ_E. Therefore the sector-wise condition
                
    e^-S_2(σ_E)+S_2( (σ_E)_O ) = D_I_E

                must hold as well as
                
    Kc_E/D_I_E e^-S_2(σ_E) = 1.

                Again, we can see that trace preservation together with isometry necessitates that the reduced state σ_E is pure.


                Additionally, as before, if we assume the state σ_E = |ϕ_E⟩⟨ϕ_E| to be pure, we reduce the isometry condition to
                
    (σ_E)_I = _I_E/∑_F D_I_F     c_E = D_I_E/∑_F D_I_F   ,

                which comes from the condition |K| = ∑_E D_I_E. This is, again, just the trace preservation condition. So also in the setting with nontrivial center, demanding purity makes TP and isometry equivalent. So once again, the three conditions give a 2-out-of-3 implication.
                
        


§ CHANNEL-STATE DUALITY

    Here, we wish to extend the usual statement of channel-state duality to the case of systems with a bipartition (_I)'=_O, but with nontrivial center in general. Therefore, we will only consider certain types of channels and states to be meaningful:
    
        
  * We only consider channels :_I→_O between the subsystems. In terms of Hilbert space representations, this implies that the channel is block diagonal in the sectors E.
        
  * We consider only operators ('states') which lie in the algebra generated by the union _I∪_O. This is the set of operators generated from operations on the subsystems. 
    
    These are meaningful as they preserve the intuition of channel-state duality making a statement about channels between subsystems, which correspond to states of the joint subsystems.
    The strongest kind of statement we can aim for is an isometric bijection
    
    BL_k(_I,_O) ≅ "ℒ_k ()" ⊆   ,

    identifying the k-positive linear maps between input and output algebras (the channel side) with a to-be-identified set of 'k-positive operators' given as a subset of the full algebra  (the quotation marks indicate this lack of precise identification at this stage).

    The easiest motivating example is given by the Choi map, which decomposes
    
    _ρ = ∑_E K c_E _ρ_E,E   .

    In each sector, the map _ρ_E,E is a CPTP map in (_I,E,_O,E) and therefore has a dual density matrix given by ρ_E,E. We can therefore already say that at least the usual set of k-positive operators ℒ_k (_E) of each sector is possible on the right hand side:
    
    ⊕_E ℒ_k (_E) ⊆ "ℒ_k ()".

    Also, using the same kind of Choi matrix τ as for usual channel-state-duality, we can restrict the tentative "ℒ_k ()" further from above:
    Given an element X = ∑_E,FX_E,F∈ with X_E,F = ∑_k_E,F a_k_E,F⊗ b_k_E,F, we can try to compute
    
    τ_(X) =  ∑_E,F∑_k_E,F_O_E[(a_k_E,F) b_k_E,F^t],

    but it's clear that (a_k_E,F) is ill-defined unless E=F. Therefore, we must restrict to the sector-diagonal operators X = ∑_E X_E,E, where
    
    τ_(X) = ∑_E τ_,E(X_E,E).

    So with both arguments, we know that the most we can expect is
    
    ℒ_k () ≅⊕_E ℒ_k (_E)    .

    However,  the maps we consider do not mix sectors, so the statement  holds by channel-state duality in the finite dimensional, bipartite case in each sector:
    
    BL_k(_I,_O) ≅⊕_E ℒ_k (_E)    .

    Therefore, the direct sum case of the duality reduces directly to the standard one. This is certainly not surprising, but it is worth stressing the ingredients that go into this statement. In principle, one might expect a correspondence on a much larger set of mappings on the left and operators on the right. However, under the assumptions we stated, and as our analysis of subalgebras has shown, the sets _I|O must be chosen such that the duality becomes a per-sector statement. So in a sense, the nontrivial part lies in the selection of subsystems and the subsequent identification of the correspondence ρ↔_ρ. 
    We can also highlight the ambiguities in this correspondence: As in the single-sector case, we need to specify reference maximally entangled state for the channel → state direction; in the current setting, this freedom is extended to a maximally entangled state per sector. However, again, once these are chosen, a bijective isomorphism can be defined with them.
    


§ GENERALISATIONS

    Here we give some indications towards the generalisation of our construction and results to the infinite dimensional setting.

    The setting of general C*-algebras, through GNS representations and Stinespring's factorisation theorem, can essentially be reduced to the study of a cospan
    
    (_I) V^†_I(-)V_I⟶() V^†_O(-)V_O⟵(_O)    ,

    where all Hilbert spaces are separable and the maps V_I|O:→_I|O are bounded. 
    We can generalize our results by taking a more general cospan of bounded operator algebras between Hilbert spaces. 
    More precisely, let us list the key ingredients in our construction and how each of them can be generalised.
    
        
  * Existence of states ρ. This can be generalised to density matrices in the folium of a state ω∈𝒮() of some C*-algebra.
        
  * Existence of the identities  for use in defining extension maps. They may be generalised to approximate identities of C*-algebras and associated nets of approximate extension maps.
        
  * Existence of partial traces. Relative to an extension map, one may either take the inverse or the adjoint generalisation. Working with the inverse is maybe simpler and requires less structure. For the adjoint variant, we can either take the Banach adjoint or, in the presence of a scalar product, use the associated Hilbert adjoint (if the partial trace is bounded or at least densely defined). Either choice, though, will generically force us to move away from the full algebra and at least into the space of compact operators, as even the identity  does not have finite trace on infinite dimensional spaces.
    
    Recall first some relevant notation and the subsets of bounded operators on an infinite dimensional Hilbert space :
    
    Fin()  ⊂ L^1() ⊂ L^2() ⊂ K() ⊂()    .

    These are the finite rank, trace class, Hilbert-Schmidt, compact and bounded operators, respectively. Hilbert-Schmidt rank ones obey the special property L^2() ≅^∗⊗. In this notation, we will find a generalised mapping between
    
    : (_I) → L^1(_O)    ,

    which may be restricted to a mapping on Hilbert-Schmidt operators.

    
    We propose here one particular generalisation for the bipartite case, focused on trace class operators. 
    Given some bounded injective *-homomorphism between the (non-unital!) C*-algebra of compact operators K(_O)i_O→ K(), we can form its Banach adjoint[The usual issues of domains apply, but as long as the extensions are bounded, we may neglect them.]
    
    i_O: K()^∗→ K(_O)^∗, i_O(f)(x) = f(i_O(x))    .

    Then, we use the fact that the map
    
    α_K: L^1()→ K()^∗, α_K(x)(y) = [xy]    ,

    turning trace class operators into associated functionals, is an isometric bijection (see Proposition 3.4 of<cit.>). We can then define a partial trace as
    
    P_O = α^-1_K ∘i_O ∘α_K : L^1()→ L^1(_O)    ,

    which satisfies
    
    [P_O(x) y] = [x i_O(y)] 
    ∀  y ∈ K(_O), x∈ L^1()    .

    Note that these extension and partial trace operations do not have compatible (co)domains, as the partial trace is only defined on a subset of the target. Therefore, we will also require that our extension maps i_I|O satisfy
    
    i_I|O(L^1(_I|O)) ⊆ L^1()    .

    With this in hand, we can proceed as in the finite-dimensional case. Given an extension map (_I)i_I→(), we can twist it by some Σ: ()→ L^1() and get
    
    _Σ: (_I)i_I→() Σ→ L^1() P_O→ L^1(_O)    ,

    which is the analogue of our transport superoperator from the finite-dimensional case. 
    In particular, we may choose
    
    Σ(x) = ρ^(t_I) x, ρ∈ L^1()    ,

    which gives the Jamiolkowski or Choi map, depending on whether we use the transpose or not. We note that this construction must be a special case of the general fact<cit.> that 
    
    BL((_I),(_O)) isometric≅ ((_I)⊗_π L^1(_O))^∗   ,

    in which mappings between operator algebras are equivalent to linear functionals on a projective tensor product algebra. We have here a case 
    
    L^1(_I⊗_O) → BL((_I),L^1(_O))    .

    So, in order to connect to the previous work, one should find a map L^1() → ((_I)⊗_π L^1(_O))^∗. This must be
    
    ρ↦τ_ρ,  τ_ρ(x⊗ y) = [(i_I(x) i_O(y)^t_O)ρ^(t_I)]    ,

    simply using the expression of the isometric isomorphism <ref>. This functional is defined on (_I)⊗_π K(_O), which what we needed. Our example fits directly into a small class of maps in the general scheme which can be represented by a density matrix ρ. Using the general statement of isometry, we even have a way to estimate the norm of :
    
    |||| = sup_v∈(_I)⊗_π L^1(_O) |τ_ρ(v)|/π(v)   ,

    with π the projective norm on (_I)⊗_π L^1(_O), under which it is a Banach space. For it to be an isometry, of course, we need that |||| = 1. In this infinite dimensional setting, this is the most we can require as there is no good notion of, say, the partial trace in the Hilbert-Schmidt setting directly.

    Now, granted the validity of the general scheme, we still need concrete injection maps. 
    The central issue to be tackled in the construction is not i_I, which we can simply take to be i_I(x) = x⊗_π_O, but rather i_O, which can not work in the same way as in the finite-dimensional case, since _I is not compact. Instead, we must content ourselves with an approximate identity
    
    {^(λ)_I ∈ K(_I)  |  ^(λ)_I  s.a., λ∈Λ}

    and thus construct an approximate family of injections i^(λ)_O(y) = ^(λ)_I⊗ y.
    This, then, results in a family of superoperators whose formal limit
    
    = lim_λ∈Λ^(λ) = lim_λ∈Λ (P^(λ)_O ∘Σ∘ i_I)

    gives us the mapping we seek. Whether this limit exists as an operator in BL((_I),L^1(_O)) is of course nontrivial, but it is a requirement for  to be an isometry.

    Now if we also restrict the mapping to Hilbert-Schmidt operators, L^2(_I), then we can in fact still speak of the same kind of isometry under Hilbert-Schmidt:
    
    (X),(Y)_L^2(_O)?=X,Y_L^2(_I)   .

    Note, though, that the Banach adjoint exists here, while the Hilbert adjoint is not defined (as L^1(_O) is not a Hilbert space). We therefore will need to make a choice on what kind of isometry we are looking for - or rather, which norm we wish to preserve.

    
    

§ CONCLUSIONS

    We have studied information transport channels for quantum systems defined by operator algebras with centers, corresponding to Hilbert spaces with a structure of a direct sum of factorized Hilbert spaces (each identifying a natural notion of subsystems), generalizing the usual situation of factorized Hilbert spaces. Operator algebras and Hilbert spaces of this type appear in a very broad and diverse range of physical contexts, from condensed matter and quantum many-body systems, to (lattice) gauge theories, to quantum gravity and holography. In particular, we have proposed a generalization of the usual channel-state duality adapted to this direct sum context. We have also sketched how our construction, given in the finite-dimensional case, could be extended to the infinite-dimensional one. These transport channels offer a powerful way to characterize and study quantum correlations (entanglement) beyond the simple definition relying on non-separability of quantum states. This conceptual and analytic power, together with the vast range of physical systems with the algebraic characterization we considered, imply the possibility of many future developments from the application of our construction. 




§ ACKNOWLEDGEMENTS

SL would like to thank D. Aliverti and L.Csillag for valuable feedback on the manuscript and useful discussions. SL and DO acknowledge funding from the Munich Center for Quantum Science and Technology. DO also acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG).


    
    
