Quantum optimal transport with convex regularization
Emanuele Caputo Augusto Gerolin Nataliia Monina Lorenzo Portinale

Constituent Automorphism Decoding of Reed–Muller Codes
    Yicheng Qu, Amir Tasbihi, and Frank R. KschischangThe
authors are with the Edward S. Rogers Sr. Dept. of Electrical and
Computer Engineering, University of Toronto, 10 King's College Road,
Toronto, Ontario M5S 3G4, Canada.  Email: ,
, .  Submitted for
publication on July 19th, 2024.
    September 9, 2024
===============================================================================================================================================================================================================================================================================================

§ ABSTRACT 

The goal of this paper is to settle the study of non-commutative optimal transport problems with convex regularization, in their static and finite-dimensional formulations. We consider both the balanced and unbalanced problem and show in both cases a duality result, characterizations of minimizers (for the primal) and maximizers (for the dual).

An important tool we define is a non-commutative version of the classical (c,ψ)-transforms associated with a general convex regularization, which we employ to prove the convergence of Sinkhorn iterations in the balanced case. Finally, we show the convergence of the unbalanced transport problems towards the balanced one, as well as the convergence of transforms, as the marginal penalization parameters go to +∞.









§ INTRODUCTION


This work provides the first complete study of convex regularized quantum optimal transport problems and their extension to the unbalanced case.  More precisely, given two finite-dimensional Hilbert spaces _1, _2,  our goal is to prove the existence and characterize the solutions of the convex regularized quantum optimal transport problem

    = min{[C Γ] + [ φ(Γ)]
                Γ↦ (ρ,σ) 
        },

where the cost C ∈(_1⊗_2) is a Hermitian matrix over _1⊗_2, and φ:[0,+∞) →ℝ is a proper, convex, superlinear at infinity and bounded from below function. The notation Γ↦(ρ,σ) means that Γ is a density matrix over _1⊗_2 having partial traces equal to density matrices ρ over _1  and  σ over _2.

The unbalanced (convex-regularized) quantum optimal transport problem is given by

    𝔉^,τ_1,τ_2 = min{[C Γ] + [φ(Γ) ]
               + τ_1 ( P_1(Γ)| ρ)+ τ_2 ( P_2(Γ)| σ)   :  Γ≥ 0
        },


where (Γ|ρ)= 
    [
            Γ(
                logΓ - logρ -  Id_)
                + ρ] 
denotes the non-commutative relative entropy functional, P_1,P_2 are, respectively, the partial traces in _2 and _1. The notation Γ≥ 0 means that Γ is a semi-definite positive operator. For the precise definitions, see equation (<ref>) and Section <ref>.

The main difference between the problems (<ref>) and (<ref>) is that instead of the marginal constraint Γ↦(ρ,σ) in (<ref>), the unbalanced quantum optimal transport introduces a soft-penalization for the partial traces of Γ to be close (in the relative entropy functional) to the density matrices ρ and σ, weighted by some trade-off parameters τ_1,τ_2 ∈ (0,+∞).

The main contributions of this paper include

    
  * A general duality result, both for the balanced (<ref>) and unbalanced case (<ref>), including the case when φ nor its Legendre's transform are nonsmooth (Theorem <ref>).
    
  * The study and characterization of the optimizers, both in (<ref>) and (<ref>), as well as their corresponding dual formulations (Theorem <ref> and <ref>). 
    
  * The definition and the proof of the convergence of Sinkhorn-type iterations in the balanced case with general convex regularization φ (Theorem <ref>).
    
  * The convergence, as the soft-penalization parameters τ_i → 
    +∞, of the unbalanced transport problem (<ref>) to the corresponding balanced formulation (<ref>). More precisely, we show that when τ_i→+∞ of  (i) the energies converge, i.e.  𝔉^,τ_1,τ_2→; (ii) Any sequence of minimizers (Γ^ε,τ_1,τ_2)_τ_1,τ_2>0 of the unbalanced quantum optimal transport (<ref>) converges to a minimizer Γ^ε of (<ref>); and finally (iii) any sequence of maximizers of the dual formulation of (<ref>) converges to a maximizer in the dual formulation of <ref> 
 (Theorem <ref>).

Moreover, if we assume that φ is strictly convex and C^1, we can show that for both primal problems (<ref>) and (<ref>) admit a unique minimizer, which can be written in the explicit form

    Γ
            =
        ψ' [ U ⊕ V - C/ε]
            ,

where ψ is the Legendre transform of φ, and (U,V) are maximizers of their associated dual problems (Theorem <ref> and Theorem <ref>). 



  §.§.§ Quantum Optimal Transport


Quantum optimal transport extends the classical theory of optimal transport for probability measures to quantum states (e.g., density matrix or operators). Mathematically, challenges arise from the intrinsic non-commutative nature of quantum mechanics, which contrasts with the commutative structures underlying classical probability theory.

The pursuit of a non-commutative version of the Monge–Kantorovich optimal transport theory dates back roughly thirty years, with pioneering works by Connes <cit.>, Biane and Voiculescu <cit.>, see also <cit.>. Their approach is primarily centered on adapting the static dual formulation of optimal transport.

More recently, several efforts have been made to extend the optimal transport theory for quantum states, starting from either the static (e.g., Kantorovich relaxation, dual) or dynamical formulation (known after Benamou–Brenier). While these formulations are equivalent in the optimal transport theory for probability measures, they can lead to distinct and not necessarily equivalent theories in the quantum setting.

In the static framework, quantum optimal transport between quantum states has been first introduced by Caglioti, Golse and Paul <cit.> and De Palma and Trevisan <cit.>. In <cit.>, the quantum optimal transport arose in the context of
the study of the semiclassical limit of quantum mechanics. In this setting, the problems correspond to (<ref>) with ε=0, the joint states Γ are defined in _1⊗_2 and the cost operator is given in terms of quadratures C=∑_i=1^2m(R_i⊗𝕀_ℋ_2 - 𝕀_ℋ_1⊗ R_i)^2.
In the alternative proposal in <cit.>, the plans instead are states on ℋ⊗ℋ^*, i.e., one acts with a partial transpose operation:



the variational problem reads similarly to the previous one, but its properties are rather different and it enjoys operational interpretations thanks to a correspondence between plans and quantum channels. Also, in <cit.>, the authors introduced a quantum optimal transport problem by generalizing the dual formulation of the so-called 𝕎_1 distance specifically for qubits.

A dynamical formulation of quantum optimal transport has been introduced by Carlen and Maas <cit.>, which the authors used to study long-behavior and fine properties of the Fermionic Fokker–Planck equations. In <cit.>, Wirth introduced a dynamical formulation for von Neumann entropy-regularized quantum optimal transport within the dynamical framework. In the same year, Monsaingeon and Vorotnikov <cit.> introduced the Fisher–Rao space of matrix-valued non-commutative probability measures, and the related Hellinger space.

The study of quantum composite systems at positive temperature – equation (<ref>) with von Neumann entropy φ(Γ)=[Γ(logΓ -  Id__1⊗_2)] – has been initiated in <cit.>, including the cases when the density matrices are fermionic, important for Electronic Structure Theory <cit.>. The von Neumann entropy regularization is fundamental for implementing quantum optimal transport functionals into efficient computational algorithms. 

Unbalanced quantum optimal transport was studied in <cit.>, motivated 
by optimal transport theory for non-negative measures introduced, independently, by Liero, Mielke and Savaré <cit.>, Chizat, Peyré Schmitzer and Vialard <cit.>, and Kondratyev, Monsaingeon and Vorotnikov <cit.>. In <cit.>,
the authors considered a particular problem (<ref>) when φ(Γ) is the von Neumann entropy regularization, proved weak duality, Γ-convergence results when either the soft-penalization parameter in (<ref>) 
 τ_1=τ_2→+∞ goes to infinity and the von Neumann entropy-regularization parameter ε→ 0^+ goes to zero.

This work addresses general convex entropy regularization problems within both the quantum optimal transport (<ref>)
and unbalanced quantum optimal transport (<ref>) frameworks. In the context of optimal transport for probability measures, convex regularization has been studied extensively (e.g., <cit.>). This list of examples include quadratic regularization as well as regularization with the Tsallis entropy. The study of these cases is primarily driven by the goal of designing computational algorithms leveraging the sparse structure of optimal transport problems.



A recent survey on the topic has been recently published <cit.>. 

Methodology: From a technical perspective, the technique and results of <cit.> often rely on the special form of the dual problem associated with the von Neumann regularization (in particular, properties of the exponential map x ↦ e^x), which are not at our disposal at this level of generality. On a second note, even the classical results provided in <cit.> regard C^1 regularization functions, which makes our general result slightly more general even with respect to the classical setting. Our results are based on suitable a priori estimates on the dual functional, which in many cases turn out to be dimension-free. Although our techniques are still finite-dimensional, the final, long-term goal is to develop all the tools and investigate ideas to obtain results in the infinite-dimensional setting. 










Organization of the paper: Section <ref> presents the setting and the statement of the main result of this manuscript. Section <ref> focus on the proofs for the (balanced) quantum optimal transport problem, the definition of (C,ψ,)-transform and its properties. In the final part, we define the Sinkhorn iterations and study their convergence. In Section <ref>, we prove strong duality in the unbalanced case, we define the (C,ψ,,τ)-transform and study its properties. Section <ref> extends the results of previous sections by removing the assumption that ψ∈ C^1. Section <ref> contains the Γ-convergence results of the unbalanced problem to the balanced one and the convergence of transforms as the size parameters τ converge to +∞.




§ SETTING AND MAIN RESULTS









Given a Hilbert space , we denote by () the vector space of Hermitian operators over .
Let us denote by _≥() the set of positive semi-definite Hermitian and by () the set of density matrices, i.e. elements of _≥() with trace equal to 1. We also write 
_>() to denote the elements A of _≥() such that ker(A)={ 0}.

Given ρ∈(_1) and Γ∈(_1 ⊗_2), we say that ρ is the first partial trace if

    [ Γ (U ⊗ Id__2) ]= [ ρ U ]    for every U ∈(_1) .

In such a case, we write P_1 Γ = __2Γ =ρ. Similarly, given σ∈(_2) and Γ as before, we denote by P_2 Γ = __1Γ =σ the second partial trace defined in an analogous way. If P_1 Γ=ρ and P_2 Γ=σ, we write Γ↦ (ρ, σ). 


For the sake of convenience, we introduce the following notation: for ∈{1, …, d }, we use λ_i(A) to denote the i-th smallest element of the spectrum of A. In particular, λ_1(A) is the smallest eigenvalue of A. 
For every continuous function φℝ→ℝ, we define the lifting of φ to the space of Hermitian matrices the operator[With a slight abuse of notation, we denote the lifting of φ also with the same letter φ.] φ() →() given by φ(A):=∑_i=1^d φ(λ_i(A)) |ξ_i⟩⟨ξ_i|, where we used the spectral decomposition A = ∑_i=1^d λ_i(A) |ξ_i⟩⟨ξ_i|. When clear from the context, we may omit the dependence on A and write λ_i instead of λ_i(A). 


For U ∈(_1) and V ∈(_2), we define U ⊕ V :=  Id__1⊗ V + U ⊗ Id__2.




The unbalanced optimal transport is obtained by replacing the constraint with two penalization terms tuned by parameters τ_1, τ_2>0. The choice of penalization we use in this paper (as typically adopted in the commutative setting as well) is given by the relative entropy with respect to the marginals. Given a finite-dimensional Hilbert space  and an operator η∈_≥(), the relative entropy functional (· | η): _≥() → [0, + ∞] is the functional defined as

    α↦(α | η)
            := 
        [
                α(
                    logα - logη -  Id_)
                    + η]
                   if η⊂α
                    , 
        
    
            + ∞   otherwise 
                    .

Here we are using the convention that α( logα - logη) = 0 on η⊂α. The nonnegativity of the relative entropy is a direct consequence of Klein's inequality <cit.> applied to the convex function t ↦ t log t. Finally, it is a (strictly) convex function on its domain, and its Legendre transform is given by 

    ^*(A|η)
            :=
        [ 
            e^A+logη  - η]
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
              ,      A ∈() 
                .

This follows from the equality 

    [ 
            e^A+logη  - η] 
            =
         g(A+logη) - [η] 
              ,   where   g(B) := [exp(B)]
                  ,

together with the fact that, for α∈(),

    g^*(α) 
            = 
        sup_A ∈(){⟨α, A ⟩ - g(A)
        } 
            = 
            [
                α(
                    logα -  Id_)
            ],    α∈_≥(),
    
            +∞,    otherwise
                .

By convexity we know that g^** = g,  hence we get  (<ref>).

Note that in general  ^*(A|η) ≠[ 
        ( 
            e^A -  Id_) 
            η], in particular when A and η do not commute.











We shall fix  φ [0,+∞) → a convex, superlinear at infinity, and bounded from below function, namely 

    φ [0,+∞) →  ,   convex  ,   lim_t → +∞φ(t)/t =+∞  ,  infφ≥ l > - ∞ 
            ,

for some l ∈. In particular we assume that φ(0)∈. 



The following operators are given and fixed throughout the whole paper: 

    
  * A cost operator C ∈(_1 ⊗_2). 
    
  * Two `marginal' operators ρ∈_>(_1) and σ∈_>(_2) (in particular, with trivial kernel). Sometimes we may assume them to be density matrices as well.  


The assumption on the kernel of the marginal is classical and not too restrictive. Similar results can be obtained by decomposing the space with respect to the kernels and their orthogonal spaces, in the very same spirit as in <cit.>. For the sake of simplicity, we won't discuss further details in this work. 

We also fix >0 a regularization parameter. 

The main objects of study of our work are primal and dual functionals. 
[Primal problems]
We define a functional _≥(_1 ⊗_2) →ℝ as

    (Γ): = [C Γ] + [ φ( Γ) ]
            ,   ∀Γ∈_≥(_1 ⊗_2) .

For every τ = (τ_1, τ_2) ∈ (0,+∞)^2, we define  : _≥(_1 ⊗_2) → as the functional 

    (Γ) :=  (Γ)+ τ_1 ( P_1(Γ)| ρ)+ τ_2 ( P_2(Γ)| σ) 
            ,

We also define the unbalanced primal problem with parameters τ_1, τ_2>0 as

    := min{ (Γ) 
        Γ∈_≥(_1 ⊗_2) }.

Moreover, if ρ∈(_1) and σ∈(_2) we define the balanced primal problem as 

    := min{(Γ) Γ↦ (ρ, σ)  }
        ,

which (formally) corresponds to the case τ_1=τ_2 = +∞.


To each primal problem, we associate the respective dual problem. Throughout the whole paper, we work with a function ψ∈ C() which is convex, superlinear at infinity, and bounded from below, i.e. 

    ψ∈ C() 
              ,   convex  ,   lim_t → +∞ψ(t)/t
        = + ∞  , 
             
        m:= infψ > - ∞ 
            .

When dealing with duality results for the non-commutative optimal transportation problems, ψ will typically be the Legendre transform of a φ satisfying (<ref>), namely of the form

    ψ(t) 
            =
        sup_x ∈ [0,+∞){
                tx - φ(x)
            }
            =
        φ^*(t)
              ,   ∀ t ∈
                ,

where when talking about Legendre transform we may implicitly extend the definition of φ on the full real line by setting φ(x) ≡ +∞, for every x < 0. The validity of (<ref>) readily follows in this case from the properties (<ref>) of φ. 
A direct computation for the relation between the primal and the dual can be seen in Appendix <ref>.

[Dual problems]
Fix ψ∈ C() satisfying (<ref>). We define the dual functional (_1) ×(_2) → as

    (U,V)
            :=
        [ U ρ] 
            + 
        [ V σ]
            -
        [ ψ(U⊕ V - C/) ]
            .



For every τ_1, τ_2>0, we also define (_1) ×(_2) → as the map

    (U,V) ↦ 
         -τ_1
         [ 
            e^-U/τ_1+logρ- ρ] 
            -
        τ_2
        [ 
           e^-V/τ_2+logσ- σ] 
            -[ 
            ψ(U⊕ V-C/)
        ] 
            .

We define (respectively) the unbalanced (respectively balanced) dual problem as 

    := 
            sup{ (U,V) 
         
            (U,V) ∈(_1) ×(_2)
        }
            ,  
    
     :=
            sup{ (U,V)
         
             (U,V) ∈(_1) ×(_2)
        }
            .



















The main results of our work can be divided in four parts: duality results and analysis of primal/dual problem for the balanced optimal transport problem under suitable regularity assumptions on ψ, the corresponding results for the unbalanced problem, a generalization of the duality for possibly nonsmooth ψ, and finally convergence results for the unbalanced problem towards the balance one was τ→ +∞. 



  §.§.§ Balanced optimal transport

The first contribution is a duality result for the balanced quantum optimal transport problem, including existence, (suitable) uniqueness, and characterization of the optimizers, both in the primal and dual problems.

[Duality and optimizers in the balanced case]

    Let φ:[0,+∞) → [0,+∞) satisfy (<ref>), and assume that ψ=φ^* is strictly convex and C^1. Assume additionally that ρ∈(_1) and σ∈(_2). Then the following statements hold: 
    
        
  1) (Duality) The dual and primal problem coincide, namely =.
        
  2) (Existence of maximizers) There exists (U^*,V^*) which maximizes D^, i.e. = D^ (U^*,V^*). Moreover, any other maximizer (Ũ, Ṽ) satisfies (Ũ - U^*, Ṽ - V^*) = (λ Id__1 , - λ Id__2 ), for some λ∈. 
        
  3) (Existence of minimizers) There exists a unique maximizer Γ^* ∈(_1 ⊗_2) for the primal problem, i.e. = (Γ^*), and it is given by 
        
        
    Γ^*
                    =
                ψ' [ U^* ⊕ V^* - C/ε]
                    ,

        where (U^*,V^*) are (any) maximizers for the dual problem. 
    


In the balanced setting, our second contribution concerns the definition of a Sinkhorn-type iterations and their convergence guarantee. To this end, a crucial tool to consider is the so-called (C,ψ,)-transform (associated with ψ). We assume once again that  ψ=φ^* is strictly convex and C^1. We define ℱ_2^(C,ψ,)(_1)→(_2) as

    ℱ_2^(C,ψ,)(U) 
            =
        {[ V σ] - [ ψ(U⊕ V - C/) ]
             
        V ∈(_2) }.

Analogously, we define the corresponding ℱ_1^(C,ψ,)(_2)→(_1) as

    ℱ_1^(C,ψ,)(V) 
            = 
        {[ U ρ] - [ ψ(U⊕ V - C/) ] 
                 
            U ∈(_1) 
        }.

For the well-posedness of the (C,ψ,)-transforms, see Section <ref>.
We use this notion to define the following algorithm. Recall that λ_1(A) denotes the first eigenvalue of an operator A.

Step 0: fix any initial (U^0,V^0)∈(_1)×(_2). 

Step 2n-1: for n ∈, for given U^n-1∈(_1), we define V^n as 

    V^n := ℱ_2^(C,ψ,)(U^n-1) - λ_1(ℱ_2^(C,ψ,)(U^n-1)) Id__2
            .


Step 2n:  for given V^n ∈(_2), we define U^n as 

    U^n := ℱ_1^(C,ψ,)(ℱ_2^(C,ψ,)(U^n-1)) + λ_1(ℱ_2^(C,ψ,)(U^n-1)) Id__1.



Then we have the following convergence result.


[Convergence of the Sinkhorn iterations]


Under the same assumptions of Theorem <ref>,
    there exists a limit point (U^*,V^*) of  {(U^n,V^n)}_n≥ 0, which is a maximizer of D^.
    
    
    
    
    
Moreover, we have that 
    
    Γ^n = ψ'
            (
                U^n⊕ V^n - C/)
                ∈(_1 ⊗_2)

    and we have that Γ^n →Γ^* ∈(_1 ⊗_2) as n →∞, where 
    
    Γ^* 
                = 
            ψ'
            ( 
                U^* ⊕ V^* - C/)
                ↦ 
            (ρ,σ)

    is the unique minimizer for the primal problem .




  §.§.§ Unbalanced optimal transport

Taking advantage of the techniques and results provided in the previous section, we next tackle the same questions in the unbalanced setting. In particular, we have the validity of the following duality result. 

[Duality and optimizers in the unbalanced case]

     Let φ:[0,+∞) → [0,+∞] satisfy (<ref>), and assume that ψ=φ^* is strictly convex and C^1. Let τ = (τ_1,τ_2) ∈^2 be trade-off parameters. Then the following holds:
    
        
  1) (Duality) The dual and primal problem coincide, namely =.
        
  2) (Existence of maximizers) There exists a unique (U^*,τ,V^*,τ) which maximizes D^,τ, i.e. = D^,τ (U^*,τ,V^*,τ). 
        
  3) (Existence of minimizers) there exists a unique minimizer Γ^*,τ∈(_1 ⊗_2) for the unbalanced primal problem, i.e. = (Γ^*,τ), and it is given by 
        
    Γ^*,τ
                    =
                ψ' [ U^*,τ⊕ V^*,τ - C/ε]
                    ,

        where (U^*,τ,V^*,τ) are the unique maximizers for the unbalanced dual problem. 
    


Note that the unbalanced case is characterized by the uniqueness of maximizers in the dual problem, in contrast with the balanced case. 



  §.§.§ General duality theorem

The assumptions on ψ are classical: in particular, the strict convexity of ψ ensures the uniqueness (up to the trivial translation) of the maximizers of the dual problem. The smoothness assumption ψ∈ C^1 ensures that φ is strictly convex in its domain, as well as it allows us to write the explicit formula for Γ^* as given in (<ref>). 
On the other hand, the validity of the duality result does not require such regularity but holds true for general convex, superlinear, and bounded from below ψ, which automatically follows from the general assumptions on φ. 
[Duality for nonsmooth ψ]

     Let φ:[0,+∞) → [0,+∞) be a convex, superlinear, and bounded from below. Then for every τ_1, τ_2 ∈ [0,+∞], the primal and dual problem coincide, i.e. =. If ρ∈(_1) and σ∈(_2), the same holds true for the balanced case, i.e. =.

It would be interesting to investigate the validity of a formula in the same style as (<ref>) for nonsmooth ψ, possibly involving the subdifferential of ψ. We leave this to future investigations. 



  §.§.§ Convergence results

Having at disposal a complete understanding of both the balanced and unbalanced setting, our final contribution is the study of the limit behavior of both primal and dual problems as the trade-off parameters τ_1, τ_2 goes to +∞. In particular, we show that not only do the unbalanced primal and dual functionals  Γ-converge to the corresponding ones in the balanced case, but we also provide the convergence of the associated optimizers (up to a suitable renormalization for the dual problem).

[From unbalanced to balanced optimal transport: convergence result]

    Under the same assumptions of Theorem <ref>, we have the following convergence results:
    
        
  * The functional   Γ-converge as τ_1,τ_2 →∞ to the functional 
        
    Γ↦(Γ)
                           if Γ↦ (ρ,σ) 
                
    
                    +∞   otherwise  .

        Similarly for the dual problem, we have that -D^,τ Γ-converge to -D^.
        
        
  * The unique minimizer Γ^*,τ∈_≥(_1 ⊗_2) of  converges (up to subsequence) as τ_1,τ_2 → +∞ to a minimizer Γ^* ∈_≥(_1 ⊗_2) of  with constraint Γ^*  ↦ (ρ,σ). 
         
  * The unbalanced optimal transport problems converge as τ_1,τ_2 → +∞  to the balanced optimal transport problem, i.e.
        
    lim_τ_1,τ_2 → +∞
                        =
                    
                        =
                     
                        =
                lim_τ_1,τ_2 → +∞ 
                        .

        
  * Let (U^*,τ, V^*,τ) ∈(_1) ×(_2) be the unique maximizer of D^,τ. Define the recentered potentials 
        
    (
                    Û^*,τ, V̂^*,τ) 
                    := 
                ( 
                    U^*,τ + λ_1(V^*,τ)
                        , 
                    V^*,τ - λ_1(V^*,τ)
                )
                    ∈(_1) ×(_2)
                        .

        Then (Û^*,τ, V̂^*,τ) converges (up to subsequence) as τ_1,τ_2 → +∞ to a maximizer (U^*,V^*) ∈(_1) ×(_2) of D^ which satisfies λ_1(V^*)=0.
    
    
    Assume additionally that ψ∈ C^1 and strictly convex. Then the convergence of the minimizers and maximizers in 2. and 4. are true without taking subsequences, and Γ^*,τ (resp. (Û^*,τ,V̂^*,τ)) converge to the unique minimizer Γ^* (resp. unique maximizer (U^*,V^*) with λ_1(V^*)=0).



Finally, the final convergence result we obtain is about the convergence of the (C,ψ,,
τ)-transforms associated with the dual functional of the unbalanced optimal transport problem.














The corresponding transform in the unbalanced case is given by the map ℱ_2^(C,ψ,,τ)(_1)→(_2)  such that

    ℱ_2^(C,ψ,,τ)(U) 
            =
        _V ∈(_2){ 
            -  τ_2 [e^-V/τ_2+logσ] - [ψ(U⊕ V -C/)] 
                
        }.

Analogously, we define ℱ_1^(C,ψ,,τ)(_2)→(_1) as

    ℱ_1^(C,ψ,,τ)(V) 
            = 
        _U ∈(_1){ 
            -  τ_1 [e^-U/τ_1+logρ] - [ψ(U⊕ V -C/)] 
                
        }.


[Convergence of C-transforms]
    Assume that ψ is strictly convex. Then we have that the associated (C,ψ,,τ)-transforms converge, as the parameter τ→ +∞. More precisely, one has that, for i=1,2, 

    ℱ_i^(C,ψ,,τ)(W_τ)
            →ℱ_i^(C,ψ,)(W_∞)
                ∀   
            W_τ→ W_∞ 
                .



A similar result could be proved as well without assuming that ψ is strictly convex. Of course, in this case, the definition of (C,ψ,)-transform is not well-defined, due to the fact that we don't have a unique maximizer in (<ref>), (<ref>). Nonetheless, it not hard to prove that unbalanced C-transforms will converge (up to subsequence) to a maximizer in (<ref>) (resp. (<ref>)).











































§ BALANCED NON-COMMUTATIVE OPTIMAL TRANSPORT





In this section, we study the non-commutative optimal transport problems, both the primal and the dual, in the balanced case. 
We begin by analyzing the dual problem, and discussing the existence of maximizers and their properties. We then use them to prove a duality result, namely the equality between the primal and the dual problem. In the last part of this section, we discuss the notion of (C,ψ,)-transform associated with a general regularization ψ.

Throughout the whole section,  _1, _2 denotes Hilbert spaces of finite dimensions d_1 and d_2, respectively. We fix C ∈(_1 ⊗_2) and consider ρ∈(_1), σ∈(_2) with trivial kernel. 

Recall the definition of the dual functional 

    (U,V)=[ U ρ] + [ V σ]-[ ψ(U⊕ V - C/) ],











where ψ∈ C() is a convex, superlinear function at infinity, and bounded from below, i.e. 

    lim_t → +∞ψ(t)/t
        = + ∞ 
        m:= infψ > - ∞ 
            .




    A crucial property that we use later in this section is that if ψ is superlinear, then the function ∋ x ↦ x - ψ(x) has superlevels which are bounded from above, namely for every α∈, 
    
    {
                x ∈ 
                x - ψ(x) ≥α}⊂{
                x ∈ 
                x ≤ R_α}
                ,
 
    for some R_α∈ (0,+∞). This directly follows from the fact that for every R>0, 
    
    {
                x ∈ 
                x - ψ(x) ≥α}∩{
                x ∈ 
                x > R 
            }⊂{
                x ∈ψ(x)/x≤max{ 
                    1 - α/R
                        , 
                    1
                }}  
                ,

    and the fact that the set on the right-hand side is bounded by the definition of superlinearity.













Note that the dual functional in the balanced case enjoys special symmetry properties. In particular, for every (U,V) ∈(_1) ×(_2) we have that


    (U + λ Id__1 , V - λ Id__2 )= (U, V)

for every λ∈ℝ. This in particular shows that  is not coercive. Nonetheless, as shown at the end of this section, this is the unique invariance of the problem, see Remark <ref>. 
















 §.§ Existence of maximizers in the balanced case




First of all, we observe that the maximization problem

    sup{  (U,V)  U∈(_1),  V∈(_2) },

is equivalent, due to the symmetry argument in Remark <ref>, to the restricted maximization 

    sup{  (U,V)  U∈(_1),  V∈(_2),  λ_1(V)=0 },

where recall that λ_1(U) denotes the smallest eigenvalue of U.

The next result shows that once restricted to the class of operators (U,V) with λ_1(V) = 0, the dual functional is coercive.









For every  U∈(_1), V∈(_2) such that 

    λ_1(V)=0 
            D^ (U,V)≥ M >-∞

for some M∈, there exists a constant 0≤𝒜=𝒜(M) <∞ such that ||U||_∞,||V||_∞≤.



For any given U∈(_1), V∈(_2), we denote by S:= U ⊕ V ∈(_1 ⊗_2) and write its spectral decomposition as S= ∑_k=1^d λ_k(S) |ξ_k ⟩⟨ξ_k | where d= d_1 d_2. Note that every eigenvalue of S is given by the sum of an eigenvalue of U and one of V, hence to show that the ||U||_∞, ||V||_∞ norms are bounded it suffices to provide an upper and a lower bound the spectrum of S, since

    λ_1(S) = λ_1(U) ≤λ_d_1(U)
              and  λ_d(S) = λ_d_1(U)+λ_d_2(V) 
            ≥max{λ_d_1(U) , λ_d_2(V) + λ_1(S) } 
            .



Writing the dual functional in terms of S, one can apply the Peierl's inequality (e.g., Theorem 2.9 in <cit.>) for the base {|ξ_k⟩}_k=1^d and obtain 


    M 
               ≤ (U,V) 
            = 
        [S (ρ⊗σ)] 
            -
        [ψ(S-C)] 
    
       ≤∑_k=1^d(                      
            ⟨ξ_k| S(ρ⊗σ) |ξ_k⟩ 
            -
                ψ(
                ⟨ξ_k|S - C|ξ_k⟩)
        ) 
    
       =
        ∑_k=1^d( 
            λ_k(S) 
            ⟨ξ_k|ρ⊗σ|ξ_k⟩ 
                -
            ψ(
                λ_k(S) - ⟨ξ_k| C |ξ_k⟩)
        )
    
       = 
        ∑_k=1^d(
            Λ_k ω_k 
                -
            ψ(
                Λ_k - C_k)
        ),

where for simplicity we set Λ_k:= λ_k(S), ω_k = ⟨ξ_k|ρ⊗σ|ξ_k⟩, and C_k=⟨ξ_k| C |ξ_k⟩. 

On the one hand, using that ∑_k=1^dω_k=1, Λ_1 ≤Λ_d, as well as ω_1 ≥λ_1(ρ⊗σ), we have that

    M 
               ≤Λ_1 ω_1 
            + 
        ∑_k=2^d 
            Λ_k ω_k 
                -
            ∑_k=1^d 
                ψ(
                    Λ_k - C_k) 
    
        ≤Λ_1 ω_1 + Λ_d ( 1-ω_1) - d m
    
       ≤Λ_1 λ_1(ρ⊗σ ) + Λ_d (1-λ_1(ρ⊗σ )) - d m
            ,


which yields the lower bound Λ_1 ≥λ_1(ρ⊗σ )^-1( M+  d m - Λ_d (1-λ_1(ρ⊗σ )) ), where we used that λ_1(ρ⊗σ ) > 0 due to the fact that ρ and σ have trivial kernel. 

Note that λ_1(ρ⊗σ) ∈ (0,1], therefore we are left to show that Λ_d is bounded from above, which follows by superlinearity of ψ. Indeed, from (<ref>) we see that

    M 
               ≤Λ_d  
            - 
        ψ(
            Λ_d - C_d) 
            -
        ∑_k=1^d-1ψ(
                Λ_k - C_k) 
    
    
    
       ≤(
             Λ_d - C_d 
                - 
            ψ(
                 Λ_d - C_d)
        ) 
            +
         C _∞ -  m (d-1)
            ,

which yields

    Λ_d - C_d 
                - 
            ψ(
                 Λ_d - C_d
        ) 
            ≥M+ m (d_1 d_2 -1) - ||C||_∞
            .

The conclusion the directly follows from the superlinearity of ψ (see in particular Remark <ref>) and the fact that |C_d| ≤ C _∞. 




    Notice that without loss of generality, due to the symmetry of the functional , the condition λ_1(V)=0 can be replaced with  λ_1(U)=0. In fact, the claimed compactness works with any other constraint of the form λ_1(U)∈ B (or V instead of U), for a given bounded set B ⊂. 




    Following the proof of the latter Proposition, one can see that  depends on C _∞, and the dimensions d_1, d_2 of the underlying Hilbert spaces. Nonetheless, it is clear from the proof that the second dependence disappears whenever ψ≥ 0, as well as in the limit as → 0, which suggests the possibility of (partially) extending these results to infinite dimensional setups.





We may now proceed to show the existence of the maximizer for the dual functional (<ref>).
[Existence of the maximizer]

Let ε>0, _1, _2 be finite-dimensional Hilbert spaces.
Let ρ∈(_1) and σ∈(_2) with trivial kernel and let ψ:→ be a superlinear convex function bounded from below. Then the dual functional D^ defined in (<ref>) admits a maximizer (U^*,V^*)∈(_1)×(_2).


    The proof follows from the direct method.
    
    
    Indeed, take a maximizing sequence {(U^n,V^n)}_n≥ 1⊂(_1)×(_2) such that  
    
    sup 
            = 
        lim_n →∞ (U^n,V^n)
                .
    
    Thanks to the observation in (<ref>), we can assume λ_1(V^n) = 0. In particular, we then have 
    
    {(U^n,V^n)}_n≥ 1⊂{(U,V)∈(_1)×(_2) λ_1(V)=0,   (U,V) ≥ M },

    for some M ∈. Due to Proposition <ref>, we may conclude that the sequence {(U^n,V^n)}_n≥ 1 is bounded, and we extract a converging subsequence (U^n_k,V^n_k) (U^*,V^*)∈(_1)×(_2). 
    By the continuity of  (<ref>), taking the limit as k →∞ we obtain that (U^*,V^*) = sup, hence a maximizer.
    
    
    
    
    
    
    
    
    
    
    






 §.§  properties of maximizers, and duality






Recall the setting introduced at the beginning of Section <ref>.

We now introduce the notion of the (C,ψ,)-transform, in analogy with the regularization given by the Shannon entropy in the quantum case in <cit.>. 


[(C,ψ,)-transform]
Assume additionally that ψ is  strictly convex. We define ℱ_2^(C,ψ,)(_1)→(_2) as

    ℱ_2^(C,ψ,)(U) 
            =
        {[ V σ] - [ ψ(U⊕ V - C/) ]
             
        V ∈(_2) }.

Analogously, we define the corresponding ℱ_1^(C,ψ,)(_2)→(_1) as

    ℱ_1^(C,ψ,)(V) 
            = 
        {[ U ρ] - [ ψ(U⊕ V - C/) ] 
                 
            U ∈(_1) 
        }.
























The definition of the transform is well-posed, as the  contains exactly one element. Indeed, due to Remark <ref>, we may provide the proof only for ℱ_1^(C,ψ,). For every given V∈(_2) we have that

    _U∈(_1){[ U ρ] - [ ψ(U⊕ V - C/) ]
        }
            =  
        _U∈(_1){ (U,V) 
        }
    
            = 
           _U∈(_1){ (U + λ_1(V)  Id__1,V - λ_1(V)  Id__2) }.

In particular, as we are looking for a maximizer, we can restrict the maximization set to 

    { U∈(_1)  (U + λ_1(V)  Id__̋1,V - λ_1(V)  Id__2)≥  M } ,

for some M ∈.

Under this condition, we have that (U + λ_1(V)  Id__̋1,V - λ_1(V)  Id__2) satisfies the assumption of the Proposition <ref>, and therefore the set (<ref>) is bounded. Arguing as in Theorem <ref>, one can show that a maximizer in (<ref>) exists.


Uniqueness follows from the strict concavity of , as a consequence of (<ref>).



 
    Under the standing assumptions of this section, let additional suppose that ψ is  strictly convex and C^1. Then, for every U ∈_1, its transform ℱ_2^(C,ψ,)(U) is the unique solution V of the equation 

    σ =_2 [ ψ'( U ⊕V - C/)].

Analogously, the characterization holds for ℱ_1^(C,ψ,)(V), replacing σ with ρ and _2 with _1. If ψ is only C^1 and convex (not necessarily strictly), then transform ℱ_2^(C,ψ,)(U) is not uniquely determined, but any (possibly not unique) maximizer of (<ref>) satisfies the same optimality condition (<ref>).



    Set V:= ℱ_2^(C,ψ,)(U) (or in general any maximizer for (<ref>)). For every δ∈ℝ and A ∈(_2), we define the map
    
    f(δ):=[ ψ(U ⊕ (V+δ A)-C/) ] - [ (V+δ A) σ]
                .

    By optimality, we know that f'(0)=0, and by Lemma <ref> this translated into 
    
    [ ψ' (U ⊕ V-C/)( Id⊗ A) ] - [ A σ] = 0
                  ,   ∀ A ∈(_2)
                ,

   which provides the sought characterization, by the very definition of partial trace _2. The second statement follows by arguing similarly. Uniqueness comes directly from the strict concavity of the function V ↦ (U,V) whenever ψ is strictly convex.


[Quadratic regularization]

A typical choice of regularization which differs from the usual von Neumann entropy is given by the quadratic case φ(t)=1/2 t^2, t ≥ 0. See for example <cit.> for the study of the regularized optimal transport problem with such ψ, including numerical methods to seek for maximizers. They are based on the fact that in this case, the associated Legendre transform is given by ψ(t) = 1/2 (t_+)^2. 
The optimality conditions therefore read as 

    σ 
            =
        _2 
        [ 
        (
            U ⊕ℱ_2^(C,ψ,)(U) - C/)_+
        ]
        
        
        
        
        
            .

Note this is an example of ψ∈ C^1 which is not strictly convex. In particular, ψ' is not globally invertible and an explicit solution cannot be found, and the maximizers of the dual problem are not necessarily determined by λ_1(U) (cfr. <cit.> for the commutative setting). 










[Equivalent characterizations for maximizers]

    Under the standing assumptions of this section, let (U^*, V^*) ∈(_1) ×(_2). Assume that ψ is  strictly convex and C^1. Then the following are equivalent:
    
        
  1) (Maximizers) (U^*,V^*) maximizes D^, i.e. = D^ (U^*,V^*).
        
  2) (Maximality condition) U^* = ℱ_1^(C,ψ,)(V^*) and V^* = ℱ_2^(C,ψ,)(U^*).
        
  3) (Complementary slackness) Let 
        Γ^*:=ψ' [ U^* ⊕ V^* - C/ε], then Γ^* ↦ (ρ,σ).
        
        
  4) (Duality) There exists Γ^* ∈(_1 ⊗_2) so that Γ^* ↦ (ρ, σ) and (Γ^*)= D^ (U^*,V^*).
    
If one (and thus all) condition holds, Γ^*, as defined in 3) is the unique minimizer to . 


As an immediate corollary of Theorem <ref> and Theorem <ref>, we obtain the sought duality.
[Duality for balanced QOT]
    Under the standing assumptions of this section, assume that ψ = φ^* is strictly convex and C^1. Then =. 



    We shall show 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 1). 

    1) ⇒ 2).  
    By definition of ℱ_1^(C,ψ,)(V^*), we have that (ℱ_1^(C,ψ,)(V^*),V^*) ≥ (U^*,V^*) =. 
    
    By uniqueness of maximizer of (·,V^*), we have that U^*=ℱ_1^(C,ψ,)(V^*). A similar argument proves the second part of the statement.

    2) ⇒ 3).  
   This holds by Proposition <ref>.

    3) ⇒ 4).  
    Denote by simplicity W^*:=(U^* ⊕ V^*- C)/ε, and compute
    
    (U^*,V^*)
                   = 
            [
                U^* ρ]+[V^* σ] -ε[ψ(U^*⊕ V^*-C/ε)
            ]
        
       3)=[
                (U^*⊕ V^*) Γ^*] -ε[ψ(U^*⊕ V^*-C/ε)
            ]
        
       =
            ε[ 
                W^* ψ'(W^*)-ψ(W^*)]+[C Γ^*
            ]
            
            
            
            
                ,

    
    As a consequence of (<ref>), we have that
    
    [ 
                W^* ψ'(W^*)-ψ(W^*)
            ]
                =
            [  φ(  ψ'(W^*) ) ]= [ φ(Γ^*)].

    Therefore, we continue the computation in (<ref>) using (<ref>) and we have
    
    (U^*,V^*) =[C Γ^* ] +ε[ φ(Γ^*)]
            = (Γ^*).

    This proves 4). 
    
    4) ⇒ 1).      
    Since (Γ^*) =  (U^*,V^*)≤ by Remark <ref>, we have that Γ^* is a minimizer for the primal problem, hence (U^*,V^*)=. 
    Moreover, once again by Remark <ref> it holds ≥ (U,V) for every (U,V) ∈(_1) ×(_2), hence we also deduce that (U^*,V^*) ≥ (U,V) for every (U,V) ∈(_1) ×(_2), which precisely shows that (U^*,V^*) is a maximizer.



    
    As we have seen in Remark <ref>, the dual functional is invariant by translations that sum up to zero. Whenever ψ is  strictly convex and C^1, we claim that every pair of maximizers can be obtained by translation of one another. More precisely, given (U_1,V_1) , (U_2,V_2) ∈(_1) ×(_2) two pairs of maximizers, then there must exist λ∈ so that (U_1,V_1) = (U_2+ λ Id__1, V_2 - λ Id__1).  To see this, we first observe that ψ being smooth implies that φ is strictly convex on the interior of its domain, see e.g. <cit.>. In particular, the primal functional admits a unique minimizer, hence thanks to Theorem <ref>, we deduce that
    
    ψ' 
            [ 
                U_1 ⊕ V_1 - C/ε]
                =
            ψ' 
            [ 
                U_2 ⊕ V_2 - C/ε]
                .

    By strict convexity, we know that ψ' is injective, hence we infer that 
    
    U_1 ⊕ V_1 - C/ε
                =
            
                U_2 ⊕ V_2 - C/ε
                  ⇒   
            U_1 ⊕ V_1
                =
            U_2 ⊕ V_2
                .

    For simplicity, for λ∈ we here adopt the  shorthand notation λ := λ Id__i. By computing the marginals from (<ref>), we see that 
    
    U_1 d_2 + d_1(V_1) = U_2 d_2 + d_1 (V_2) 
                  ,
        
    (U_1) d_2 + d_1 V_1 = (U_2) d_2 + d_1 V_2
                  ,
        ⇒ 
            U_1 - U_2 
                = 
            d_1/d_2( 
                (V_2) - (V_1) 
            ) 
                =: λ_1
                  ,
        
    
            V_1 - V_2 
                = 
            d_2/d_1( 
                (U_1) - (U_2) 
            )
                =: λ_2
                  .

      The fact that λ_1 = -λ_2 clearly follows by using these relations together with (<ref>), due to the fact that (λ_1  Id__1) ⊕ (λ_2  Id__2 ) = ( λ_1 + λ_2 )  Id__1 ⊗_2.


 §.§ Sinkhorn iterations and their convergence

In this section, we define the Sinkhorn iterations associated with a general regularization ψ and show it provides a maximizing sequence for the dual functional  defined in (<ref>). Thus, we prove the qualitative convergence of the iteration.




Under the standing assumptions of the section, assume additionally that  ψ is strictly convex and C^1. Let ℱ_1^(C,ψ,):(_2)→(_1) and ℱ_2^(C,ψ,):(_1)→(_2) be defined as in (<ref>) and (<ref>), respectively. We define the following algorithm: for n ∈, 

Step 0: fix any initial (U^0,V^0)∈(_1)×(_2). 

Step 2n-1: for given U^n-1∈(_1), we define V^n as 

    V^n := ℱ_2^(C,ψ,)(U^n-1) - λ_1(ℱ_2^(C,ψ,)(U^n-1)) Id__2
            .


Step 2n: we define U^n as 

    U^n := ℱ_1^(C,ψ,)(ℱ_2^(C,ψ,)(U^n-1)) + λ_1(ℱ_2^(C,ψ,)(U^n-1)) Id__1.














Then we have the convergence of the iteration. More precisely:

    
  (a) There exists a limit point (U^*,V^*) of  {(U^n,V^n)}_n≥ 0, which is a maximizer of D^.
    
    
    
    
    
    
  (b) The operator 
    Γ^n := ψ'
        (
            U^n⊕ V^n - C/) belongs to (_1 ⊗_2), and moreover we have that Γ^n →Γ^* ∈(_1 ⊗_2) as n →∞, where 
    
    Γ^* 
                = 
            ψ'
            ( 
                U^* ⊕ V^* - C/)
                ↦ 
            (ρ,σ)

    is a minimizer of the primal problem .



    The iteration is defined in the following way. We first start with an arbitrary initialization (U^0,V^0). Then, we compute iteratively (U^n,V^n) as follows. Denote by Ṽ^n the (C,ψ,ε)-transform of U^n-1. On one hand, 
    V^n is obtained recentering Ṽ^n so that λ_1(V^n)=0. 
    On the other hand, U^n is obtained by computing the (C,ψ,ε)-transform of Ṽ^n and translating it so that (U^n,V^n) =   (ℱ_1^(C,ψ,ε)(Ṽ^n),Ṽ^n). Intuitively, this procedure enjoys good compactness properties by Proposition <ref>, by ensuring that λ_1(V^n)=0, for every n ∈. Finally, the fact that limit points are maximizers for the dual functional is suggested by the properties of the (C,ψ,)-transform. Note indeed that by the optimality conditions (<ref>) 
    
    _1
            [
                ψ'
                (
                    U^n⊕ V^n - C/)
            ]
                = 
            ρ  whereas  _2
            [
                ψ'
                (
                    U^n⊕Ṽ^n+1 - C/)
            ]
                = 
            σ
                ,

    
    and compare with 3) in Theorem <ref>.

    We set M:=  (U_0,V_0) ∈. 

    Proof of (a).  
    See that, by construction, for any n≥ 0 we have λ_1(V^n) = 0 and additionally, due to the invariance by translation (Remark (<ref>)), from the definition of ℱ_1^(C,ψ,) and the one of ℱ_2^(C,ψ,), we obtain that
    
    (U^n,V^n)  =    ( ℱ_1^(C,ψ,)(ℱ_2^(C,ψ,)(U^n-1)) , ℱ_2^(C,ψ,)(U^n-1) ) 
    ≥    ( U^n-1 , ℱ_2^(C,ψ,)(U^n-1) ) 
    ≥    ( U^n-1 , V^n-1 ) ≥…≥ ( U^0 , V^0 ) = M.

    We conclude that the sequence {(U^n,V^n)}_n≥ 0 satisfies the assumption of Proposition <ref>, and therefore is bounded. Thus, there exist an accumulation point (U^*, V^*)∈(_1)×(_2) and a subsequence (U^n_k, V^n_k) (U^*, V^*). Now, using the continuity of :
    
    (U^*,V^*) =   lim sup_k→∞ (U^n_k, V^n_k)  ≥lim sup_k→∞ (U^n_k-1+1, V^n_k-1+1) 
    
            =   lim sup_k→∞ ( ℱ_1^(C,ψ,)(ℱ_2^(C,ψ,)(U^n_k-1)) , ℱ_2^(C,ψ,)(U^n_k-1) ) 
    ≥   lim sup_k→∞ ( U^n_k-1 , ℱ_2^(C,ψ,)(U^n_k-1) ) 
    ≥   lim sup_k→∞ ( U^n_k-1 , ℱ_2^(C,ψ,)(U^*) ) =  ( U^* , ℱ_2^(C,ψ,)(U^*) ),

    where in the last inequality, we used once again that ℱ_2^(C,ψ,)(U^n_k-1) is a maximizer of V ↦ (U^n_k-1,V).
    
    This, and a similar computation for U^*, implies that 
    
    V^* = ℱ_2^(C,ψ,)(U^*),     and     U^* = ℱ_1^(C,ψ,)(V^*),
 
    hence (U^*, V^*) is a maximizer due to Theorem  <ref>. 
    
    In addition, one observes that λ_1(V^*)= 0, as if A_k → A then λ_1(A_k) →λ_1(A) as k →∞. 
    
    
    
    
    
    
    
    
    
    
    
    Let (U^n_j, V^n_j) be any converging subsequence of (U^n, V^n) and denote by (Ũ, Ṽ) their limit point. For the first part of the proof, we obtain that (<ref>) also holds for (Ũ, Ṽ),  thus it is also a maximizer. On the other hand, due to Remark <ref>, we must then have (Ũ,Ṽ) = (U^*+λId__1, V^*-λId__2) for some λ∈. However, λ_1(Ṽ)=0 as well, and thus λ=0 or (Ũ,Ṽ) = (U^*, V^*).

    
    
    
    
    
    
    
    
    

    

    Proof of (b).     
    It follows by the optimality conditions that _1
        [
            ψ'
            (
                U^n⊕ V^n - C/)
        ]
            = 
        ρ, therefore, by taking the trace at both sides we get that [Γ^n]=1, thus proving Γ^n ∈(_1 ⊗_2). The convergence Γ^n →Γ follows from (a) and the fact that ψ' is continuous.





§ UNBALANCED NON-COMMUTATIVE OPTIMAL TRANSPORT



In this section, we turn our attention to the non-commutative optimal transport problems, both the primal and the dual, in the unbalanced case. 
We begin by analyzing the dual problem, discussing the existence of maximizers.

We work under the same standing assumptions of Section <ref>, namely throughout the whole section,  _1, _2 denotes Hilbert spaces of finite dimensions d_1 and d_2, respectively. We fix C ∈(_1 ⊗_2) and consider ρ∈_>(_1), σ∈_>(_2) (in particular with trivial kernel). 







 §.§ Existence of the maximizers to the unbalanced dual problem 


In this section, we prove the existence of maximizers in the dual problem for the unbalanced case for a general regularization.

We recall the definition of the dual functional 

    (U,V)
           = -τ_1
         [ 
            e^-U/τ_1+logρ- ρ] 
            -
        τ_2
        [ 
           e^-V/τ_2+logσ- σ] 
            -[ 
            ψ(U⊕ V-C/)
        ]
            .

In order to show the existence of maximizers, we show directly that the functional  is coercive, differently from what happens in the balance case with  (due to the presence of the symmetries, cfr. Remark <ref>). Nonetheless, we significantly simplify the proof by observing that the unbalanced dual functional is in fact always bounded from above by the balanced dual functional, as we show in the proof below.

[Existence of the maximizer, unbalanced]

    Let _1, _2 be finite-dimensional Hilbert spaces.
    Let ρ∈_>(_1) and σ∈_>(_2) with trivial kernel, ε, τ_1, τ_2 >0 and let ψ:→ be a superlinear convex function bounded from below. Then the unbalanced dual functional D^,τ defined in (<ref>) admits a (unique) maximizer (U^*,V^*)∈(_1)×(_2).



    First of all, we observe that the functional  is continuous over (_1) ×(_2), arguing in the same way as in (<ref>). Therefore, the existence of a maximizer follows by showing that  is coercive, i.e. it has bounded superlevel sets. Moreover, without loss of generality, we can assume that ψ≥ 0.
    
    We introduce the following notation: for (U,V) ∈(_1) ×(_2), we set 
    
    F_1(U) = τ_1 [e^-U/τ_1+logρ] ≥ 0
                ,       F_2(V) = τ_2 [e^-V/τ_2+logσ]≥ 0
                ,
        
    G(U,V) = [ψ(U⊕ V -C/)] ≥ 0
                ,

    so that by construction, for every (U,V) ∈(_1) ×(_2) one has 
    
    (U,V) 
                = 
            -F_1(U) - F_2(V) - G(U,V)
                + τ_1 [ρ] + τ_2 [σ] 
            .

    
    We start by claiming that 
    
    (U,V)  ≥ (U,V) 
                  ,   for all  (U,V) ∈(_1) ×(_2)
                  , τ_1, τ_2 ∈_+ 
                    .

    This is consequence of Klein's inequality <cit.> applied to the exponential function, which implies (for F_1, and similarly for F_2)
    
    F_1(U)
            - τ_1 [ρ]
            =
        τ_1
         [ 
            e^-U/τ_1+logρ- e^logρ]
            
            ≥τ_1
         [ 
           e^logρ(
                - U/τ_1)
        ]
            = 
        -  [ U ρ ] 
            ,

    thus providing the claimed lower bound (<ref>).

    Now, we want to show that, given M_0 ∈ℝ, the set
    
    𝒮_M_0:= { (U,V) ∈(_1) ×(_2):   (U,V) ≥ - M_0 }

    is bounded. In particular, up to a translation that depends only on the marginals, there exists M∈ so that 𝒮_M_0 is a subset of 
    
    { (U,V) ∈(_1) ×(_2):  F_1(U)+F_2(V)+G(U,V) ≤ M }.

    For (U,V) ∈𝒮_M, we have that 0≤F_1(U)+F_2(V) + G(U,V) ≤ M, which, by the positivity of all the terms, gives that F_1(U) ≤ M, F_2(V) ≤ M, G(U,V) ≤ M. 
    

    We define Û:=-U/τ_1+logρ.
    We write the spectral decomposition Û =∑_i λ_i|ξ_i⟩⟨ξ_i|, where λ_i := λ_i(Û). Therefore
    
    M 
                ≥F_1(U) 
                = 
            τ_1 [e^Û] 
                = 
            τ_1 ∑_i e^λ_i≥τ_1 e^λ_d  ⇒  λ_d_1≤log(M/τ_1).

     
     Since U=-τ_1 Û+τ_1logρ, we have λ_1(U)≥λ_1(-τ_1 Û)+λ_1(τ_1logρ)=-τ_1λ_d_1(Û)+τ_1log(λ_1(ρ)).
     Combining the previous inequalities, we conclude that 
    
    λ_1(U) 
                ≥ 
            -τ_1 log( M/λ_1(ρ) τ_1)
                =: _1 > -∞ 
                    .

    Arguing analogously for V and F_2(V), we also obtain that
    
    λ_1(V) 
                ≥ 
            -τ_2 log( M/λ_1(σ) τ_2)
                =: _2 > -∞ 
                    .

    In order to obtain the sought upper bound on the spectrum of U and V, we take advantage of (<ref>), which implies (U,V) ≥ M_0. In particular, we apply Proposition <ref> and we get ||U+λ_1(V)Id_H_1|| ≤ and ||V-λ_1(V)Id_H_2|| ≤, hence in particular for all i=1,…,d_1 and j=1,…,d_2, the two-sided bounds
    
    -≤λ_i(U) +λ_1(V) 
                ≤
                ,      
            0
                ≤λ_j(V) -λ_1(V) 
                ≤
                .

    Along with (<ref>) and (<ref>) one easily obtains 
    
    _1
                ≤λ_i(U)
                ≤ - _2
                ,     _2 
                ≤λ_j(V)
                ≤ 
            2 - _1
                ,

    which yields 𝒮_M_0 is bounded, thus the sought coercivity property for .








 §.§ The -transform and its properties



Recall the setting introduced at the beginning of Section <ref>.

Similarly as done in the balanced case, we now introduce the corresponding notion of (C,ψ,)-transform associated to . Note that in this case, we do not require ψ to be strictly convex, due to the presence of the exponential (which is strictly convex) in the marginal penalization terms. In particular, the existence of the maximizer in the definition below follows from the concavity and continuity of , together with the coercivity proved within the proof of Theorem <ref>, whereas uniqueness comes exactly from the strict concavity of . 

For simplicity of notation, we set τ := (τ_1,τ_2). Recall the decomposition as given in (<ref>), in particular the definitions of F_1, F_2, and G in (<ref>).



[(C,ψ,,τ)-transform]

We define ℱ_2^(C,ψ,,τ)(_1)→(_2) as

    ℱ_2^(C,ψ,,τ)(U) 
            =
        { 
            - F_2(V) - G(U,V)
                 
            V ∈(_2) 
        }.

Analogously, we define the corresponding ℱ_1^(C,ψ,,τ)(_2)→(_1) as

    ℱ_1^(C,ψ,,τ)(V) 
            = 
        { 
            - F_1(U) - G(U,V)
                 
            U ∈(_2)
        }.


The definition of the transform is well-posed, as the  contains exactly one element, as mentioned above. The next step is to characterize the optimality condition in the same spirit as done in the balanced case in Proposition <ref>.



 
    Under the standing assumptions of this section, assume additionally that ψ∈ C^1. Then, for every U ∈(_1), its transform ℱ_2^(C,ψ,,τ)(U) is the unique solution V of the equation 

    e^-V/τ_2+logσ =_2 [ ψ'( U ⊕V - C/)].

Analogously, the characterization holds for ℱ_1^(C,ψ,,τ)(V), replacing σ with ρ, _2 with _1.






    Set V:= ℱ_2^(C,ψ,,τ)(U). For every δ∈ℝ and A ∈(_2), arguing as in the proof of Proposition <ref>, from the fact that the map δ↦ - F_2(V+δ A) - G(U,V+δ A) has vanishing derivative in δ =0, we obtain that 
    
    [ 
                e^-V/τ_2 + logσ A
            ]
                -
            [ 
                ψ' (U ⊕ V-C/)( Id⊗ A) 
            ] 
                = 
            0
                  ,   ∀ A ∈(_2)
                ,

   which provides the sought (<ref>), by the very definition of partial trace _2. The second statement follows by arguing similarly. Uniqueness comes directly from the strict concavity of the function V ↦ (U,V).




















Recall the definition of the primal functional (<ref>) and problem (<ref>) in the unbalanced case.


[Equivalent characterizations for maximizers, unbalanced]

    Under the standing assumptions of this section, assume ψ∈ C^1. Let (U^*, V^*) ∈(_1) ×(_2). Then the following are equivalent:
    
        
  1) (Maximizers) (U^*,V^*) maximizes D^,τ, i.e. = D^,τ (U^*,V^*).
        
  2) (Maximality condition) U^* = ℱ_1^(C,ψ,,τ)(V^*) and V^* = ℱ_2^(C,ψ,,τ)(U^*).
        
        
        
        
        
  3) (Duality) There exists Γ^* ∈_≥(_1 ⊗_2) so that (Γ^*)= D^,τ (U^*,V^*).
    
If one (and thus all) condition holds, the unique Γ^* which satisfies 3) is given by 

    Γ^*:=ψ' [ U^* ⊕ V^* - C/ε]
            ,

and it is the unique minimizer to . 


As in the balanced case, from Theorem <ref> and Theorem <ref>, we obtain the sought duality.
[Duality for unbalanced QOT]
    Under the standing assumptions of this section, assume that ψ = φ^* is C^1. Then =. 


    We shall show 1) ⇒ 2) ⇒ 3) ⇒ 1). 

    1) ⇒ 2) .  
   This follows as in the proof of Theorem <ref>.

    2) ⇒ 3).  
    Consider Γ^* as given in (<ref>). 
    From the properties of the Legendre transform and (<ref>) (by computing the optimality condition in the definition of ^*(·|η)), we infer that 
    
    (α|η) = [  
                η - e^A+logη] 
                + ⟨ A, α⟩  ,     if α = e^A+logη
                        .
 
    By applying the above equation to η= ρ (resp. η = σ), A = -U^*/τ_1 (resp. A=-V^*/τ_2), noting that α = _1(Γ^*) (resp. α=_2(Γ^*)) from the optimality condition Proposition <ref>, we obtain 
    
    τ_1 (_1(Γ^*)|ρ) 
                = 
            τ_1 [  
                ρ - e^ -U^*/τ_1+logρ] 
                - 
             [
                U^* _1(Γ^*) 
            ]
                        .
        
    τ_2 (_2(Γ^*)|σ) 
                = 
            τ_2 [  
                σ - e^-V^*/τ_2+logσ] 
                - 
            [
                V^* _2(Γ^*)
            ] 
                        .

    On the other hand, the same computation as in (<ref>) shows that 
    
    [ C Γ^*] 
                +
            [ 
                φ(Γ^*)
            ]
                = 
            [
                U^* _1(Γ^*) 
            ]
                +
            [
                V^* _2(Γ^*)
            ] 
                - 
            ε[
                ψ(U^*⊕ V^*-C/ε)
            ]
                .

    Putting the last three equalities together we end up with
    
    (Γ^*) 
                   =
            τ_1 [  
                ρ - e^ -U^*/τ_1+logρ]      
                + 
            τ_2 [  
                σ - e^-V^*/τ_2+logσ] 
                -
            ε[
                ψ(U^*⊕ V^*-C/ε)
            ]
        
       =
             (U^*,V^*)
                ,

    which is the sought equality. 
    
    3) ⇒ 1).      
    The proof is completely analogous to the one provided in the proof of Theorem <ref>, including the fact that Γ^* as given in (<ref>) is the unique minimizer for the primal problem.







§ EXTENSIONS OF THE DUALITY THEOREM




We finally show that the duality result holds with the weaker assumption of ψ being (not necessarily strictly) convex, superlinear, and bounded from below. In particular, no additional smoothness is required.

[Duality for non-smooth ψ]

    Let ψ be a convex function satisfying (<ref>). Under the standing assumptions of Section <ref>, duality holds, i.e. 
    
    = 
              ,

    for every τ∈ (0,+∞] (where τ = +∞ denotes the balanced problem).


Before proving the theorem, we discuss some preliminary constructions. Let ψ be a superlinear function, bounded from below, and convex (but not necessarily strictly convex nor C^1). 
Fix a parameter n ∈, a smooth, nonnegative, monotone increasing and strictly convex function  ψ̅, and a function ρ∈ C_c^∞(), strictly positive, symmetric, and supported on [-1,1], and such that ∫ρ x = 1. Construct the sequence of mollifiers ρ_n in the usual way, i.e. ρ_n(x) := n ρ(nx), so that ρ_n satisfies the same properties of ρ (on the interval [-1/n, 1/n]), for every n ∈. Finally, define

    ψ_n: →  ,   ψ_n := 
        (
            ψ + 1/nψ̅)
            *
        ρ_n
             .

Note that in the unbalanced case τ_1,τ_2 <∞, one could choose ψ̅=0 (due to the fact that no strict convexity is required in Theorem <ref>). 
First of all, we note that ψ_n ∈ C^∞, and if ψ≥ m for some m ∈_+, then ψ_n ≥ m for every n ∈. Furthermore, ψ_n is also superlinear at +∞, since by Fatou's lemma

    lim inf_x → +∞ψ_n(x)/x≥lim inf_x → +∞∫ψ(x-z)/xρ_n(z)
                ≥∫lim inf_x → +∞( ψ(x-z)/x)
                ρ_n(z)
                = 
            +∞ 
                .

Finally, observe ψ_n = ψ̅_n * ρ_n, where ψ̅_n := ψ + n^-1ψ̅ is a strictly convex function (being a sum of a convex and a strictly convex function). As a consequence, using that ρ_n ≥ 0, we see that for every x,y ∈, λ∈ [0,1] 

    ψ_n(λ x + (1-λ) y )
               =
        ∫ψ̅_n
            (
                λ (x-z) + (1-λ) (y-z)
            ) 
                ρ_n(z) 
    
       ≤λ∫ψ̅_n(x-z) 
                ρ_n(z) 
            +
        (1-λ) ∫ψ̅_n(y-z) 
                ρ_n(z) 
    
       =
        λψ_n(x)  
            +
        (1-λ) ψ_n(y)    
              ,

with equality if and only if, for every z ∈, 

    ψ̅_n 
        (
            λ (x-z) + (1-λ) (y-z)
        )
            ρ_n(z)
                =
        (
             λψ̅_n(x-z) 
                  +
            (1-λ) ψ̅_n(y-z)
        ) 
            ρ_n(z)
                .

Using that ρ_n >0 on [-1/n,1/n], and that ψ̅_n is strictly convex, we infer that x=y, which precisely shows that ψ_n is strictly convex as well.



We thus constructed a smooth, bounded from below, strictly convex function ψ_n, which we claim to be a good approximation of ψ, in a suitable uniform sense.


First of all, note that ψ being the Legendre transform of a convex map φ satisfying (<ref>) implies that ψ is monotone non-decreasing as well as 

    lim_x → -∞ψ(x) 
                =
        inf_x ∈ψ(x)
                = 
            -φ(0) ∈ 
                ,

see Remark <ref> in the appendix for a proof. In particular, this implies that 

    ψ∈ L^∞(I) ∩Lip(I)
              ,   for every interval  I = (-∞, b]
              ,   b ∈ 
                .

The same holds by construction for ψ̅. 
As a consequence, we deduce that, for every b ∈, 

    sup_x ∈ (-∞,b] 
            | ψ_n(x) - ψ(x)|
               ≤sup_x ∈ (-∞,b]
            | ψ̅_n * ρ_n(x) - ψ * ρ_n(x) |
                +
            | ψ * ρ_n(x) - ψ(x) |
    
       = 
        sup_x ∈ (-∞,b]1/n 
            | ψ̅* ρ_n(x) |  
                +
            |  ψ * ρ_n(x) - ψ(x) |
    
       ≤1/nsup_x ∈ (-∞,b+1]
            |ψ̅(x)| 
                +
            Lip(ψ)( (-∞,b+1]) 
                ∫ |y| ρ_n(y) 
    
       ≤1/n(
            sup_x ∈ (-∞,b+1]
            |ψ̅(x)| 
                +
            Lip(ψ)( (-∞,b+1]) 
        )
                    .

which clearly goes to 0 as n →∞.

This in particular shows that ψ_n →ψ in L^∞_loc().




Note due to the fact that ρ (hence ρ_n) is symmetric (and thus in particular ∫ z ρ_n(z) = 0) and ψ̅≥ 0, we have that 

    ψ_n (x) 
            ≥ψ * ρ_n (x) 
            Jensen≥ψ(
            ∫ (x-z) ρ_n(z)
        )
            = 
        ψ(x) 
              ,     ∀ x ∈  .


We consider the non-commutative optimal transport problems associated with ψ_n and φ_n := ψ_n^*, and denote by ,  respectively the dual and the primal functional associated to ψ_n, and with ,  the dual and the primal problem, respectively.

By definition of dual functional, together with (<ref>), this shows that ≤≤, for every n ∈. In particular, it is straightforward to check that the validity of Proposition <ref> for   implies the following equi-coercivity property (up to translation) for .





For every  U∈(_1), V∈(_2) such that 

    D^,τ,n (U,V)≥ M >-∞  (
            and λ_1(V)=0   if τ = +∞)

for some M∈, there exists a constant 0≤𝒜=𝒜(M) <∞ independent of n such that ||U||_∞,||V||_∞≤.  In particular, whenever a sequence (U_n,V_n) ∈(_1) ×(_2) satisfies, for every n∈,

    inf_n ∈D^,τ,n (U_n,V_n)≥ M >-∞  (
                and λ_1(V_n)=0 
               if τ = +∞)
                ,

then { (U_n,V_n)}_n is bounded, and therefore pre-compact.


We are ready to prove Theorem <ref>. 


We apply  Theorem <ref>, Theorem <ref> to ψ_n and deduce that

    =   ,     ∀ n ∈ 
                .

We claim we can pass to the limit the previous equality as n →∞ to conclude the duality result for ψ. We start by observing that for every (U_n,V_n) → (U, V) ∈(_1) ×(_2), we have that 

    lim_n →∞ (U_n,V_n) 
                = 
         (U,V)
            ,

which is a stronger statement than Γ-convergence (every approximating sequence is a recovery sequence).
This follows from the fact that converging sequences in (_1) ×(_2) are bounded and that ψ_n →ψ in L^∞_loc(). 




This in particular shows that -Γ→ -. Together with (<ref>) and Proposition <ref>, by the fundamental theorem of Γ-convergence, it provides the claimed convergence at the level of the dual problems

    lim_n →∞ 
        =   
            .


We are left to show the convergence of the primal problems, which follows from very similar arguments, namely by showing that Γ→ plus suitable compactness. 



In order to show the first statement, we first claim that ψ_n →ψ in  L^∞_loc(I) for I bounded from above (cfr. (<ref>)) implies φ_n →φ in  L^∞_loc(_+). To prove this, we start observing that for every compact K ⊂_+, φ is continuous, and thus bounded, on K. Therefore, for every p ∈ K, we have that 

    - 
        φ_L^∞(K)≤φ(p) 
            =        
        sup_x ∈{
            p x - ψ(x) 
        }
        
            .

This in particular shows that the sup in the previous equation can be taken over the set 

    C(p)
            := 
        {
            x ∈
        
        
        p x - ψ(x)
            ≥ 
        - φ_L^∞(K) -1 
        }
            .





As a  consequence of the Fenchel–Young inequality, we infer that, for p ∈ K, 

    φ(p) 
            = 
        sup_x ∈ C(p){
            p x - ψ(x) 
        }   ≤sup_x ∈ C(p){φ_n(p) + ψ_n(x) - ψ(x) 
        }
       ≤φ_n(p) + ψ - ψ_n _L^∞(C(p))
            .

On the other hand, we know that φ_n ≤φ everywhere (as ψ≤ψ_n), hence we conclude that 

    φ - φ_n _L^∞(K)≤sup_p ∈ Kψ - ψ_n _L^∞(C(p))  ,   ∀ n ∈ 
            .

Let p_K be the max of K. Then, by definition of the sets themselves, we have that for every p ∈ K, 

    C(p) ∩_+ ⊂ C(p_K)
             
        C(p_K) ⊂ (-∞ , α_K] 
            ,

for some α_K < +∞, by superlinearity of ψ, see Remark <ref>. This, together with (<ref>), provides the estimate

    φ - φ_n _L^∞(K)≤ψ - ψ_n _L^∞ ((-∞,α_K] )  ,   ∀ n ∈ 
            .


Recalling that from (<ref>) we know that ψ_n →ψ uniformly on every set which is bounded from above, we conclude that φ_n →φ in  L^∞_loc(_+).


As in (<ref>), we now take advantage of this fact  to ensure that

    lim_n →∞(Γ_n) 
                = 
        (Γ)
              ,   ∀ Γ_n →Γ∈(_1 ⊗_2)
            .





Secondly, let (U_n^*,V_n^*) be the maximizer of  (with λ_1(V_n^*)=0 if τ = +∞). By Theorem <ref>, Theorem <ref>  we know that 

    Γ_n^* 
            := 
        ψ_n' [ U_n^* ⊕ V_n^* - C/ε]

is the unique minimizer of .

Thanks to Proposition <ref>, we know that, up to a non-relabeled subsequence, (U_n^*,V_n^*) → (U^*,V^*) as n →∞. Moreover, with ψ being convex, we have that ψ is locally Lipschitz. In particular, by the properties of the convolution this implies that, for every bounded set I ⊂, one has that 

    sup_n ∈ψ_n' _L^∞(I)≤Lip (ψ;B_1(I)) + Lip (ψ̅;B_1(I)) 
                < ∞ 
            .

This, together with the boundedness of (U_n^*,V_n^*), implies that {Γ_n^* }_n is also bounded, and therefore pre-compact. 
Thanks to the Γ-convergence showed above in (<ref>), this in particular shows that every limit point (which exists) must necessarily be a minimizer (the marginal constraints are closed by uniform convergence), thus concluding the proof of Theorem <ref> passing to the limit in (<ref>).
















§ CONVERGENCE RESULTS: FROM UNBALANCED TO BALANCED NON-COMMUTATIVE OPTIMAL TRANSPORT



In this section, we discuss limit behaviors for the unbalanced optimal transport as the parameters τ_1, τ_2 → +∞. Recall that we set τ=(τ_1,τ_2), and use the slight abuse of notation τ→ +∞. We discuss convergence results both at the level of primal and dual problems, as well as the convergence of minimizers/maximizers. 


[Convergence as τ→ +∞]

    Under the standing assumption of Section <ref>, we have the following convergence results:
    
        
  * The functionals   Γ-converge as τ→∞ to the functional 
        
    Γ↦(Γ)
                           if Γ↦ (ρ,σ) 
                
    
                    +∞   otherwise  .

        
        
  * The unique minimizer  Γ^*,τ∈(_1 ⊗_2) of  converges (up to subsequence) as τ→ +∞ to a minimizer Γ^* ∈(_1 ⊗_2) of , under the constraint Γ^*  ↦ (ρ,σ). 
        
  * We have the following convergence result for the dual:
        for every (U_τ,V_τ) → (U,V) ∈(_1) ×(_2) as τ→ +∞ one has that 
        
    lim_τ→ +∞D^,τ (U_τ,V_τ) = D^ (U,V)
                        .

        In particular, -D^,τ Γ-converges to -D^.
        
  * The unbalanced optimal transport problems converge as τ→ +∞  to the balanced optimal transport problem, i.e.
        
    lim_τ_1,τ_2 → +∞
                        =
                    
                        =
                     
                        =
                lim_τ_1,τ_2 → +∞ 
                        ,

        and the minimizers Γ^*,τ converge (up to subsequence) to a minimizer of  with marginal constraints.
        
  * Let (U^*,τ, V^*,τ) ∈(_1) ×(_2) be the unique maximizer of D^,τ. Define the recentered potentials 
        
    (
                    Û^*,τ, V̂^*,τ) 
                    := 
                ( 
                    U^*,τ + λ_1(V^*,τ)  Id__1
                        , 
                    V^*,τ - λ_1(V^*,τ)
                 Id__2)
                    ∈(_1) ×(_2)
                        .

        Then (Û^*,τ, V̂^*,τ) converges (up to subsequence)  as τ→ +∞ to a maximizer (U^*,V^*) ∈(_1) ×(_2) of D^ which satisfies λ_1(V^*)=0.
    
    If we assume additionally that ψ∈ C^1 and strictly convex, then the Γ^*,τ converge to the unique minimizer in , while (Û^*,τ, V̂^*,τ) converges to the unique maximizer (U^*,V^*) ∈(_1) ×(_2) of D^ which satisfies λ_1(V^*)=0.


Note that  is not coercive, since the sequence {(a Id,-a Id)}_a ∈ℝ is such that (a Id,-a Id) =C_0 ∈, but it does not admit a convergent subsequence. By Theorem <ref>, and by the fundamental theorem of Γ-convergence, this shows that the family of functionals  is not equi-coercive (note indeed that the bounds obtained in the proof of Proposition <ref> are not uniform in τ_1,τ_2). This explains why it is not directly possible to infer the main results for the balanced dual problem directly from the unbalanced setting by sending the parameter to infinity. For a similar reason, the Γ-convergence result proved in 1. in the above theorem does not directly provide the validity of 3., which instead requires a different argument.















1.  
First, we show the claimed Γ-convergence for the primal functionals. We prove the lim inf and the lim sup inequality, starting from the first one. Let Γ_τ→Γ∈(_1 ⊗_2) as τ→ +∞, and assume without loss of generality that sup_τF^,τ (Γ_τ) <∞. By the continuity of , we know that 

    lim_τ→ +∞(Γ_τ) 
                =
            (Γ) < ∞  ,

and therefore we also infer that 

    sup_τ_1, τ_2 > 0τ_1 ( P_1(Γ_τ)| ρ)+ τ_2 ( P_2(Γ_τ)| σ) 
                < ∞ 
            .

Note that the partial trace is a continuous operation with respect to the Hilbert-Schmidt convergence of operators. Therefore we have that P_i(Γ_τ) → P_i(Γ) as τ→∞. In  particular, we deduce that 

    sup_τ_1, τ_2 > 0( P_1(Γ_τ)| ρ) +  ( P_2(Γ_τ)| σ) 
                < ∞ 
            .

Combining the estimates in (<ref>) and (<ref>), from the convergence of the marginals and the continuity of the relative entropy we conclude that 

    ( P_1(Γ)| ρ)
            +
        ( P_2(Γ)| σ)
            =
        lim_τ→ + ∞( P_1(Γ_τ)| ρ)
                +
            ( P_2(Γ_τ )| σ)
            =
        0   
            .

By using that the relative entropy is a nonnegative functional, this shows that both relative entropies must vanish, which in particular yields Γ↦ (ρ,σ). 
Combining this with (<ref>), we end up with

    lim inf_τ→ +∞F^,τ (Γ_τ)
        ≥(Γ) 
                Γ↦ (ρ,σ)
            ,

which concludes the proof of the lim inf inequality.

For the lim sup inequality, it is enough to choose the constant recovery sequence. 














2.  
Note that {F^,τ}_τ is equi-coercive, due to the fact that F^,τ≥, for every τ>0, together with the fact that  has bounded sublevels.
As a consequence, 2. follows directly from 1. by the fundamental theorem of Γ-convergence.


3.  
Let us now show the convergence claimed in (<ref>). 
To this purpose, we observe that 

    lim_τ_1 → +∞
            -τ_1 [ 
                e^-U_τ_1/τ_1 + logρ - ρ] 
            =
        lim_t → 0^-[ 
                    e^logρ + t Ũ_t - e^logρ] 
                /t
            =  
        / t |_t=0
            g(logρ + t U) 
                ,

where we set Ũ_t := U_-1/t and g(A):= [e^A]. The last equality is a consequence of the following observation: let B⊂(_1) be the ball of radius 1 centered in logρ∈(_1). For t sufficiently small, both logρ + t Ũ_t and logρ + t U belong to B, hence by (<ref>) we can estimate

    |
         1/t[ 
            e^logρ + t Ũ_t - e^logρ +  t U]
        |
            ≤Lip (g ; B)
        
             (Ũ_t -U)
        _HS→ 0
                ,

as t → 0^-.  The claimed equality readily follows.








































An application of Lemma <ref> yields  ∇ g(A) = e^A as well as that the lim inf is in fact a limit, and it holds

    lim_τ_1 → +∞
            -τ_1 [ 
                e^-U/τ_1 + logρ - ρ] 
            =
       ⟨ 
            (∇ g)(logρ)
                , 
            U
       ⟩
            =
         [ ρ U ]
            .

































The very same argument applied on _2 ensures that 

    lim_τ_2 → +∞
            -τ_2 [ 
                e^-V/τ_2 + logσ - σ] 
            =
         [ σ V ]
            .

The latter two equalities prove exactly that 
lim_τ→ +∞D^,τ (U_τ,V_τ) =  (U,V), hence providing the claimed convergence.



4.   
The convergence of the primal problems follows from the same argument above in the proof of 2. The convergence (and equality) of the dual problems follows by duality, in particular Theorem <ref>.

5.  
We take advantage of Remark <ref> and once again of (<ref>), to deduce that

    ( 
            Û^*,τ, V̂^*,τ)
            =
        ( 
            U^*,τ,  V^*,τ)
            ≥D^,τ( 
            U^*,τ,  V^*,τ)
            = 
        𝔇^,τ 
            =
        𝔉^,τ
            ,

where in the last step we used the duality result proved in Theorem <ref>. We use the convergence proved in 3. to infer

    lim inf_τ→ +∞( 
                Û^*,τ, V̂^*,τ)
                ≥lim inf_τ→ +∞𝔉^,τ   
                =
       
                > 
        - ∞ 
            .

This implies that the sequence 
{( 
    Û^*,τ, V̂^*,τ) 
}_τ
belongs to a superlevel set of , and thus by construction satisfies the hypothesis of Proposition <ref>. It follows that it is a bounded sequence of operators, and therefore it is pre-compact. We conclude the proof of 4. by showing that any limit point is necessarily a maximizer (U^*,V^*) of  with λ_1(V^*)=0. Assume indeed that 
( 
    Û^*,τ, V̂^*,τ) → (U^*,V^*) as τ→ +∞. The fact that λ_1(V^*,τ)=0 for every τ>0 ensures that λ_1(V^*)=0. Finally, by (<ref>) together with the continuity of  we deduce that

    (U^*,V^*)
            =
        lim_τ→ +∞( 
                Û^*,τ, V̂^*,τ)   
                ≥
            = 
         
            ,

where at last we used the duality result for the balanced case, i.e. Theorem <ref>. We thus conclude that (U^*,V^*) is indeed a maximizer with λ_1(V^*)=0.









 §.§ Convergence of 


Assume that ψ is strictly convex and C^1.
In this short section, we take advantage of the convergence results proved in Theorem <ref> to show that also the associated (C,ψ,,τ)-transforms converge, as the parameters τ→ +∞. More precisely, we claim that 

    ℱ_1^(C,ψ,,τ)(V_τ)
            →ℱ_1^(C,ψ,)(V)
                ∀   
            V_τ→ V 
                .

Recall from the previous section that the following property (stronger than Γ-convergence) holds: one has that (<ref>)

    lim_τ→ +∞D^,τ (U_τ,V_τ) 
        = 
             (U,V)
                  ,     for every    (U_τ, V_τ) → (U,V)
                  .

In particular, we have the Γ-convergence of the one-parameter functionals, given by 

    - D^,τ (·, V_τ) 
            
        -D^ (·,V)
            ,

for every V_τ→ V. Using once again the inequality (<ref>) together with the coercivity of the dual functional in the balanced case (cfr. Proposition <ref>, see also Remark <ref>), it readily follows that the sequence of functionals {- D^,τ (·, V_τ) }_τ is equi-coercive. This, together with the aforementioned Γ-convergence, provides the convergence of the associated minimizers, which precisely means 

    lim_τ→ +∞ℱ_1^(C,ψ,,τ)(V_τ)
        = 
            ℱ_1^(C,ψ,)(V)
                    ∀   V_τ→ V 
                    ,

as claimed.














































§ LEGENDRE'S TRANSFORM IN A NON-COMMUTATIVE SETTING




In this appendix, we study the properties and formula for the Legendre transform of functions obtained by composing the trace operator with (the lifting of) a convex map on the real line. More specifically and throughout all the section, let φ [0,+∞) → [m,+∞) be convex, superlinear at infinity, namely lim_t → +∞φ(t)/t =+∞, and m > -∞. Recall that, given such φ, its Legendre transform is given by 

    ψ: →  ,   ψ(x) := 
        sup_t ∈{
            x t - φ(t) 
        }  ,   ∀ x ∈ 
            ,

where we extend φ to +∞ over the negative half-line. 

    

    
    



    Under the aforementioned assumptions on φ, we claim that ψ is monotone non-decreasing and satisfies 
    
    lim_x → -∞ψ(x) = 
            inf_x∈ψ(x) 
            = - φ(0) ∈  .

Indeed, any non-constant real-valued convex function ψ on  is either 

    
  (i) monotone non-decreasing
    
  (ii) monotone non-increasing
    
  (iii) admits a global minimizer x^*∈.

For (i) the conclusion trivially follows, and we claim that both (ii) and (iii) fail under the assumptions on φ, since (ii)-(iii) must imply that lim_x→-∞ψ(x) = +∞ or/and there exist x_1<x_2 such that ψ(x_1)>ψ(x_2). We can exploit this fact to obtain that φ(p)<+∞ for some p <0, contradicting the original assumption on φ.

Indeed, let p_1∈∂ψ(x_1), then ψ(x)≥ψ(x_1) + p_1(x-x_1). Since x_1 is not a minimizer by construction, then p_1≠ 0, and in fact p_1 must be strictly negative, as 

    ψ(x_2)≥ψ(x_1) +p_1(x_2-x_1) > ψ(x_2) + p_1(x_2-x_1).

However, then we have φ(p_1) = ψ^*(p_1) = p_1 x_1 - ψ(x_1) < ∞, contradicting that φ(p)=+∞ for all p<0.




 §.§ Properties of operator liftings

Given a d-dimensional Hilbert space ℋ, the map that sends f ∈ C(ℝ) into Lin((ℋ),(ℋ)) via spectral calculus is a vector space homomorphism and preserves the natural composition and product operations in C(ℝ) and Lin((ℋ),(ℋ)), namely given f_1,f_2  ∈ C() and λ∈ℝ we have

    (f_1+f_2)(A) = f_1(A) + f_2(A),          (λ f_1)(A) = λ f_1(A),
        (f_1· f_2)(A) = f_1(A) f_2(A),        (f_1 ∘ f_2)(A) =f_1(f_2(A)),

for every A ∈().

It can be proved (see for instance <cit.>) that, if φ [0,+∞) →ℝ is convex (respectively strictly convex), then _≥()∋Γ↦[φ(Γ)] ∈ℝ is convex (respectively strictly convex).

In particular, since real-valued convex functions on ℝ^n are locally Lipschitz we have that 

    _≥()∋Γ↦[φ(Γ)] ∈ℝ     is locally Lipschitz.


Additionally, using that () ∋ W ↦[Wξ] is continuous for every ξ∈(), we infer  

    (_1) ×(_2) ∋(U,V) ↦ (U,V) ∈ℝ  is (strictly) concave and continuous

whenever ψ is (strictly) convex. 

Note that Γ↦φ(Γ) is not necessarily convex (e.g. φ(x) = x^3, see <cit.>). 






    By (<ref>) with W:= (U ⊕ V - C)/, we obtain   
    
    ⟨ U ⊕ V - C, Γ⟩≤[ φ(Γ)] 
                + 
            [ ψ(
                    U ⊕ V - C/)
            ]
                .

    Therefore, for every Γ↦ (ρ,σ), we have that 
    
    (Γ) 
                   = 
            [C Γ] + [φ(Γ)]
                ≥⟨ U ⊕ V , Γ⟩ 
                - 
            [ ψ(
                    U ⊕ V - C/)
            ] 
        
       =
            [U ρ] + [V σ]  
                - 
            [ ψ(
                    U ⊕ V - C/)
            ] 
                =
             (U,V)
                ,

    for every (U,V) ∈(_1) ×(_2). The same holds true for the unbalanced case, using the fact that the Legendre transform of the relative entropy is the exponential function, cfr. (<ref>).





Let φ satisfy the standing assumption of the section and let us denote the lifting of φ to the space of Hermitian matrices with a slight abuse of notation by φ. For every W ∈()
    
    sup_Γ∈_≥()( ⟨ W, Γ⟩ - [ φ(Γ)] )
                = 
            max_Γ∈_≥()( ⟨ W, Γ⟩ - [ φ(Γ)] )




Given M ∈ℝ, we claim that the superlevel set 𝒮_M:={Γ : ⟨ W,Γ⟩ -[ φ(Γ)] ≥ M} is bounded. By contradiction, there exists a sequence {Γ^n }_n ⊂𝒮_M such that, denoting {λ_i^n}_i the eigenvalues of Γ^n with λ_1^n ≤λ_2^n ≤…≤λ_d^n, we have that λ_d^n → +∞. We estimate

    ⟨ W, Γ^n ⟩ -[ φ(Γ^n) ] 
           ≤ d W λ_d^n -∑_i=1^d φ(λ_i^n)= d W λ_d^n - φ(λ_d^n)-∑_i=1^d-1φ(λ_i^n)
    
       ≤ d W λ_d^n - φ(λ_d^n)-min{φ}(d-1),

which converges to -∞, using the property that φ is superlinear at infinity.
Thus, the claim is proved. Moreover, notice that the map Γ↦⟨ W, Γ⟩ - [ φ(Γ)], is continuous by (<ref>), hence the maximum exists.




    Let φ satisfy the standing assumption of the section. We have that 
    Ψ(W):=sup_Γ∈_≥()( ⟨ W, Γ⟩ - [ φ(Γ)] )
 satisfies Ψ(W)=[ψ'(W)], where ψ' is the lifting of ψ=φ^* to the space of Hermitian matrices. 



    We claim that
        
    ⟨ W, Γ⟩≤[ φ(Γ)] + [ ψ'(W)].

    To do this, let us write Γ = ∑_i=1^d Γ_i |γ_i⟩⟨γ_i| and W = ∑_i=1^d W_i |ξ_j⟩⟨ξ_j| and compute
    
    ⟨ W, Γ⟩ = [ W Γ] 
               = ∑_iΓ_i ⟨γ_i|W|γ_i⟩ = ∑_i∑_j Γ_i W_j |⟨ξ_j |γ_i⟩|^2
        
       ≤∑_i ∑_j (φ(Γ_i) + ψ(W_j)) |⟨ξ_j |γ_i⟩|^2
        
       = ∑_i  φ(Γ_i) (∑_j|⟨ξ_j |γ_i⟩|^2) +  ∑_j ψ(W_j) (∑_i |⟨ξ_j |γ_i⟩|^2) 
        
       =  ∑_i  φ(Γ_i) +  ∑_j ψ(W_j)  = [φ(Γ)]+ [ ψ'(W) ].

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    This gives that Ψ(W)≤[ ψ'(W) ]. We prove that Ψ(W)≥[ ψ'(W) ]. To do so, we choose a specific Γ̅:=∑_j Γ_j |ξ_j⟩⟨ξ_j| with Γ_j ∈∂ψ(W_j). This in particular shows that the inequality in (<ref>) is an equality, proving the claimed inequality.


Incidentally, during the proof above, we explicitly construct a maximizer. 












The next lemma considers maps induced via functional calculus by convex functions on the real line, and discusses their properties.


    Let ψ∈ C^1() a convex function and define the map Ψ: () → as Ψ(A) := [ ψ (A)]. Then we have that
    
    / t|_t=0Ψ(A+tB)
                = 
            ⟨∇Ψ(A) , B
            ⟩
                =
            [
                ∇Ψ(A) B 
            ]
                  ,   where  ∇Ψ(A) = ψ'(A)
                ,

    for every A ∈() and B ∈(). In particular, for every A ∈() it holds
    
    [A  ψ'(A)] 
                =
             [ ψ(A)]
                +
            [ ψ^* (  ψ'(A))]
                .


The proof of (<ref>) can be found e.g. in  <cit.>.  The equality in (<ref>) follows directly from  Proposition <ref> and the fact that

    ⟨ A, ∇Ψ(A) ⟩
                = 
            Ψ(A) + Ψ^* (∇Ψ(A))
                ,

which is a property of real, convex functions and their Legendre transform <cit.>.
















 




 §.§ Acknowledgements


E.C. acknowledges the support of the New Frontiers in Research Fund (NFRFE-2021-00798) and the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 948021). Both A.G. and N.M. acknowledge the support of the Canada Research Chairs Program, the Natural Sciences and Engineering Research Council of Canada and the New Frontiers in Research Fund (NFRFE-2021-00798).  L.P. gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. We thank Luca Tamanini for the critical reading of a preliminary version of the manuscript.

abbrv




