Conditions for the compatibility of channels in general probabilistic theory and their connection to steering and Bell nonlocality
aa r X i v : . [ qu a n t - ph ] N ov Conditions for the compatibility of channels in general probabilistic theory and theirconnection to steering and Bell nonlocality
Martin Pl´avala ∗ Mathematical Institute, Slovak Academy of Sciences, ˇStef´anikova 49, Bratislava, Slovakia
We derive general conditions for the compatibility of channels in general probabilistic theory. Weintroduce formalism that allows us to easily formulate steering by channels and Bell nonlocalityof channels as generalizations of the well-known concepts of steering by measurements and Bellnonlocality of measurements. The generalization does not follow the standard line of thinkingstemming from the Einstein-Podolsky-Rosen paradox, but introduces steering and Bell nonlocalityas entanglement-assisted incompatibility tests. We show that all of the proposed definitions are, inthe special case of measurements, the same as the standard definitions, but not all of the knownresults for measurements generalize to channels. For example, we show that for quantum channels,steering is not a necessary condition for Bell nonlocality. We further investigate the introducedconditions and concepts in the special case of quantum theory and we provide many examples todemonstrate these concepts and their implications.
I. INTRODUCTION
Incompatibility of measurements is the well-knownquantum phenomenon that gives rise to steering and Bellnonlocality. Historically, the idea of measurement incom-patibility dates back to Bohr’s principle of complemen-tarity. Steering was first described by Schr¨odinger [1] andBell nonlocality was first introduced by Bell [2], both asa reply to the paradox of Einstein, Podolsky and Rosen[3]. It is known that incompatibility of measurements isnecessary and in some cases sufficient for both steeringand Bell nonlocality, but the operational connection be-tween incompatibility, steering and Bell nonlocality wasso far not described in general terms that would also fitchannels, not only measurements.There was extensive research into properties of quan-tum incompatibility of measurements [4, 5], quantum in-compatibility of measurements and its noise robustness,or degree of compatibility [6, 7], connection of quan-tum incompatibility of measurements and steering [8–12], connection of quantum incompatibility of measure-ments and Bell nonlocality [13–16] and connection be-tween steering and Bell nonlocality [17, 18], for a recentreview see [19]. In recent years, the problems of incom-patibility of measurements on channels [20], compatibil-ity of channels [21], the connection of channel steering tomeasurement incompatibility [22] and incompatibility ingeneral probabilistic theory [23–26] were all studied.The aim of this paper is to heavily generalize the re-cent results of [26], where compatibility, steering and Bellnonlocality of measurements were formulated using con-vex analysis and the geometry of tensor products. In thispaper, we will generalize the ideas and results of [26] forthe case of two channels in general probabilistic theory.The generalizations are not straightforward and we willhave to introduce several new operational ideas and def-initions, e.g. we introduce the operational interpretation ∗ [email protected] of direct products of state spaces and we define steer-ing and Bell nonlocality as very simple entanglement-assisted incompatibility test, that boil down to the prob-lem whether there exists a multipartite state with givenmarginal states.During all of our calculations we will restrict ourselvesto finite-dimensional general probabilistic theory and toonly the case of two channels. We will restrict to only twochannel just for simplicity, as one may easily formulatemany of our results for more than two channels using thesame operational ideas as we will present.The paper is organized as follows: in Sec. II we de-scribe our motivation for using general probabilistic the-ory. We provide several references to known applicationsand their connections to each other. In Sec. III we intro-duce general probabilistic theory. Note that in subsectionIII D we introduce the operational interpretations of di-rect products in general probabilistic theory. In Sec. IVwe define compatibility of channels and we derive a con-dition for compatibility of channels. In Sec. V we showthat our condition for compatibility of channels yieldsthe condition for compatibility of measurements that waspresented in [26]. In Sec. VI we derive specific condi-tions for the compatibility of quantum channels. In Sec.VII we propose an idea for a test of incompatibility ofchannels, that will not work at first, but will eventuallylead to both steering and Bell nonlocality. In Sec. VIIIwe define steering by channels as one-side entanglementassisted incompatibility test and we derive some basicresults. In Sec. IX we show that for the special case ofmeasurements our definition of steering leads to the stan-dard definition of steering [27] in the formalism of [26].In Sec. X we derive the specific conditions for steeringby quantum channels, we show that every pair of incom-patible channels may be used for steering of maximallyentangled state and that there are entangled states thatare not steerable by any pair of channels, among otherresults. In Sec. XI we define Bell nonlocality of channelsas a two-sided entanglement assisted incompatibility testand we derive some basic results, then in Sec. XII weshow that, when applied to measurement, the generaldefinition of Bell nonlocality yields the standard defini-tion of Bell nonlocality [27] in the formalism of [26] andwe also show that for measurements steering is a neces-sary condition for Bell nonlocality. In Sec. XIII we deriveconditions for the Bell nonlocality of quantum channels,we formulate a generalized version of the CHSH inequal-ity, we show that for such inequality Tsirelson bound [28]both holds and is reached, we show an example of viola-tion of the generalized version of CHSH inequality and webuild on the example from Sec. X of an entangled statenot steerable by any pair of channels to show that, eventhough the state is not steerable by any pair of channels,it leads to Bell nonlocality, which shows that steering isnot a necessary condition for Bell nonlocality for quan-tum channels. In Sec. XIV we conclude the paper bypresenting the many open questions and possible areasof research opened by our paper. II. MOTIVATIONS FOR USING GENERALPROBABILISTIC THEORY
There are few motivations to using general probabilis-tic theory. The first motivation is mathematical as gen-eral probabilistic theory is a unified framework capableof describing both classical and quantum theory, as wellas other theories. In the current manuscript the mathe-matical motivation is (according to the personal opinionof the author) even stronger as some of the formulationsof the presented ideas and some of the proofs of the pre-sented theorems turn out to be clearer in the frameworkof general probabilistic theory.The second motivation comes from foundations ofquantum theory as general probabilistic theory providesinsight into the structure of entanglement and incompat-ibility.The third and most promising motivation comes frominformation theory. There were developed several models[29–31] that have very interesting information-theoreticproperties and that can be described by general proba-bilistic theory, albeit sometimes it needs to be extendedeven more [32]. Apart from the well-known results onthe properties of Popescu-Rohrlich boxes [33, 34], it wasshowed that there are theories in which one can search a N -item database in O ( √ N ) queries [35] and that thereis a general probabilistic theory that can be simulatedby a probabilistic classical computer that can performDeutsch-Jozsa and Simon’s algorithm [36].The aforementioned results show that studying gen-eral probabilistic theory is interesting even from practicalviewpoint and that it could have potential applicationsin information processing. III. INTRODUCTION TO GENERALPROBABILISTIC THEORY
General probabilistic theory is a unified framework todescribe the kinematics of different systems in a mathe-matically unified fashion. The may idea of general prob-abilistic theory is an operational approach to setting theaxioms and then carrying forward using convex analy-sis. Useful bookns on convex analysis are [37, 38]. Thebeautiful aspect of general probabilistic theory is thatit is only little bit more general than dealing with thedifferent systems on their own, but we do not have to ba-sically rewrite the same calculations over and over againfor different theories.During our calculations we will use two recurring ex-amples, one will be finite-dimensional classical theoryand other will be finite-dimensional quantum theory. Thefinite-dimensional classical theory is closely tied to theknown results about incompatibility, steering and Bellnonlocality of measurements and we will mainly use itto verify that the definitions we will propose are, in thespecial case of measurements, the same as the known def-initions. The quantum theory is our main concern as thisis the theory we are mostly interested in. Some results,that we will only prove for quantum theory, may be gen-eralized for general probabilistic theory, but we will limitthe generality of our calculations to make them more un-derstandable to readers that are not so far familiar withgeneral probabilistic theory.Given that we will work with many different spaces,their duals, their tensor products and many isomorphicsets, all isomorphisms will be omitted unless explicitlystated otherwise.
A. The state space and the effect algebra ofgeneral probabilistic theory
There are two central notions in general probabilistictheory: the state space that describes all possible statesof the system and the effect algebra that describes themeasurements on the system. We will begin our con-struction from the state space and then define the effectalgebra, but we will show how one can go the other wayand start from an effect algebra and obtain state space af-terwards. We will restrict ourselves to finite-dimensionalspaces and always use the Euclidean topology.Let V denote a real, finite-dimensional vector spaceand let X ⊂ V , then by conv ( X ) we will denote theconvex hull of X , by aff ( X ) we will denote the affinehull of X . We will proceed with the definition of relativeinterior of a set X ⊂ V . Definition 1.
Let X ⊂ V , then the relative interior of X , denoted ri ( X ) is the interior of X when it is consid-ered as a subset of aff ( X ).For a more throughout discussion of relative interiorsee [37, p. 44].Let K be a compact convex subset of V , then K is astate space. The points x ∈ K represent the states ofsome system and their convex combination is interpretedoperationally, that is for x, y ∈ K , λ ∈ [0 , ⊂ R thestate λx + (1 − λ ) y corresponds to having prepared x with probability λ and y with probability 1 − λ .To define measurements we have to be able to assignprobabilities to states, that is we have to have a map f : K → [0 ,
1] such that, to follow the operational in-terpretation of convex combination, we have assign theconvex combination of probabilities to the convex combi-nation of respective states. In other words for x, y ∈ K , λ ∈ [0 ,
1] we have to have f ( λx + (1 − λ ) y ) = λf ( x ) + (1 − λ ) f ( y ) , which means that f is an affine function. Such functionsare called effects because they correspond to assigningprobabilities of measurement outcomes to states. We willproceed with a more formal definition of effects and ofeffect algebra.Let A ( K ) denote the set of affine functions K → R . A ( K ) is itself a real linear space, moreover it is orderedas follows: let f, g ∈ A ( K ), then f ≥ g if f ( x ) ≥ g ( x ) forevery x ∈ K . There are two special functions 0 and 1 in A ( K ), such that 0( x ) = 0 and 1( x ) = 1 for all x ∈ K .The set A ( K ) + = { f ∈ A ( K ) : f ≥ } is the convex,closed cone of positive functions. The cone A ( K ) + isgenerating, that is for every f ∈ A ( K ) we have f + , f − ∈ A ( K ) + such that f = f + − f − , and it is pointed, that isif f ≥ − f ≥
0, then f = 0.Although we will provide a proper definition of mea-surement in subsection III E, we will now introduce theconcept of yes/no measurement, or two-outcome mea-surement, that will motivate the definition of the effectalgebra. Our notion of measurement might seem differ-ent to the standard understanding and one may arguethat what we will refer to as measurements are shouldbe called entanglement-breaking maps, but this way ofdefining measurement is standard in general probabilis-tic theory, hence we will use it. A measurement is aprocedure that assigns probabilities to possible outcomesbased on the state that is measured. If we have only twooutcomes and we know that the probability of the firstoutcome is p ∈ [0 , − p . This showsthat a two-outcome measurement needs to assign onlyprobability to one outcome and the other probability fol-lows.Since assigning probabilities to states is a function f : K → [0 ,
1] and due to our operational interpretation ofconvex combination we want such function to be affine.Traditionally the functions that assign probabilities tostates are called effects and the set of all effect is calledeffect algebra.
Definition 2.
The set E ( K ) = { f ∈ A ( K ) : 0 ≤ f ≤ } is called the effect algebra. In general, one may define effect algebra in more gen-eral fashion, using the partially defined operation of ad-dition and a unary operation ⊥ , that would in our casecorrespond to f ⊥ = 1 − f , see [39] for a more throughouttreatment.Let f ∈ E ( K ) then the two outcome measurement m f corresponding to the effect f is the procedure that for x ∈ K assigns the probability f ( x ) to the first outcome andthe probability 1 − f ( x ) to the second outcome. Note thatwe did not mention any labels of the outcomes. Usuallythe outcomes are labeled yes and no, or 0 and 1, or − Example . In classical theory, thestate space K is a simplex, that is the convex hull ofa set of affinely independent points x , . . . , x n . The spe-cial property of the simplex is that every point x ∈ K can be uniquely expressed as convex combination of thepoints x , . . . , x n , due to their affine independence. Example . Let H denote a finite-dimensional complex Hilbert space, let B h ( H ) denote thereal linear space of self-adjoint operators on H , for A ∈ B h ( H ) let Tr( A ) denote the trace of the operator A andlet A ≥ A is positive semi-definite. Wesay that A ≤ B if 0 ≤ B − A . Let B h ( H ) + = { A ∈ B h ( H ) : A ≥ } denote the cone of positive semi-definiteoperators.In quantum theory the state space is given as D H = { ρ ∈ B h ( H ) : ρ ≥ , Tr( ρ ) = 1 } which is the set of density operators on H . The effectalgebra E ( D H ) is given as E ( D H ) = { M ∈ B h ( H ) : 0 ≤ M ≤ } . The value of the effect M ∈ E ( D H ) on the state ρ ∈ D H is given as M ( ρ ) = Tr( ρM ) . B. The structure of general probabilistic theory
This subsection will be rather technical, but we willintroduce several mathematical results that we will uselater on.Let x ∈ K and consider the map x : A ( K ) → R , that to f ∈ A ( K ) assigns the value f ( x ). This is clearly a linearfunctional on A ( K ). Moreover for x, y ∈ K , λ ∈ [0 ,
1] wehave λx + (1 − λ ) y = λx + (1 − λ ) y as the functions in A ( K ) are affine by definition. We con-clude that the state space K must be affinely isomorphicto some subset of the dual of A ( K ). Since the afore-mentioned isomorphism is going to be extremely usefulin later calculations we will describe it in more detail.Let A ( K ) ∗ denote the dual of A ( K ), that is the spaceof all linear functionals on A ( K ). For ψ ∈ A ( K ) ∗ and f ∈ A ( K ) we will denote the value the functional ψ reaches on f as h ψ, f i . The dual cone to A ( K ) + is thecone A ( K ) ∗ + = { ψ ∈ A ( K ) ∗ : h ψ, f i ≥ , ∀ f ∈ A ( K ) + } that gives rise to the ordering on A ( K ) ∗ given as fol-lows: let ψ, ϕ ∈ A ( K ) ∗ , then ψ ≥ ϕ if and only if( ψ − ϕ ) ∈ A ( K ) ∗ + , i.e. if ψ − ϕ ≥ K is iso-morphic to a subset of the cone A ( K ) ∗ + , moreover itis straightforward to realize that the functionals isomor-phic to K must map the function 1 ∈ A ( K ) to the value1. Definition 3.
Let S K = { ψ ∈ A ( K ) ∗ + : h ψ, i = 1 } .We call S K the state space of the effect algebra E ( K ).It might be confusing at this point why we call S K astate space, but this will be cleared by the following. Proposition 1. S K is affinely isomorphic to K .Proof. It is clear that the map x → x maps K to a convexsubset of S K . It is easy to show the inclusion of S K inthe image of K using Hahn-Banach separation theorem,see [40, Chapter 1, Theorem 4.3] for a proof.We will omit the isomorphism between K and S K , sofor any x, y ∈ K , α ∈ R we will write αx + y insteadof αx + y ∈ A ( K ) ∗ . Still, one must be careful whenomitting this isomorphism, beacuse if 0 ∈ V denotes thezero vector and 0 ∈ K , then 0 ∈ A ( K ) ∗ is not the zerofunctional as by construction we have h , i = 1. We willdo our best to avoid such possible problems by choosingappropriate notation.There are two more result we will heavily rely on: Proposition 2. S K is a base of A ( K ) ∗ + , that is forevery ψ ∈ A ( K ) ∗ + , ψ = 0 there is a unique x ∈ K and λ ∈ R , λ ≥ such that ψ = λx .Proof. Let ψ ∈ A ( K ) ∗ + , ψ = 0, then h ψ, i 6 = 0 asif h ψ, i = 0 and ψ ≥
0, then ψ = 0, because 1 ∈ ri ( A ( K ) + ). Let ψ ′ = h ψ, i ψ . It is straightforward that ψ ′ ∈ S K . Proposition 3. A ( K ) ∗ + is a generating cone in A ( K ) ∗ ,that is for every ψ ∈ A ( K ) there are ϕ + , ϕ − ∈ A ( K ) ∗ + such that ψ = ϕ + − ϕ − .Proof. The result follows from the fact that A ( K ) + is apointed cone, see [38, Section 2.6.1]. C. Tensor products of state spaces and effectalgebras
Tensor products are a way to describe joint systems ofseveral other systems. There are several approaches to introducing a tensor product in general probabilistic the-ory. There is a category theory based approach [41] thatis a viable way to introduce the tensor products, but wewill use a simpler, operational approach. Note that thestate space of the joint system will be a compact convexsubset of a real, finite-dimensional vector space as it itselfmust be a state space of some general probabilistic the-ory. Also keep in mind that describing a tensor productof state spaces K A , K B is equivalent to describing thetensor product of the cones A ( K A ) ∗ + , A ( K B ) ∗ + . This isgoing to be useful as some things are easier to express interms of the positive cones.Let V , W be real finite-dimensional vector spaces andlet v ∈ V , w ∈ W . v ⊗ w will refer to the element of thealgebraic tensor product V ⊗ W , see e.g. [42]. We willfirst describe the minimal and maximal tensor productsof state spaces that set bounds on the real state spaceof the joint system. Note that when describing the jointstate space of two state spaces or states of two systems,we will refer to them as bipartite state space or bipartitestates.Let K A , K B denote two state spaces of Alice and Bobrespectively. For every x A ∈ K A , x B ∈ K B there must bea state of the joint system describing the situation thatAlice’s system is in the state x A and Bob’s system is inthe state x B . We will denote such state x A ⊗ x B and wewill call it a product state. Since the state space must beconvex, the state space of the joint system must containat least the convex hull of the product states. This leadsto the definition of minimal tensor product. Definition 4.
The minimal tensor product of statespaces K A and K B , denoted K A ˙ ⊗ K B is the compactconvex set K A ˙ ⊗ K B = conv ( { x A ⊗ x B : x A ∈ K A , x B ∈ K B } ) . The bipartite states y ∈ K A ˙ ⊗ K B are also called sepa-rable states. For the positive cones we get A ( K A ˙ ⊗ K B ) ∗ + = conv ( { ψ A ⊗ ψ B : ψ A ∈ A ( K A ) ∗ + ,ψ B ∈ A ( K B ) ∗ + } ) . Example . In quantum theory, the minimal tensor prod-uct D H ˙ ⊗ D H is the set of all separable states, that is of allstates of the form P ni =1 λ i ρ i ⊗ σ i for n ∈ N and ρ i ∈ D H , σ i ∈ D H , 0 ≤ λ i for i ∈ { , . . . , n } , P ni =1 λ i = 1.In a similar fashion, let f A ∈ E ( K A ), f B ∈ E ( K B ),then we can define a function f A ⊗ f B as the uniqueaffine function such that for x A ∈ K A , x B ∈ K B we have( f A ⊗ f B )( x A ⊗ x B ) = f A ( x A ) f B ( x B ) . This function is used in the most simple measurement onthe joint system, such that Alice applies the two-outcomemeasurement m f A and Bob applies the two outcome mea-surement m f B , so f A ⊗ f B must be an effect on the jointstate space. This leads to the definition of the maximaltensor product. Definition 5.
The maximal tensor product of the statespaces K A and K B , denoted K A ˆ ⊗ K B , is defined as K A ˆ ⊗ K B = { ψ ∈ A ( K A ) ∗ ⊗ A ( K B ) ∗ : ∀ f A ∈ A ( K A ) + , ∀ f B ∈ A ( K B ) + , h ψ, f A ⊗ f B i ≥ } States in K A ˆ ⊗ K B \ K A ˙ ⊗ K B are called entangledstates. Equivalent definition, in terms of the positivecones would be A ( K A ˆ ⊗ K B ) ∗ + = ( A ( K A ) + ˙ ⊗ A ( K B ) + ) ∗ + where A ( K A ) + ˙ ⊗ A ( K B ) + = conv ( { f A ⊗ f B : f A ∈ A ( K A ) + ,f B ∈ A ( K B ) + } ) . As we see, the definition of tensor product of cones ofpositive functionals goes hand in hand with the definitionof tensor product of cones of positive functions.
Example . In quantum theory, the maximal tensorproduct of the cones B h ( H ) + ˆ ⊗ B h ( H ) + is the cone ofentanglement witnesses [43, Section 6.3.1], i.e. W ∈ B h ( H ) + ˆ ⊗ B h ( H ) + if for every ρ ∈ D H , σ ∈ D H we haveTr( W ρ ⊗ σ ) ≥
0. Note that this does not imply thepositivity of W .From the constructions it is clear that the state spaceof the joint system has to be a subset of the maximaltensor product and it has to contain the minimal tensorproduct. But there is no other specification of the statespace of the joint system in general, it has to be providedby the theory we are working with. Definition 6.
We will call the joint state space of thesystems described by the state spaces K A and K B thereal tensor product of K A and K B and we will denote it K A ˜ ⊗ K B . We always have K A ˙ ⊗ K B ⊆ K A ˜ ⊗ K B ⊆ K A ˆ ⊗ K B . Example . In quantum theory, the real tensor productof the state spaces is defined as the set of density matriceson the tensor product of the Hilbert spaces, that is D H ˜ ⊗ D H = D H⊗H . It is tricky to work with the tensor products in gen-eral probabilistic theory as the real tensor product is notalways specified, or it may not be clear what it shouldbe. We will always assume that every tensor product weneed to be defined is defined. Moreover when workingwith a tensor product of more than two state spaces, say K A , K B , K C we will always assume that( K A ˜ ⊗ K B ) ˜ ⊗ K C = K A ˜ ⊗ ( K B ˜ ⊗ K C )and we will simply write K A ˜ ⊗ K B ˜ ⊗ K C . In the appli-cations of general probabilistic theory to quantum andclassical theory it will always be clear how to construct the needed tensor products and we consider this sufficientfor us since we are mainly interested in the applicationsof our results.We will state and prove a result about classical statespaces that we will use several times later on. Proposition 4.
Let S be a simplex with the extremalpoints x , . . . , x n , i.e. S = conv ( { x , . . . , x n } ) and let K be any state space, then we have S ˙ ⊗ K = S ˆ ⊗ K. Proof.
Let S be a simplex and let x i ∈ A ( S ) ∗ + , i ∈{ , . . . , n } , be the extreme points of S . The points x , . . . , x n form a basis of A ( S ) ∗ . Let ψ ∈ S ˆ ⊗ K thenwe have ψ = n X i =1 x i ⊗ ϕ i , for some ϕ i ∈ A ( K ) ∗ . Our aim is to prove that ϕ i ∈ A ( K ) ∗ + then ψ ∈ S ˙ ⊗ K follows by definition.Let b , . . . , b n denote the basis of A ( S ) dual to the basis x , . . . , x n of A ( S ) ∗ , i.e. we have b i ( x j ) = δ ij , where i, j ∈ { , . . . , n } and δ ij is the Kronecker delta. We have b i ∈ E ( S ) because S is a simplex. For any f ∈ E ( K ) wehave 0 ≤ ( ψ, b i ⊗ f ) = ( ϕ i , f )for all i ∈ { , . . . , n } , which implies ϕ i ∈ A ( K ) ∗ + .Note that tensor product of the simplexes S , S isalso a simplex, so we have K ˆ ⊗ S ˆ ⊗ S = K ˙ ⊗ S ˙ ⊗ S . D. Direct product of state spaces and effectalgebras
For certain reasons we will need to use direct productstogether with tensor product. The idea of why they willbe used is going to be clear in the end, but now we willpresent several of their properties that will be requiredlater. As in the Subsec. III C we will work mostly withthe cones of the positive functionals.Let K A , K B be two state spaces. Given A ( K A ) ∗ + and A ( K B ) ∗ + there are two ways to define the direct productof these cones. The first is to use the cone A ( K A ) ∗ + × A ( K B ) ∗ + . The second is to realize that we can construct K A × K B that will be a compact and convex set, i.e. astate space that gives rise to the cone A ( K B × K B ) ∗ + .It may seem that these cones are fairly similar, butthey are not and they have different physical interpreta-tions. Let ψ ∈ A ( K A × K B ) ∗ + , then there are unique λ ∈ R , x A ∈ K A , x B ∈ K B such that ψ = λ ( x A , x B ).Now let ϕ ∈ A ( K A ) ∗ + × A ( K B ) ∗ + , then there are y A ∈ K A , y B ∈ K B , α A , α B ∈ R , α A , α B ≥ ϕ = ( α A y A , α B y B ). In other words the normaliza-tion may be different in every component of the product.This can be rewritten as ϕ = ( α A y A , α B y B )= ( α A + α B ) (cid:18) α A α A + α B y A , α B α A + α B y B (cid:19) = ( α A + α B ) (cid:18) α A α A + α B ( y A ,
0) + α B α A + α B (0 , y B ) (cid:19) that shows that every element of A ( K A ) ∗ + × A ( K B ) ∗ + can be uniquely expressed as a multiple of a convex com-bination of elements of the form ( y A ,
0) and (0 , y B ). Theoperational interpretation of such states is that we donot even know which system we are working with, butwe know that with some probability p we have the firstsystem and with probability 1 − p we have the secondsystem.The operational interpretation of A ( K A × K B ) ∗ + is abit harder to grasp. We may understand ψ ∈ A ( K A × K B ) ∗ + as a (multiple of) conditional state. That is, wewill interpret the object ( x A , x B ) as a state that cor-responds to making a choice in the past between thesystems K A and K B and keeping track of both of theoutcomes at once. The cone A ( K A × K B ) ∗ + will play acentral role in our results on incompatibility, steering andBell nonlocality, because in the problem of incompatibil-ity we wish to implement two channels at the same timeand in steering and Bell nonlocality we are choosing be-tween two incompatible channels.At last we will need to describe the set A ( K A × K B )and its structure with respect to the sets A ( K A ) and A ( K B ). We will show that A ( K A × K B ) corresponds toa certain subset of A ( K A ) × A ( K B ) by using the follow-ing two ideas: since all of the vector spaces are finitedimensional we have that A ( K A ) × A ( K B ) is the dual to A ( K A ) ∗ × A ( K B ) ∗ and A ( K A × K B ) ∗ can be identifiedwith a subset of A ( K A ) ∗ × A ( K B ) ∗ . Note that this iden-tification holds only between the vector spaces and notbetween the corresponding state spaces. Proposition 5.
We have A ( K B × K B ) ∗ + ⊂ A ( K B ) ∗ + × A ( K B ) ∗ + . Proof.
The idea of the proof is that if we have ϕ ∈ A ( K A ) ∗ + × A ( K B ) ∗ + such that ϕ = ( α A y A , α B y B ) then ϕ ∈ A ( K A × K B ) ∗ + if and only if α A = α B . There-fore we can identify A ( K A × K B ) ∗ + with the set { ψ ∈ A ( K A ) ∗ + × A ( K B ) ∗ + : h ψ, (1 , − i = 0 } . It is easy toverify this constraint on the positive cones and since it islinear it must hold everywhere else.The above proof shows that the function (1 , − ∈ A ( K A ) × A ( K B ) is equal to zero when restricted to A ( K A × K B ) ∗ , or in other words (1 ,
0) = (0 ,
1) whenrestricted to A ( K A × K B ) ∗ . We introduce a relation ofequivalence on A ( K A ) × A ( K B ) as follows: for f, g ∈ A ( K A ) × A ( K B ) we say that f and g are equivalent and we write f ∼ g if f − g = k (1 , −
1) for some k ∈ R .Equivalently, f ∼ g if for every ψ ∈ A ( K A × K B ) ∗ wehave h ψ, f i = h ψ, g i . A ( K A × K B ) corresponds to the setof equivalence classes of A ( K A ) × A ( K B ) with respect tothe relation of equivalence ∼ .To demonstrate this, consider the constant function1 ∈ E ( K A × K B ) and let x ∈ K A , y ∈ K B , then we have h ( x, y ) , (1 , i = h x, i = 1 = h ( x, y ) , i , h ( x, y ) , (0 , i = h y, i = 1 = h ( x, y ) , i . This is not a coincidence, because (1 , − (0 ,
1) = (1 , − , ∼ (0 , E. Channels and measurements in generalprobabilistic theory
It is not easy to define channels in general probabilistictheory as we would like all of the channels to be com-pletely positive. We will use the following definition:
Definition 7.
Let K A , K B be state spaces, then channelΦ is a linear mapΦ : A ( K A ) ∗ → A ( K B ) ∗ that is positive, i.e. for every ψ ∈ A ( K A ) ∗ + we haveΦ( ψ ) ∈ A ( K B ) ∗ + and that for ψ ∈ K A we have Φ( ψ ) ∈ K B .One may also require a channel to be completely pos-itive, that is if K C is some state space such that we candefine K C ˜ ⊗ K A , then we can consider the map id ⊗ Φ : K C ˜ ⊗ K A → K C ˆ ⊗ K B and require it to be positive. Inthe applications of general probabilistic theory to classi-cal and quantum theories, we always know how to createjoint systems of given two systems so in the examples wewill always require complete positivity of channels, butone still has to bear in mind that in the general case,complete positivity is not a well-defined concept.One can identify the channel Φ : A ( K A ) ∗ → A ( K B ) ∗ with an element of A ( K A ) ⊗ A ( K B ) ∗ as follows: let x ∈ K A and f ∈ A ( K B ), then the expression h Φ( x ) , f i givesrise to a linear functional on A ( K A ) ∗ ⊗ A ( K B ). Thismeans that we have Φ ∈ A ( K A ) ⊗ A ( K B ) ∗ , where we omitthe isomorphism between the channel and the functional.If we also consider the positivity of the channel on theelements of the form x ⊗ f ∈ K A ˙ ⊗ E ( K B ) we getΦ ∈ A ( K A ) + ˆ ⊗ A ( K B ) ∗ + . This is a well known construction that may be also usedto define the tensor product of linear spaces [42, Chapter1.3].There is one more construction with channels that willbe important in our formulation of compatibility of chan-nels: compositions with effect. Let Φ : K A → K B be achannel and let f ∈ E ( K B ), then they give rise to aneffect ( f ◦ Φ) ∈ E ( K A ) defined for x A ∈ K A as h x A , ( f ◦ Φ) i = h Φ( x A ) , f i . By the same idea we can define a map f ⊗ id : A ( K B ) ∗ ⊗ A ( K C ) ∗ → A ( K C ) ∗ such that for x B ∈ K B and x C ∈ K C we have ( f ⊗ id )( x B ⊗ x C ) = f ( x B ) x C and we extendthe map by linearity. Also given a channel Φ : K A → K B ˜ ⊗ K C we can compose the map f ⊗ id with the channelΦ to obtain ( f ⊗ id ) ◦ Φ ′ : A ( K A ) ∗ → A ( K C ) ∗ such thatthe corresponding functional on A ( K A ) ⊗ A ( K C ) ∗ is for x A ∈ K A and g ∈ A ( K C ) given as h ( f ⊗ id ) ◦ Φ , x A ⊗ g i = h Φ( x A ) , f ⊗ g i . Specifically we will be interested in the expressions (1 ⊗ id ) ◦ Φ and ( id ⊗ ◦ Φ. If Φ is a channel then (1 ⊗ id ) ◦ Φand ( id ⊗ ◦ Φ are channels as well and they are calledmarginal channels of Φ.A special type of channel is a measurement.
Definition 8.
A channel m : K A → K B is called ameasurement if K B is a simplex.The interpretation is simple: the vertices of the sim-plex correspond to the possible measurement outcomesand the resulting state is a probability distribution overthe measurement outcomes, i.e. an assignment of prob-abilities to the possible outcomes. Since we require allstate spaces to be finite-dimensional this implies that weconsider only finite-outcome measurements. Let K B bea simplex with vertices ω , . . . , ω n , then we can identify ameasurement m with an element of A ( K A ) + ˙ ⊗ A ( K B ) ∗ + of the form m = n X i =1 f i ⊗ δ ω i where for i ∈ { , . . . , n } we have f i ∈ E ( K A ), P ni =1 f i =1 and δ ω i ∈ S ( K B ) are the functionals corresponding tothe extreme points of K B (where we have not omitted theisomorphism this time). This expression has an opera-tional interpretation that for x ∈ K A the measurement m assigns the probability f i ( x ) to the outcome ω i . Example . Quantum channels are completely positive,trace preserving maps Φ : D H → D H . The completepositivity means that for any ρ ≥ id ⊗ Φ)( ρ ) ≥
0. We denote the set of channels Φ : D H → D H as C H→H .Let | i , . . . , | n i be an orthonormal base of H . To everyquantum channel we may assign its unique Choi matrix C (Φ) defined as C (Φ) = (Φ ⊗ id ) n X i,j =1 | ii ih jj | , where we use the shorthand | ii i = | i i ⊗ | i i . Note that C (Φ) ≥ ( C (Φ)) = , where Tr denotes the partial trace. Also every matric C ∈ B h ( H ⊗ H ) suchthat C ≥ ( C ) = is a Choi matrix of somechannel, see [43, Section 4.4.3].The Choi matrix C (Φ) is isomorphic to a state H ) C (Φ), which corresponds to the channel Φ ⊗ id acting on the maximally entangled state | ψ + ih ψ + | , where | ψ + i = 1 p dim( H ) n X i =1 | ii i . IV. COMPATIBILITY OF CHANNELS
Definition 9.
Let K A , K B , K B be state spaces andlet Φ , Φ be channelsΦ : K A → K B , Φ : K A → K B . We say that Φ , Φ are compatible if and only if thereexists a channel Φ : K A → K B ˜ ⊗ K B such that Φ and Φ are the marginal channels of Φ, i.e.we have Φ = ( id ⊗ ◦ Φ , (1)Φ = (1 ⊗ id ) ◦ Φ . (2)The channel Φ is also called the joint channel of the chan-nels Φ , Φ .The operational meaning of compatibility of channelsis that if the channels Φ , Φ are compatible, then we canapply them both to the input state at once and selectingwhich one we actually want the output from later. If thechannels are incompatible we have to choose from whichone we want the output before applying anything. For amore in-depth explanation see [19]. The important thingis that there is a choice from which channel we want toget the output so we can expect to see A ( K B × K B ) ∗ + come up in the calculations.Consider the channel Φ : K A → K B ˜ ⊗ K B . One canrealize that the maps ( id ⊗
1) : Φ ( id ⊗ ◦ Φ and(1 ⊗ id ) : Φ (1 ⊗ id ) ◦ Φ are linear maps of channels.Moreover the Eq. (1), (2) both have Φ on the right handside in the same position. We are going to exploit this toobtain simpler condition for compatibility of the channelsΦ , Φ . To do so we have to introduce a new map J .Let us define a map J : A ( K A ) ⊗ A ( K B ) ∗ ⊗ A ( K B ) ∗ → A ( K A ) ⊗ A ( K B × K B ) ∗ given for Ξ ∈ A ( K A ) ⊗ A ( K B ) ∗ ⊗ A ( K B ) ∗ as J (Ξ) = (( id ⊗ ◦ Ξ , (1 ⊗ id ) ◦ Ξ) . For Ξ = f ⊗ ψ ⊗ ϕ we have J (Ξ) = f ⊗ ( h ϕ, i ψ, h ψ, i ϕ ) . Proposition 6. J is a linear mapping.Proof. Let Ξ , Ξ ∈ A ( K A ) ⊗ A ( K B ⊗ K B ) ∗ and λ ∈ R ,then we have J ( λ Ξ + Ξ ) = ( λ ( id ⊗ ◦ Ξ + ( id ⊗ ◦ Ξ , , λ (1 ⊗ id ) ◦ Ξ + (1 ⊗ id ) ◦ Ξ )= λ (( id ⊗ ◦ Ξ , (1 ⊗ id ) ◦ Ξ )+ (( id ⊗ ◦ Ξ , (1 ⊗ id ) ◦ Ξ )= λJ (Ξ ) + J (Ξ ) . Assume that the channels Φ , Φ are compatible andthat Φ is their joint channel then we must have J (Φ) = (Φ , Φ )which is just a more compact form of the Eq. (1), (2). Proposition 7.
The channels Φ , Φ are compatible ifand only if there is Φ ∈ A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + suchthat J (Φ) = (Φ , Φ ) . (3) Proof.
If the channels Φ , Φ are compatible then Eq. (3)must hold for their joint channel Φ. If Eq. (3) holds forsome Φ ∈ A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + , then the channelsΦ , Φ are compatible and Φ is their joint channel.The operational interpretation is that (Φ , Φ ) repre-sents a conditional channel in the same way as the statesfrom A ( K B × K B ) ∗ + represent conditional states thatkeep track of some choice made in the past. If the chan-nels are compatible, then we actually do not have to makethe choice of either using Φ or Φ , but we can use theirjoint channel, that has the property that its marginalsreproduce the outcomes of the two channels Φ , Φ . Wewill investigate several of the properties of the map J . Proposition 8.
For every ( ξ , ξ ) ∈ A ( K A ) ⊗ A ( K B × K B ) ∗ there is a Ξ ∈ A ( K A ) ⊗ A ( K B ) ∗ ⊗ A ( K B ) ∗ suchthat J (Ξ) = ( ξ , ξ ) . Moreover if we have (1 , ◦ ( ξ , ξ ) = 1 then (1 ⊗ ◦ Ξ = 1 . Proof.
Let f , . . . , f n be a basis of A ( K A ), then we have ξ = n X i =1 f i ⊗ ψ i ξ = n X i =1 f i ⊗ ϕ i for some ψ i ∈ A ( K B ) ∗ and ϕ i ∈ A ( K B ) ∗ . Since wemust have (1 , ◦ ( ξ , ξ ) = (0 , ◦ ( ξ , ξ )we obtain n X i =1 h ψ i , i f i = n X i =1 h ϕ i , i f i which implies h ψ i , i = h ϕ i , i = k i for all i ∈ { , . . . , n } as f , . . . , f n is linearly independent.Let Ξ = n X i =1 k − i f i ⊗ ψ i ⊗ ϕ i then we have J (Ξ) = n X i =1 k − i f i ⊗ ( h ϕ i , B i ψ i , h ψ i , B i ϕ i )= n X i =1 f i ⊗ ( ψ i , ϕ i ) . If we have 1 ◦ ( ξ , ξ ) = 1 then n X i =1 k i f i = 1and we get(1 ⊗ ◦ Ξ = (1 ⊗ ◦ ( n X i =1 k − i f i ⊗ ψ i ⊗ ϕ i )= n X i =1 k − i h ψ i , ih ϕ i , i f i = 1 . Proposition 9.
We have J ( A ( K A ) + ˙ ⊗ A ( K B ) ∗ + ˙ ⊗ A ( K B ) ∗ + ) == A ( K A ) + ˙ ⊗ A ( K × K B ) ∗ + . Proof.
Let ( ξ , ξ ) ∈ A ( K A ) + ˙ ⊗ A ( K B × K B ) ∗ + then asin the proof of Prop. 8 we have ξ = n X i =1 f i ⊗ ψ i ξ = n X i =1 f i ⊗ ϕ i but now we have f i ≥ ψ i ≥ ϕ i ≥ i ∈{ , . . . , n } . It follows by the same construction as in theproof of Prop. 8 that we can construct Ξ = P ni =1 k − i f i ⊗ ψ i ⊗ ϕ i and we get Ξ ∈ A ( K A ) + ˙ ⊗ A ( K B ) ∗ + ˙ ⊗ A ( K B ) ∗ + .Let Ξ ∈ A ( K A ) + ˙ ⊗ A ( K B ) ∗ + ˙ ⊗ A ( K B ) ∗ + , then wehave Ξ = P ni =1 f i ⊗ ψ i ⊗ ϕ i such that f i ≥ ψ i ≥ ϕ i ≥ i ∈ { , . . . , n } , moreover without lack ofgenerality we can assume h ψ i , B i = h ϕ i , B i = 1. Wehave J (Ξ) = n X i =1 f i ⊗ ( ψ i , ϕ i ) ∈ A ( K A ) + ˙ ⊗ A ( K B × K B ) ∗ + which concludes the proof.It would be very useful to know what is the im-age of the cone A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + when mappedby J . We will denote the resulting cone Q = J ( A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + ). The cone is importantdue to the following: Corollary 1.
The channels Φ , Φ are compatible if andonly if (Φ , Φ ) ∈ Q = J ( A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + ) . Proof.
Follows from Prop. 7.
Proposition 10. A ( K A ) + ˙ ⊗ A ( K B × K B ) ∗ + ⊂ Q .Proof. Since A ( K A ) + ˙ ⊗ A ( K B ˙ ⊗ K B ) ∗ + ⊂ A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + we must have J ( A ( K A ) + ˙ ⊗ A ( K B ˙ ⊗ K B ) ∗ + ) ⊂ Q. The result follows from Prop. 9.
Proposition 11. Q ⊂ A ( K A ) + ˆ ⊗ A ( K B × K B ) ∗ + .Proof. Since we have A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + ⊂ A ( K A ) + ˆ ⊗ A ( K B ˆ ⊗ K B ) ∗ + we must have Q ⊂ J ( A ( K A ) + ˆ ⊗ A ( K B ˆ ⊗ K B ) ∗ + ) . Let Ξ ∈ A ( K A ) + ˆ ⊗ A ( K B ˆ ⊗ K B ) ∗ + , then for ψ ∈ A ( K A ) ∗ + and ( f , f ) ∈ A ( K B × K B ) + we get h J (Ξ) , x ⊗ f i = h (( id ⊗ ◦ Ξ , (1 ⊗ id ) ◦ Ξ) , x ⊗ f i = h Ξ( x ) , f ⊗ i + h Ξ( x ) , ⊗ f i ≥ , that shows we have J ( A ( K A ) + ˆ ⊗ A ( K B ˆ ⊗ K B ) ∗ + ) ⊂ A ( K A ) + ˆ ⊗ A ( K B × K B ) ∗ + which concludes the proof.We can also construct Q as the cone we get whenwe factorize the cone A ( K A ) + ˆ ⊗ A ( K B ˜ ⊗ K B ) ∗ + withrespect to the relation of equivalence given as follows:Ξ ≈ Ξ if and only if J (Ξ ) = J (Ξ ), or equivalently ifand only if Ξ = Ξ + Ξ, such that J (Ξ) = 0.Note that since J is a linear map, as we showed inProp. 6, it is clear that Q is a convex cone. For twogiven channels Φ : K A → K B , Φ : K A → K B onemay write a primal linear program that would check thecondition for compatibility given by Cor. 1. We willwrite such linear program for quantum channels later. V. COMPATIBILITY OF MEASUREMENTS
We will apply the results of Sec. IV to the problemof compatibility of measurements. We will obtain thesame results that were recently presented in [26], thatare generalization a of [44].Let K A be a state space and let S , S be simplexesand let m : K A → S , m : K A → S be measure-ments. According to Prop. 7 the measurements m , m are compatible if and only if( m , m ) ∈ J ( A ( K A ) + ˆ ⊗ A ( S ˜ ⊗ S ) ∗ + ) . Since both S and S are simplexes, then we have S ˜ ⊗ S = S ˙ ⊗ S and the condition for compatibility re-duces according to Prop. 9 to( m , m ) ∈ A ( K A ) + ˙ ⊗ A ( S × S ) ∗ + . Due to the simpler structure of simplexes one may geteven more specific results about measurements, see [26].For demonstration of the derived conditions we willreconstruct the result of [44] about compatibility of two-outcome measurements. According to our definition, ameasurement is two-outcome if the simplex it has as atarget space has two vertexes, i.e. it is a line segment.Let K be a state space, f, g ∈ E ( K ) and m f : K → S , m g : K → S be two-outcome measurements given as m f = f ⊗ δ ω + (1 − f ) ⊗ δ ω ,m g = g ⊗ δ ω + (1 − g ) ⊗ δ ω . The state space given by A ( S × S ) ∗ + is a square givenas conv (( δ ω , δ ω ) , ( δ ω , δ ω ) , ( δ ω , δ ω ) , ( δ ω , δ ω )), that isjust affinely isomorphic to S × S . We have( m , m ) = f ⊗ ( δ ω ,
0) + (1 − f ) ⊗ ( δ ω , g ⊗ (0 , δ ω ) + (1 − g ) ⊗ (0 , δ ω )= f ⊗ ( δ ω , δ ω ) + (1 − f ) ⊗ ( δ ω , δ ω )+ g ⊗ (0 , δ ω − δ ω ) , where in the second step we have used the basis ( δ ω , δ ω ),( δ ω , δ ω ), (0 , δ ω − δ ω ) of A ( S × S ) ∗ to express ( m , m )in a more reasonable form. To have ( m , m ) ∈ A ( K ) + ˙ ⊗ A ( S × S ) ∗ + we must have( m , m ) = h ⊗ ( δ ω , δ ω ) + h ⊗ ( δ ω , δ ω )+ h ⊗ ( δ ω , δ ω ) + h ⊗ ( δ ω , δ ω )= ( h + h ) ⊗ ( δ ω , δ ω )+ ( h + h ) ⊗ ( δ ω , δ ω )+ ( h + h ) ⊗ (0 , δ ω − δ ω ) , for some h , h , h , h ∈ E ( K ). This implies thestandard conditions for the compatibility of two-outcomemeasurements m f , m g : f = h + h , − f = h + h ,g = h + h , see e.g. [45].0 VI. COMPATIBILITY OF QUANTUMCHANNELS
In this section we will derive results more specific to thecompatibility of quantum channels. Let Φ : D H → D H ,Φ : D H → D H be quantum channels, then accordingto Prop. 7 they are compatible if and only if there is achannel Φ : D H → D H⊗H such that for all ρ ∈ D H wehave (Φ ( ρ ) , Φ ( ρ )) = (Tr (Φ( ρ )) , Tr (Φ( ρ ))) . (4)This is equivalent to the definition of compatibility ofquantum channels already stated in [21]. It is straight-forward that Eq. (4) implies that( C (Φ ) , C (Φ )) = (Tr ( C (Φ)) , Tr ( C (Φ))) , we will show that they are equivalent. This will help usto get rid of the state ρ in Eq. (4). Proposition 12.
The channels Φ : D H → D H , Φ : D H → D H are compatible if and only if there exists achannel Φ : D H → D H⊗H such that ( C (Φ ) , C (Φ )) = ( Tr ( C (Φ)) , Tr ( C (Φ))) . Proof.
Let ρ ∈ D H , then we haveTr (Φ( ρ )) = Tr ,E ( C (Φ) ⊗ ⊗ ρ T )= Tr E (Tr ( C (Φ)) ⊗ ρ T )= Tr E ( C (Φ ) ⊗ ρ T ) = Φ ( ρ ) . The same follows for Φ .As we already showed in Sec. IV, the cone Q = J ( A ( D H ) + ˆ ⊗ A ( D H⊗H ) ∗ + ) is of interest for the compat-ibility of channels. In the case of quantum channels wewill use Prop. 12 to formulate similar cone in terms ofChoi matrices of the channels and we will write a semi-definite program for the compatibility of quantum chan-nels based on this approach.Denote P = { (Tr ( C ) , Tr ( C )) : C ∈ C H→H⊗H } , thenaccording to Prop. 12 the channels Φ : D H → D H ,Φ : D H → D H are compatible if and only if( C (Φ ) , C (Φ )) ∈ P. Note that, by our definition, P is not a cone, but it gen-erates some cone just by adding all of the operators ofthe form λC , where C ∈ P and λ ∈ R , λ ≥ P , but the task is not trivial.To make it simpler we will investigate the structure ofthe dual cone P ∗ given as P ∗ = { ( A, B ) ∈ B h ( H ) × B h ( H ) : h C, ( A, B ) i ≥ , ∀ C ∈ P } . Notice that (
A, B ) ∈ B h ( H ) × B h ( H ) is simply a block-diagonal matrix having blocks A and B . Also every C ∈ P is a block diagonal matrix, let C = ( C , C ), then h ( C , C ) , ( A, B ) i = Tr( C A ) + Tr( C B ) . Let C ∈ P , then by definition there exist a channel Φ : D H → D H⊗H such that C = (Tr ( C (Φ)) , Tr ( C (Φ))) . Let (
A, B ) ∈ P ∗ , then we have h C, ( A, B ) i = Tr (cid:0) Tr ( C (Φ)) A + Tr ( C (Φ)) B (cid:1) = Tr( C (Φ)( ˜ A + ⊗ B )) ≥ , where ˜ A is the operator such that Tr(Tr ( C (Φ)) A ) =Tr( C (Φ) ˜ A ). If A = A ⊗ A , then ˜ A = A ⊗ ⊗ A . Ingeneral one can write A as a sum of factorized operatorsand express ˜ A in such way, because the map A ˜ A islinear.The result is that ˜ A + ⊗ B must correspond to apositive function on quantum channels, hence we musthave ˜ A + ⊗ B ≥
0, see [46, 47]. We have proved thefollowing:
Proposition 13.
The channels Φ : D H → D H , Φ : D H → D H are compatible if and only ifTr ( C (Φ ) A ) + Tr ( C (Φ ) B ) ≥ for all A, B ∈ B h ( H ⊗ H ) such that ˜ A + ⊗ B ≥ . This allows us to formulate the semi-definite program[38] for the compatibility of quantum channels as follows:
Proposition 14.
Given channels Φ : D H → D H , Φ : D H → D H , the semi-definite program for the com-patibility of quantum channels is inf A,B Tr ( C (Φ ) A ) + Tr ( C (Φ ) B )˜ A + ⊗ B ≥ , where ˜ A is given as above.If the reached infimum is negative, then the channelsare incompatible, if the reached infimum is then thechannels are compatible.Proof. The result follows from Prop. 13. One may seethat the infimum is at most 0 because one may alwayschose A = B = 0. VII. PRELUDE TO STEERING AND BELLNONLOCALITY
We will propose a possible test for the compatibilityof channels that will not work, but it will motivate ourdefinitions of steering and Bell nonlocality.1Let K A , K B , K B be state spaces and let Φ : K A → K B , Φ : K A → K B be channels. The channels Φ ,Φ are compatible if Eq. (3) is satisfied for some channelΦ : K A → K B ˜ ⊗ K B . This is the same as saying thechannels Φ , Φ are compatible if for all x ∈ K A we have(Φ ( x ) , Φ ( x )) = ((( id ⊗ ◦ Φ)( x ) , ((1 ⊗ id ) ◦ Φ)( x )) . (5)If the channels Φ and Φ are compatible, then for every x ∈ K A there must exist a state y ∈ K B ˜ ⊗ K B such thatΦ ( x ) = ( id ⊗ y ) , (6)Φ ( x ) = (1 ⊗ id )( y ) . (7)Would it be a reasonable test for the compatibility of thechannels Φ and Φ if we considered the state x ∈ K A fixed and we would test whether, for the fixed state x ,there exists y ∈ K B ˜ ⊗ K B such that Eg. (6), (7) aresatisfied? It would not, because for a fixed x ∈ K A onealways has Φ ( x ) ⊗ Φ ( x ) ∈ K B ˜ ⊗ K B that satisfies Eg.(6), (7).Still, throwing away this line of thinking would notbe a good choice, because going further, on may ask: ifthere would be another system K C , such that K C ˜ ⊗ K A is defined, then what if we would use the entanglementbetween the systems K A and K C to obtain a better con-dition for the compatibility of the channels Φ , Φ usingthe very same line of thinking? As we will see, this ap-proach leads to the notions of steering and Bell nonlocal-ity. VIII. STEERING
Steering is one of the puzzling phenomena we find inquantum theory but not in classical theory. It is usuallydescribed as a two party protocol, that allows one sideto alter the state of the other in a way that would notbe possible in classical theory by performing a measure-ment and announcing the outcome. Although originallydiscovered by Schr¨odinger [1], steering was formalised in[27]. Recently there was introduced a new formalism forsteering in [26].So far it was always only considered that during steer-ing one party performs a measurement. Since a measure-ment is a special case of a channel, one may ask whetherit is possible to define steering by channels. We will useour formalism for compatibility of channels to introducesteering by channels by continuing the line of thoughtspresented in Sec. VII. We will have to formulate steeringin a little different way than it usually is formulated formeasurements, but we will show that for measurementswe will obtain the known results.Let K A , K B , K B , K C be finite-dimensional statespaces, such that K C ˜ ⊗ K A is defined and letΦ : K A → K B , Φ : K A → K B , be channels. We can construct channels id ⊗ Φ : A ( K C ) ∗ + ˜ ⊗ A ( K A ) ∗ + → A ( K C ) ∗ + ˆ ⊗ A ( K B ) ∗ + ,id ⊗ Φ : A ( K C ) ∗ + ˜ ⊗ A ( K A ) ∗ + → A ( K C ) ∗ + ˆ ⊗ A ( K B ) ∗ + . Moreover we can construct the conditional channel id ⊗ (Φ , Φ ) : A ( K C ) ∗ + ˜ ⊗ A ( K A ) ∗ + →→ A ( K C ) ∗ + ˆ ⊗ A ( K B × K B ) ∗ + . These channels play a central role in steering and wewill keep this notation throughout this section. First,we will introduce a handy name for the output state of id ⊗ (Φ , Φ ). Definition 10.
Let ψ ∈ K C ˜ ⊗ K A be a bipartite state,then we call ( id ⊗ (Φ , Φ ))( ψ ) a bipartite conditionalstate.Steering may be seen as a three party protocol thattests the compatibility of channels. The parties in ques-tion will be named Alice, Bob and Charlie. Alice andCharlie share a bipartite state ψ ∈ K C ˜ ⊗ K A and Alicehas the channels Φ and Φ at her disposal, that wouldsend her part of the state ψ to Bob. Since Alice canchoose between the channels Φ and Φ , she will be, inour formalism, applying the conditional channel (Φ , Φ )and the resulting state will be a bipartite state from A ( K C ) ∗ + ˜ ⊗ A ( K B × K B ) ∗ + . The structure of the result-ing bipartite conditional state ( id ⊗ (Φ , Φ ))( ψ ) will notonly depend on the input state ψ , but also on the com-patibility of the channels Φ and Φ . Let us assume thatthe channels Φ and Φ are compatible, then there is achannel Φ : K A → K B ˜ ⊗ K B such that (Φ , Φ ) = J (Φ)and we have( id ⊗ (Φ , Φ ))( ψ ) = ( id ⊗ J (Φ))( ψ )= ( id ⊗ J ′ )(( id ⊗ Φ)( ψ ))where J ′ : A ( K B ˜ ⊗ K B ) ∗ → A ( K B × K B ) ∗ , J ′ ( ψ ) =(( id ⊗ ψ ) , (1 ⊗ id )( ψ )). The calculation shows that ifthe channels Φ , Φ are compatible, then we must have( id ⊗ (Φ , Φ ))( ψ ) ∈ ( id ⊗ J ′ )( K C ˜ ⊗ K B ˜ ⊗ K B )which does not have to hold in general if the channelsare not compatible. This shows that we can define steer-ing of a state by channels as an entanglement assistedincompatibility test. Definition 11.
The bipartite state ψ ∈ A ( K C ) ∗ + ˜ ⊗ A ( K A ) ∗ + is steerable by channels Φ : A ( K A ) ∗ + → A ( K B ) ∗ + , Φ : A ( K A ) ∗ + → A ( K B ) ∗ + if( id ⊗ (Φ , Φ ))( ψ ) / ∈ ( id ⊗ J ′ )( K C ˜ ⊗ K B ˜ ⊗ K B )Now we present the standard result about the connec-tion between compatibility of the channels and steering.The result follows from our definition immediately.2 Corollary 2.
The bipartite state ψ ∈ A ( K C ) ∗ + ˜ ⊗ A ( K A ) ∗ + is not steerable by channels Φ : A ( K A ) ∗ + → A ( K B ) ∗ + , Φ : A ( K A ) ∗ + → A ( K B ) ∗ + if the channels Φ and Φ are compatible.Proof. If the channels Φ , Φ are compatible, then wehave (Φ , Φ ) = J (Φ) for some Φ : K A → K B ˜ ⊗ K B andfor every ψ ∈ K C ˜ ⊗ K A we have( id ⊗ (Φ , Φ ))( ψ ) ∈ ( id ⊗ J ′ )( K C ˜ ⊗ ( K B ˜ ⊗ K B )) . Proposition 15.
The bipartite state ψ ∈ A ( K C ) ∗ + ˜ ⊗ A ( K A ) ∗ + is not steerable by channels Φ : A ( K A ) ∗ + → A ( K B ) ∗ + , Φ : A ( K A ) ∗ + → A ( K B ) ∗ + if ψ ∈ A ( K C ) ∗ + ˙ ⊗ A ( K A ) ∗ + , i.e. if ψ is separable.Proof. Every separable state is by definition a convexcombination of product states, i.e. of states of the form x C ⊗ x A , where x A ∈ K A , x C ∈ K C . Since the maps id ⊗ (Φ , Φ ) and id ⊗ J ′ are linear it is sufficient to provethat for every product state x C ⊗ x A ∈ K C ˙ ⊗ K A we have( id ⊗ (Φ , Φ ))( x C ⊗ x A ) ∈ ( id ⊗ J ′ )( K C ˜ ⊗ K B ˜ ⊗ K B ).It follows by our construction in Sec. VII that productstates are not steerable by any channels as one can alwaystake x C ⊗ Φ ( x A ) ⊗ Φ ( x A ). Remember that during steer-ing, we fix not only the channels, but also the bipartitestate, so the presented construction is valid. IX. STEERING BY MEASUREMENTS
We will show that the definition of steering given byDef. 11 follows the standard definition of steering [27] inthe formalism introduced in [26], when we replace mea-surements by channels.
Proposition 16.
Let S , S be simplexes and let m : K A → S , m : K A → S be measurements, then a state ψ ∈ K C ˜ ⊗ K A is steerable by m , m if and only if ( id ⊗ ( m , m ))( ψ ) / ∈ K C ˙ ⊗ ( S × S ) . Proof.
The result follows from the fact that K C ˜ ⊗ S ˜ ⊗ S = K C ˙ ⊗ S ˙ ⊗ S .To obtain the standard definition of steering, one onlyneeds to note that if ξ ∈ K C ˙ ⊗ ( S × S ), then there are x i ∈ K C , s i ∈ S × S and 0 ≤ λ i ≤ i ∈ { , . . . , n } such that ξ = n X i =1 λ i x i ⊗ s i , (8)where the interpretation of s i is that it is a conditionalprobability, conditioned by the choice of the measure-ment. At this point it is straightforward to see that Eq.(8) corresponds to [27, Eq. (5)]. X. STEERING BY QUANTUM CHANNELS
Steering plays an important role in quantum theory.It has found so far applications in quantum cryptogra-phy [48] as an intermediate step between quantum keydistribution and device-independent quantum key distri-bution.We will prove several results and present a simple ex-ample of steering by quantum channels. Given the stan-dard, operational, interpretation of steering by measure-ments the example may seem strange, but rather ex-pected.Let Φ : D H → D H , Φ : D H → D H be channels andlet | ψ + i = (dim( H )) − P dim( H ) i =1 | ii i be the maximallyentangled vector. We will show that the maximally en-tangled state | ψ + ih ψ + | is steerable by the channels Φ ,Φ whenever they are incompatible.The proof is rather simple as the bipartite conditionalstate we obtain is ( id ⊗ (Φ , Φ ))( | ψ + ih ψ + | ). If the chan-nels Φ , Φ are compatible then the state | ψ + ih ψ + | isnot steerable by compatible channels. Now let us assumethat there is a state in ρ ∈ D H⊗H⊗H such that we have( id ⊗ (Φ , Φ ))( | ψ + ih ψ + | ) = ( id ⊗ J ′ )( ρ ) , (9)i.e. that the state state | ψ + ih ψ + | is not steerable by thechannels Φ , Φ . Eq. (9) implies that we must have( id ⊗ Φ )( | ψ + ih ψ + | ) = Tr ( ρ ) , that, after taking trace over the second Hilbert space,gives 1dim( H ) = Tr ( ρ ) . (10)Now the picture becomes clear: ( id ⊗ Φ )( | ψ + ih ψ + | ) isisomorphic to the Choi matrix C (Φ ) and Eq. (10) im-plies that the state ρ must be isomorphic to a Choi ma-trix of some channel Φ. This together with Prop. 12means that Eq. (9) holds if and only if the channels arecompatible. Thus we have proved: Proposition 17.
The maximally entangled state | ψ + ih ψ + | is steerable by channels Φ : D H → D H , Φ : D H → D H if and only if they are incompatible. We will investigate steering by unitary channels. Wewill see a phenomenon that is impossible to happen forsteering by measurements - it is possible to steer a statewhen the two channels we are testing for incompatibilityare two copies of the same channel. Let
U, V be unitarymatrices, i.e.
U U ∗ = V V ∗ = , where U ∗ denotes theconjugate transpose matrix to U and let Φ U , Φ V be thecorresponding unitary channels, that is for ρ ∈ D H wehave Φ U ( ρ ) = U ρU ∗ , Φ V ( ρ ) = V ρV ∗ . Note that we have Φ = id , i.e. the unitary channelsgiven by an identity matrix is the identity channel.3 Proposition 18.
The bipartite state ρ ∈ D H⊗H is steer-able by the unitary channels Φ U , Φ V if and only if it issteerable by two copies of the identity channel id .Proof. The state ρ ∈ D H⊗H is steerable by the channelsΦ U , Φ V if and only if there is a state σ ∈ D H⊗H⊗H suchthat Tr ( σ ) = ( id ⊗ Φ U )( ρ ) , Tr ( σ ) = ( id ⊗ Φ V )( ρ ) . If such state σ exists, then for ˜ σ = ( id ⊗ Φ U ∗ ⊗ Φ V ∗ )( σ )we have Tr (˜ σ ) = ρ, Tr (˜ σ ) = ρ, i.e. the state ρ is not steerable by two copies of id .The same holds other way around by almost the sameconstruction; if the state ρ is not steerable by two copiesof id then it is not steerable by any unitary channels Φ U ,Φ V .Note that similar result would hold if only one of thechannels would be unitary, but then only that one unitarychannel would be replaced by the identity map id .Clearly if the state ρ would be separable, then it wouldnot be steerable by any channel. The converse does nothold, even if the state ρ is entangled it still may not besteerable by any channels. We will provide a useful con-dition for the steerability of a given state ρ ∈ D H⊗H thatwill help us to show that even if the state ρ is entangled,it does not have to be steerable by any pair of channelsΦ : D H → D H , Φ : D H → D H . Proposition 19.
The state ρ ∈ D H⊗H is steerable bythe channels Φ : D H → D H , Φ : D H → D H only ifit is steerable by two copies of the identity channel id : D H → D H .Proof. Assume that the state ρ ∈ D H⊗H is not steerableby two copies of the identity channel id : D H → D H ,then there exists a state σ ∈ D H⊗H⊗H such thatTr ( σ ) = ρ, Tr ( σ ) = ρ. Let Φ : D H → D H , Φ : D H → D H be any two channelsand denote ˜ σ = ( id ⊗ Φ ⊗ Φ )( σ ) , then we have Tr (˜ σ ) = ( id ⊗ Φ )( ρ ) , Tr (˜ σ ) = ( id ⊗ Φ )( ρ ) , so the state ρ is not steerable by the channels Φ , Φ . Note that one may get other conditions for steering byreplacing only one of the channels by the identity map id .One may generalize this result to the general proba-bilistic theory but it may be rather restrictive and not asgeneral as one would wish. One may also use the idea ofthe proof of Prop. 19 together with the result of Prop.17 to obtain the results on compatibility of channels thatare concatenations of other channels, similar to the re-sults obtained in [21].We will present an example of an entangled state thatis not steerable by any pair of channels. Example . Let dim( H ) = 2 with the standard basis | i , | i and let | W i ∈ H ⊗ H ⊗ H be given as | W i = 1 √ | i + | i + | i ) . The projector | W ih W | ∈ D H⊗H⊗H is known as W state.We have ρ W = Tr ( | W ih W | ) = Tr ( | W ih W | ) ∈ D H⊗H , that shows that the state ρ W is not steerable by a pair ofthe identity channels id : D H → D H , which as a result ofProp. 19 means that it is not steerable by any channelsΦ : D H → D H , Φ : D H → D H . Moreover it is knownthat the state ρ W is entangled [43, Example 6.70].Since it will be useful in later calculations we will showthat the state | W ih W | is the only state from D H⊗H⊗H such that ρ W = Tr ( | W ih W | ) = Tr ( | W ih W | ). Let | ϕ i = √ ( | i + | i ), then we have ρ W = 13 | ih | + 23 | ϕ ih ϕ | . Let σ ∈ D H⊗H⊗H denote the state such that ρ W =Tr ( σ ) = Tr ( σ ). We have ρ W | i = 0 that impliesTr( σ | ih | ⊗ ) = Tr( σ | ih | ⊗ ⊗ | ih | ) = 0 thatimplies h | σ | i = h | σ | i = h | σ | i = 0 as σ ≥
0. We will show that this implies σ | i = σ | i = σ | i = 0.Let A ∈ B h ( H ), A ≥ | ψ i ∈ H . Let k ψ k = p h ψ | ψ i denote the norm given by inner product. Assumethat we have h ψ | A | ψ i = 0, then k√ Aψ k = h√ Aψ |√ Aψ i = h ψ | A | ψ i = 0and in conclusion we have √ A | ψ i = 0 and A | ψ i = √ A ( √ A | ψ i ) = 0 . Finally let us denote | ϕ ⊥ i = √ ( | i − | i ). We have ρ W | ϕ ⊥ i = 0 that implies Tr( σ | ϕ ⊥ ih ϕ ⊥ | ⊗ ) = 0 whichyields σ | ϕ ⊥ i = σ | ϕ ⊥ i = 0. We still use the shorthand | ϕ ⊥ i = | ϕ ⊥ i ⊗ | i .The eight vectors | i , | i , | ϕ i , | ϕ i , | ϕ ⊥ i , | ϕ ⊥ i , | i , | i form an orthonormal basis of H ⊗ H ⊗ H . Wehave already showed that we must have σ | ϕ ⊥ i = σ | ϕ ⊥ i = σ | i = σ | i = 04so in general we must have σ = a | ih | + a | ih | + a ϕ | ϕ ih ϕ | + a ϕ | ϕ ih ϕ | + b | ih | + ¯ b | ih | + b | ih ϕ | + ¯ b | ϕ ih | + b | ih ϕ | + ¯ b | ϕ ih | + b | ih ϕ | + ¯ b | ϕ ih | + b | ih ϕ | + ¯ b | ϕ ih | + b | ϕ ih ϕ | + ¯ b | ϕ ih ϕ | . Using the above expression for σ we getTr ( σ ) = a | ih | + a | ih | + a ϕ ⊗ | ih | + a ϕ ⊗ | ih | + b | ih | + ¯ b | ih | + b √ | ih | + ¯ b √ | ih | + b √ | ih | + ¯ b √ | ih | + b √ | ih | + ¯ b √ | ih | + b √ | ih | + ¯ b √ | ih | + b ⊗ | ih | + ¯ b ⊗ | ih | , that implies a = a ϕ = 0, a ϕ = , a = , b = b = b = b = b = 0 and b = √ . In conclusion we have σ = 13 ( | ih | + 2 | ϕ ih ϕ | + √ | ih ϕ | + √ | ϕ ih | )= | W ih W | . XI. BELL NONLOCALITY
Bell nonlocality is, similarly to steering, a phe-nomenom that we do not find in classical theory, butis often used in quantum theory. Bell nonlocality [2] wasformulated as a response to the well-known EPR paradox[3]. Although in the original formulation, the operationalidea was different than the one we will present, we willsee that Bell nonlocality may be understood as an incom-patibility test, in the same way as steering.Let us assume that we have four parties: Alice, Bob,Charlie and Dan. Alice has two channels Φ A : K A → K B and Φ A : K A → K B that she can use to senda state to Bob and Charlie has two channels Φ C : K C → K D and Φ C : K C → K D that he can useto send a state to Dan. Assume that K C ˜ ⊗ K A is de-fined and let ψ ∈ K C ˜ ⊗ K A be a bipartite state sharedby Alice and Charlie. The idea that we use to defineBell nonlocality is very simple: if we were able to use( id ⊗ (Φ A , Φ A ))( ψ ) and ((Φ C , Φ C ) ⊗ id )( ψ ) as non-trivialincompatibility test, we may as well investigate whether((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ ) provides an incompatibilitytest in the same manner. Definition 12.
Let ψ ∈ K C ˜ ⊗ K A and letΦ A : K A → K B , Φ A : K A → K B , Φ C : K C → K D , Φ C : K C → K D be channels. We call the state ((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ )bipartite biconditional state.Assume that the channels Φ A and Φ A are compati-ble, so that we have (Φ A , Φ A ) = J (Φ A ) for some chan-nel Φ A : K A → K B ˜ ⊗ K B and also that the chan-nels Φ C and Φ C are compatible, so there is a channelΦ C : K C → K D ˜ ⊗ K D such that (Φ A , Φ A ) = J (Φ A ).Let ψ ∈ K C ˜ ⊗ K A , then we have((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ ) = ( J ′ ⊗ J ′ )((Φ C ⊗ Φ A )( ψ )) , where the maps J ′ are defined as before, with the ex-ception that we denote them the same even though theymap different spaces.We present a definition of Bell nonlocality using thesame line of thinking as we used in Def. 11. For simplicitywe will denote Q DC = ( J ′ ⊗ J ′ )( K D ˜ ⊗ K D ˜ ⊗ K C ˜ ⊗ K C ) . Definition 13.
Let ψ ∈ K C ˜ ⊗ K A be a bipartite stateand let Φ A : K A → K B , Φ A : K A → K B , Φ C : K A → K C and Φ C : K A → K D be channels. We say that thebipartite biconditional state ((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ ) isBell nonlocal if((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ ) / ∈ Q DC . Otherwise we call the bipartite biconditional state Belllocal.The following result follows immediatelly from Def. 13.
Corollary 3.
Let ψ ∈ K C ˜ ⊗ K A be a bipartite state andlet Φ A : K A → K B , Φ A : K A → K B , Φ C : K A → K C and Φ C : K A → K D be channels. The bipartite bicon-ditional state ((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ ) is Bell nonlocalonly if the channels Φ A , Φ A and Φ C , Φ C are incompati-ble. We will show that entanglement plays a key role in Bellnonlocality.
Proposition 20.
Let ψ ∈ K C ˙ ⊗ K A be a separable bipar-tite state and let Φ A : K A → K B , Φ A : K A → K B , Φ C : K A → K C and Φ C : K A → K D be channels. Thebipartite biconditional state ((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ ) isBell local.Proof. It is again sufficient to consider ψ = x C ⊗ x A for x A ∈ K A , x C ∈ K C due to the linearity of the5maps (Φ A , Φ A ) and (Φ C , Φ C ). Consider the state ϕ ∈ K D ˜ ⊗ K D ˜ ⊗ K C ˜ ⊗ K C given as ϕ = Φ C ( x C ) ⊗ Φ C ( x C ) ⊗ Φ A ( x A ) ⊗ Φ A ( x A ) , then we have((Φ C , Φ C ) ⊗ (Φ A , Φ A ))( ψ ) = ( J ′ ⊗ J ′ )( ϕ ) . XII. BELL NONLOCALITY OFMEASUREMENTS
We will again show that the Def. 13 follows the stan-dard definition of Bell nonlocality [27] in the formalismof [26].
Proposition 21.
Let S A , S A , S C and S C , be sim-plexes and let m A : K A → S A , m A : K A → S A , m C : K C → S C , m C : K C → S C be measurements.Let ψ ∈ K C ˜ ⊗ K A , then the bipartite biconditional state (( m C , m C ) ⊗ ( m A , m A ))( ψ ) is Bell nonlocal if (( m C , m C ) ⊗ ( m A , m A ))( ψ ) / ∈ ( S C × S C ) ˙ ⊗ ( S A × S A ) . Proof.
By direct calculation we have Q CD = ( J ′ ⊗ J ′ )( S C ˙ ⊗ S C ˙ ⊗ S A ˙ ⊗ S A )= ( S C × S C ) ˙ ⊗ ( S A × S A ) . One may again use the interpretation that both S C × S C and S A × S A are spaces of conditional measurementprobabilities, so if we have ψ ∈ ( S C × S C ) ˙ ⊗ ( S A × S A )then we must have 0 ≤ λ i ≤
1, for i ∈ { , . . . , n } , P ni =1 λ i = 1, such that ψ = n X i =1 λ i s Ci ⊗ s Ai where in standard formulations both s Ci ∈ S C × S C and s Ai ∈ A × S A are represented by probabilities, i.e. by num-bers, so the tensor product between them is omitted.We will provide proof of the standard and well-knownresult about connection of steering and Bell nonlocalityof measurements. Proposition 22.
Let S A , S A , S C and S C , be simplexesand let m A : K A → S A , m A : K A → S A , m C : K C → S C , m C : K C → S C be measurements. Let ψ ∈ K C ˜ ⊗ K A .If ( id ⊗ ( m A , m A ))( ψ ) ∈ K C ˙ ⊗ ( S A × S A ) , i.e. if the bipartite state is not steerable by measurements m A , m A , then (( m C , m C ) ⊗ ( m A , m A ))( ψ ) ∈ ( S C × S C ) ˙ ⊗ ( S A × S A ) . Proof.
Let( id ⊗ ( m A , m A ))( ψ ) ∈ K C ˙ ⊗ ( S A × S A ) , then for n ∈ N , i ∈ { , . . . , n } , there are 0 ≤ λ i ≤ x i ∈ K C and s i ∈ S A × S A , P ni =1 λ i = 1, such that wehave ( id ⊗ ( m A , m A ))( ψ ) = n X i =1 λ i x i ⊗ s i . We get(( m C , m C ) ⊗ ( m A , m A ))( ψ ) = n X i =1 λ i ( m C , m C )( x i ) ⊗ s i and since we have ( m C , m C )( x i ) = ( m C ( x i ) , m C ( x i )) ∈ S C × S C we have(( m C , m C ) ⊗ ( m A , m A ))( ψ ) ∈ ( S C × S C ) ˙ ⊗ ( S A × S A ) . Note that the same result would also hold for steeringby the measurements m C , m C .One may think that steering is somehow half of Bellnonlocality, or that it is some middle step towards Bellnonlocality as even our constructions in Sec. VIII and XIwould point to such a result. We will show that this isnot true in general, as we will provide a counter-exampleusing quantum channels in example 9. XIII. BELL NONLOCALITY OF QUANTUMCHANNELS
Bell nonlocality of quantum measurements is a deeplystudied topic in quantum theory, with several applica-tions in various device-independent protocols [49–52],randomness generation and randomness expansion [53,54] and others, for a recent review on Bell nonlocalitysee [55].Bell nonlocality of quantum channels follows very simi-lar rules to steering by quantum channels. We will deriveresults specific for quantum theory in the same mannersas in Sec. X.
Proposition 23.
Let ρ ∈ D H⊗H and let Φ : D H → D H , Φ : D H → D H , Φ : D H → D H , Φ : D H → D H bechannels. The bipartite biconditional state ((Φ , Φ ) ⊗ (Φ , Φ ))( ρ ) is Bell nonlocal only if the bipartite bicondi-tional state (( id, id ) ⊗ ( id, id ))( ρ ) is Bell nonlocal.Proof. If the bipartite biconditional state (( id, id ) ⊗ ( id, id ))( ρ ) is Bell local, then there exist σ ∈ D H⊗H⊗H⊗H such that Tr ( σ ) = ρ, Tr ( σ ) = ρ, Tr ( σ ) = ρ, Tr ( σ ) = ρ. σ = (Φ ⊗ Φ ⊗ Φ ⊗ Φ )( σ )then Tr (˜ σ ) = (Φ ⊗ Φ )( ρ ) , Tr (˜ σ ) = (Φ ⊗ Φ )( ρ ) , Tr (˜ σ ) = (Φ ⊗ Φ )( ρ ) , Tr (˜ σ ) = (Φ ⊗ Φ )( ρ ) . Note that again we do not have to replace all of thechannels by the identity channels id , but we may replaceonly some. Proposition 24.
Let ρ ∈ D H⊗H and let Φ : D H → D H , Φ : D H → D H , Φ : D H → D H , Φ : D H → D H be channels, moreover let Φ = Φ U be a unitarychannel given by the unitary matrix U , then the bipar-tite biconditional state ((Φ U , Φ ) ⊗ (Φ , Φ ))( ρ ) is Bellnonlocal if and only if the bipartite biconditional state (( id, Φ ) ⊗ (Φ , Φ ))( ρ ) is Bell nonlocal.Proof. Using the very same idea as before, if the bipartitebiconditional state ((Φ U , Φ ) ⊗ (Φ , Φ ))( ρ ) is Bell local,then there is σ ∈ D H⊗H⊗H⊗H such thatTr ( σ ) = (Φ U ⊗ Φ )( ρ ) , Tr ( σ ) = (Φ U ⊗ Φ )( ρ ) , Tr ( σ ) = (Φ ⊗ Φ )( ρ ) , Tr ( σ ) = (Φ ⊗ Φ )( ρ ) . Let ˜ σ = (Φ U ∗ ⊗ id ⊗ id ⊗ id )( σ )then we get Tr (˜ σ ) = ( id ⊗ Φ )( ρ ) , Tr (˜ σ ) = ( id ⊗ Φ )( ρ ) , Tr (˜ σ ) = (Φ ⊗ Φ )( ρ ) , Tr (˜ σ ) = (Φ ⊗ Φ )( ρ ) . One may obtain similar results if some other of thechannels Φ , Φ , Φ , Φ is unitary as well as if more oreven all of them are unitary.The most iconic and most studied aspect of Bell non-locality are the Bell inequalities. We are going to presenta version of CHSH inequality for quantum channels.Assume that dim( H ) = 2 and let | i , | i denote anyorthonormal basis of H . We will use the shorthand | i = | i ⊗ | i . Let i, j ∈ { , } and let E (Φ i , Φ j ) = h | (Φ i ⊗ Φ j )( ρ ) | i− h | (Φ i ⊗ Φ j )( ρ ) | i− h | (Φ i ⊗ Φ j )( ρ ) | i + h | (Φ i ⊗ Φ j )( ρ ) | i = Tr((Φ i ⊗ Φ j )( ρ ) A )where A = | ih | − | ih | − | ih | + | ih | . The quantity E (Φ i , Φ j ) is to be interpreted as the cor-relation between the marginals Tr ((Φ i ⊗ Φ j )( ρ )) andTr ((Φ i ⊗ Φ j )( ρ )). Since we have − ≤ A ≤ it isstraightforward that we have − ≤ E (Φ i , Φ j ) ≤
1. De-fine a quantity X ρ = E (Φ , Φ ) + E (Φ , Φ ) + E (Φ , Φ ) − E (Φ , Φ ) , we will show that X ρ corresponds to the quantity usedin CHSH inequality. It is straightforward to see that − ≤ X ρ ≤ X ρ . Proposition 25.
If the biconditional bipartite state ((Φ , Φ ) ⊗ (Φ , Φ ))( ρ ) is Bell local, then we have − ≤ X ρ ≤ .Proof. If the biconditional bipartite state ((Φ , Φ ) ⊗ (Φ , Φ ))( ρ ) is Bell local then there is σ ∈ D H⊗H⊗H⊗H such that Tr ( σ ) = (Φ ⊗ Φ )( ρ ) , Tr ( σ ) = (Φ ⊗ Φ )( ρ ) , Tr ( σ ) = (Φ ⊗ Φ )( ρ ) , Tr ( σ ) = (Φ ⊗ Φ )( ρ ) . This yields E (Φ , Φ ) = Tr((Φ ⊗ Φ )( ρ ) A ) = Tr(Tr ( σ ) A )= Tr( σ ( | ih | ⊗ ⊗ | ih | ⊗ − | ih | ⊗ ⊗ | ih | ⊗ − | ih | ⊗ ⊗ | ih | ⊗ + | ih | ⊗ ⊗ | ih | ⊗ )) . In the same manner we get E (Φ , Φ ) = Tr( σ ( | ih | ⊗ ⊗ ⊗ | ih |−| ih | ⊗ ⊗ ⊗ | ih |−| ih | ⊗ ⊗ ⊗ | ih | + | ih | ⊗ ⊗ ⊗ | ih | )) ,E (Φ , Φ ) = Tr( σ ( ⊗ | ih | ⊗ | ih | ⊗ − ⊗ | ih | ⊗ | ih | ⊗ − ⊗ | ih | ⊗ | ih | ⊗ + ⊗ | ih | ⊗ | ih | ⊗ ))7and E (Φ , Φ ) = Tr( σ ( ⊗ | ih | ⊗ ⊗ | ih |− ⊗ | ih | ⊗ ⊗ | ih |− ⊗ | ih | ⊗ ⊗ | ih | + ⊗ | ih | ⊗ ⊗ | ih | )) . Together we get X ρ = 2Tr( σ ( | ih | + | ih |−| ih | − | ih | + | ih | − | ih | + | ih | − | ih |−| ih | + | ih |−| ih | + | ih |−| ih | − | ih | + | ih | + | ih | ))that implies − ≤ X ρ ≤ √ Proposition 26.
For any state ρ ∈ D H⊗H and any fourchannels Φ : D H → D H , Φ : D H → D H , Φ : D H → D H , Φ : D H → D H we have X ρ ≤ √ . Proof.
We define the adjoint channel Φ ∗ to channel Φ as the the linear map Φ ∗ : B h ( H ) → B h ( H ) such thatfor all σ ∈ D H and E ∈ B h ( H ), 0 ≤ E ≤ we haveTr(Φ ( σ ) E ) = Tr( σ Φ ∗ ( E )) . Since Φ is a channel we have 0 ≤ Φ ∗ ( E ) ≤ andΦ ∗ ( ) = . This approach of mapping effects instead ofstates is called the Heisenberg picture.Let i, j ∈ { , } , then we haveTr((Φ i ⊗ Φ j )( ρ ) | ih | ) = Tr( ρ Φ ∗ i ( | ih | ) ⊗ Φ ∗ j ( | ih | )) . Denoting M i = Φ ∗ i ( | ih | ) M j = Φ ∗ j ( | ih | )we see that we have E (Φ i , Φ j ) = Tr( ρM i ⊗ M j ) − Tr( ρ ( − M i ) ⊗ M j ) − Tr( ρM i ⊗ ( − M j ))+ Tr( ρ ( − M i ) ⊗ ( − M j ))= E ( M i , M j ) , where E ( M i , M j ) is a correlation for the two-outcomemeasurements given by the effects M i and M j . It is wellknown result [28] that we always have E ( M , M ) + E ( M , M )+ E ( M , M ) − E ( M , M ) ≤ √ . It is very intuitive that the Tsirelson bound, reach-able by measurements, will be also reachable by chan-nels. To prove this, let
M, N ∈ B h ( H ), 0 ≤ M ≤ ,0 ≤ N ≤ and define channels Φ M : B h ( H ) → B h ( H ),Φ N : B h ( H ) → B h ( H ) such that for σ ∈ D H we haveΦ M ( σ ) = Tr( σM ) | ih | + Tr( σ ( − M )) | ih | , Φ N ( σ ) = Tr( σN ) | ih | + Tr( σ ( − N )) | ih | . It is easy to verify that the maps Φ M , Φ N are quan-tum channels and that they are also measurementsas they map the state space D H to the simplex conv {| ih | , | ih |} . Let ρ ∈ D H⊗H , then we haveTr((Φ M ⊗ Φ N )( ρ ) A ) = Tr( ρ ( M ⊗ N − ( − M ) ⊗ N ) − Tr( ρ ( M ⊗ ( − N )))+ Tr( ρ (( − M ) ⊗ ( − N )))= E ( M, N ) . This proves that any set of correlations and any viola-tion of CHSH inequality reachable by measurements isalso reachable by quantum channels as a violation of thebound given by Prop. 25.To generalize the proposed inequality one may replacethe projectors | ih | and | ih | by any pair of effects M, N ∈ B h ( H ), 0 ≤ M ≤ , 0 ≤ N ≤ and have A = M ⊗ N − ( − M ) ⊗ N − M ⊗ ( − N )+( − M ) ⊗ ( − N ) . From now on we will consider a special case. Keepdim( H ) = 2 and let | ψ + ih ψ + | = 12 ( | ih | + | ih | + | ih | + | ih | )be the maximally entangled state, let U , U , V , V beunitary matrices and let Φ = Φ U , Φ = Φ U , Φ = Φ V ,Φ = Φ V be unitary channels given by the respectiveunitary matrices. We will consider the bipartite bicon-ditional state ((Φ U , Φ U ) ⊗ (Φ V , Φ V ))( | ψ + ih ψ + | ) andwe will show that the correlations for the given bipartitebiconditional state are of a particular nice form. We have(Φ U i ⊗ Φ V j )( | ψ + ih ψ + | ) = ( id ⊗ Φ V j U Ti )( | ψ + ih ψ + | )where i, j ∈ { , } and for U T denotes the transpose ofthe matrix U . For the correlation we have E (Φ U i , Φ V j ) = Tr(( id ⊗ Φ V j U Ti )( | ψ + ih ψ + | ) A )= 12 ( |h | V j U Ti | i| + |h | V j U Ti | i| − |h | V j U Ti | i| − |h | V j U Ti | i| ) . (11)8 ϑ X | ψ + ih ψ + | X | ψ + ih ψ + | √ Figure 1. The blue solid line is X | ψ + ih ψ + | as a function of theparameter ϑ ∈ [1 ,
10] when we consider the bipartite bicondi-tional state (Φ U , Φ U ) ⊗ (Φ V , Φ V )( | ψ + ih ψ + | ) from example8. The red dashed line corresponds to the Tsirelson bound2 √ We will provide an example of a violation of the boundgiven by Prop. 25 by incompatible unitary channels.
Example . Let dim( H ) = 2 and let ϑ ∈ R be a pa-rameter. Let U , U , V , V be unitary matrices givenas U = 1 √ (cid:18) − (cid:19) ,U = (cid:18) (cid:19) ,V = 1 √ ϑ (cid:18) √ ϑ −√ ϑ (cid:19) ,V = 1 √ ϑ (cid:18) √ ϑ √ ϑ − (cid:19) . Consider the bipartite biconditional state (Φ U , Φ U ) ⊗ (Φ V , Φ V )( | ψ + ih ψ + | ). Using Eq. (11) we can obtain X | ψ + ih ψ + | as a function of ϑ . The function is plotted inFig. 1, where it is shown that for certain values of ϑ thebipartite biconditional state violates the bound given byProp. 25.It is also easy to see that the bipartite biconditionalstate (( id, id ) ⊗ ( id, id ))( | ψ + ih ψ + | ) does not violate thebound given by Prop. 25, because all of the correlationsare the same, yet according to Prop. 24 we know thatit must be a Bell nonlocal bipartite biconditional state.This shows that not all Bell nonlocal bipartite bicondi-tional states violate the inequality given by Prop. 25.One may wonder whether there is or is not a connec-tion between steering and Bell nonlocality. As we havealready showed in Prop. 22, for measurements Bell non-locality implies steering. We will show that for channelsthe same does not hold. Example . Let dim( H ) = 2. Let ρ W ∈ D H⊗H be givenas in example 7 as a partial trace over the state | W ih W | .We already know that the state ρ W is not steerable byany pair of channels. Consider the bipartite biconditionalstate (( id, id ) ⊗ ( id, id ))( ρ W ), if it is Bell local, then theremust be a state σ ∈ D H⊗H⊗H⊗H such thatTr ( σ ) = Tr ( σ ) = Tr ( σ ) = Tr ( σ ) = ρ W . Observe that Tr ( σ ) ∈ D H⊗H⊗H is such thatTr (Tr ( σ )) = Tr (Tr ( σ )) = ρ W which implies that,according to our calculations in example 7, we must haveTr ( σ ) = | W ih W | . According to [56, Lemma 3] this implies that there is astate ρ ∈ D H such that σ = ρ ⊗ | W ih W | . This impliesthat we have Tr ( σ ) = ρ ⊗
13 (2 | ih | + | ih | ) which isclearly a separable state. This is a contradiction as weshould have had Tr ( σ ) = ρ W , which is an entangledstate. XIV. CONCLUSIONS
We have introduced the general definition of compati-bility of channels in general probabilistic theory throughthe idea of conditional channels. We have also shownthat a naive idea for a compatibility test leads to a sim-ple and straightforward formulation of steering and Bellnonlocality. These formulations of steering and Bell non-locality are overall new even when we consider only mea-surements instead of channels. Throughout the paper wehave shown that all of our definitions and result are incorrespondence with the known result for measurementsand we have also provided several examples and resultsabout the introduced concepts in quantum theory.The paper has opened several new questions and ar-eas of research. For example, a possible area of researchwould be to look at the structure of conditional statesand conditional channels and to try to connect them toBayesian theory.Concerning the compatibility of channels, one may for-mulate different notions of degree of (in)compatibility orof robustness of compatibility in general probabilistic the-ory and look at their properties, in a similar way as itwas already done in quantum theory [57]. For quantumchannels one may wonder which types of channels arecompatible. This would generalize the no broadcastingtheorem [56, 58] which states that two unitary channelscan not be compatible.One may also consider our formulations of steering andBell nonlocality as a case of the problem of finding a mul-tipartite state with given marginals. Such problems werestudied in recent years [59, 60], but not in the form thatwould be applicable to the problems of steering and Bellnonlocality as incompatibility tests. This opens questionswhether one may characterize the structure of the cone9 Q CD and of other cones of interest in quantum theory.From a geometrical viewpoint this question is closely tiedto the question of existence of other Bell inequalities forchannels than the one we presented. Existence and ex-act form of the generalized Bell inequalities is also a veryinteresting possible area of research.We may also consider the use of steering and Bell non-locality of channels in the context of quantum informa-tion theory and quantum communication. Both steeringand Bell nonlocality of measurement were used to for-mulate new quantum protocols and it is of great interestwhether exploiting the steering and Bell nonlocality ofchannels may lead to even better or more useful applica-tions.One may also try and clarify the lack of connectionbetween steering and Bell nonlocality of channels. Aswe have showed in example 9, even if two channels cannot steer a state, when applied to both parts of the statethe resulting biconditional bipartite state may be Bellnonlocal. This may even have interesting applicationsin quantum theory of information as so far steering hasbeen considered to lead to one-side device-independent protocols that were seen as a middle step between theoriginal protocol and device-independent protocol.It may also be interesting to consider the resource the-ories of channel incompatibility, of steering by channelsand of Bell nonlocality of channels. Several similar re-source theories were already constructed, see [61] for areview. ACKNOWLEDGMENTS
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