aa r X i v : . [ qu a n t - ph ] F e b Submitted to:QPL 2021 © A.J. ParzygnatThis work is licensed under theCreative Commons Attribution License.
Conditional distributions for quantum systems
Arthur J. Parzygnat
Institut des Hautes ´Etudes ScientifiquesBures-sur-Yvette, France [email protected]
Conditional distributions, as defined by the Markov category framework, are studied in the settingof matrix algebras (quantum systems). Their construction as linear unital maps are obtained via acategorical Bayesian inversion procedure. Simple criteria establishing when such linear maps arepositive are obtained. Several examples are provided, including the standard EPR scenario, wherethe EPR correlations are reproduced in a purely compositional (categorical) manner. A comparisonbetween the Bayes map and the Petz recovery map is provided, illustrating some key differences.
Keywords.
Bayes, inference, Markov category, operator system, positive map, quantum informationtheory, quantum probability, recovery map
Contents
There is a one-to-one correspondence between stochastic maps (conditional probabilities) on finite setsand positive unital maps on finite-dimensional commutative C ∗ -algebras. This correspondence is madeprecise categorically via a stochastic variant of Gelfand duality [14, 19]. Hence, any concept describedcategorically at the level of stochastic maps can be instantiated on arbitrary (not necessarily commutative) C ∗ -algebras. In particular, the notion of Bayesian inversion, disintegrations, and conditioning have beenformulated categorically [5, 6, 8, 9, 12, 13, 15, 23], and the first two have been explored in the setting offinite-dimensional C ∗ -algebras in [23, 24] through a generalization of Markov categories [5, 13] to theirquantum variants [21]. However, conditioning in this setting remains unexplored, as far as I am aware. The purpose of the present work is to begin the systematic study of quantum conditionals as positivemaps between finite-dimensional C ∗ -algebras. Although this goal is not fully realized here, we are con-tent with achieving it on bi-partite systems of matrix algebras equipped with states whose marginals arefaithful. Even though this sounds quite restrictive, it already includes many cases of interest, includingthe fully entangled EPR state on a two qubit system [4, 10]. The case of multi-partite states, non-faithfulmarginals, and more general hybrid classical-quantum systems will be addressed in future work. The conditioning in [15] does not use the multiplication map in its formulation of conditioning in the quantum setting.
Conditional distributionsIn this work, we use category theory to define what we mean by quantum conditionals. Then, weprove a purely categorical theorem indicating how one can construct quantum conditionals through theusage of
Bayes maps (whose definition is motivated by categorical probability theory). We then im-plement this construction in the setting of matrix algebras. In general, the resulting conditional doesnot define a positive map. As such, we find necessary and sufficient conditions for conditionals to bepositive. A positive conditional need not be completely positive, and EPR provides an example illus-trating this point. We end by introducing the conditional domain , which is the largest operator systemfor which a conditional is positive (in the Heisenberg picture). Typically, this operator system is not a C ∗ -subalgebra. Examples are provided throughout. This section briefly reviews the abstract theory of quantum CD and Markov categories [21], which aregeneralizations of CD and Markov categories [5, 13]. String diagrams are reviewed in these mentionedpapers, but see [27] for a more thorough exposition. Time will always go up the page. The compositionwill go up the page for definitions and the example
FinStoch , while the composition will go down thepage for C ∗ -algebra maps (in the Heisenberg picture). Definition 2.1. A classical CD category is a symmetric monoidal category ( M , ⊗ , I ) , with ⊗ the tensorproduct and I the unit (associators and unitors are excluded from the notation), and where each object X in M is equipped with morphisms ! X ≡ X : X → I , called the discarder/grounding , and ∆ X ≡ : X → X ⊗ X , called the copy/duplicate , all satisfying the following conditions = = = = (2.2) X ⊗ Y = X Y I = X ⊗ Y = X Y I = (2.3)expressed using string diagrams. A classical Markov category is a classical CD category for which everymorphism X f −→ Y is unital , i.e. natural with respect to in the sense that f = . A state on X is amorphism I p −→ X , which is drawn as p X . Example 2.4.
Let
FinStoch be the category whose objects are finite sets and where a morphism X f −→ Y is a stochastic map/conditional probability from X to Y , which, by definition, assigns to each element x ∈ X a probability measure f x on Y , whose value on y is written as f yx . The composite of a composablepair X f −→ Y g −→ Z is defined by the Chapman–Kolmogorov equation ( g ◦ f ) zx : = ∑ y ∈ Y g zy f yx . The tensorproduct is the cartesian product of sets and the product X × X ′ f × f ′ −−−→ Y × Y ′ of stochastic maps X f −→ Y and X ′ f ′ −→ Y ′ , and is given by ( f × f ′ ) ( y , y ′ )( x , x ′ ) : = f yx f y ′ x ′ . The tensor unit is the single element set,often denoted by {•} . Functions are special kinds of stochastic maps whose probability measures are { , } -valued. In particular, the maps ∆ X and ! X are the stochastic maps associated to the functions ∆ X ( x ) : = ( x , x ) and ! X ( x ) = • . A state on X encodes a probability measure on X . The reader will notice that the /o/o / / notation is not used in this article, unlike in our earlier works [19, 21, 23, 24]. Thereason is because we do not need to emphasize the distinction between deterministic maps and stochastic maps in this work. .J.Parzygnat 3The conditions described in (2.2) suggest that classical Markov categories cannot be extended to thequantum setting due to the universal no-broadcasting theorem [2, 18]. However, there is a way aroundthese conditions by working with a larger class of morphisms, adding an even and odd grading formorphisms, and substituting the commutativity condition for another closely-related condition [21]. Definition 2.5. A quantum CD category is a Z -graded symmetric monoidal category M , and whereeach object X is equipped with an even discarder, an even copy map, and an odd involution ∗ X ≡ X : X → X satisfying the same conditions as a classical CD category, except the last condition in (2.2), andalso satisfying the additional conditions = = X ⊗ Y = X Y X = X (2.6)A quantum Markov category is a quantum CD category in which every morphism is unital. A morphism X f −→ Y is said to be ∗ -preserving iff f ◦ ∗ X = ∗ Y ◦ f . Example 2.7.
From now on, all C ∗ -algebras will be assumed unital. Although the category of finite-dimensional C ∗ -algebras and positive unital maps (cf. Notation 3.11) does not form a quantum Markovcategory (essentially due to the no-broadcasting theorem), this category naturally embeds into a quantumMarkov category, allowing the structure of the ambient quantum Markov category to be utilized [21]. Let fdC *- AlgU op Y be the category whose objects are finite-dimensional C ∗ -algebras (see [19, Section 2.3] fora review of C ∗ -algebras within a categorical setting). For example, a matrix algebra will be written as M n ( C ) indicating the C ∗ -algebra of complex n × n matrices. A morphism from A to B in fdC *- AlgU op Y is either a linear (even) or conjugate-linear (odd) unital map B F −→ A . Notice that the function goesbackwards because of the superscript op . The tensor product (over C ) is the tensor product of finite-dimensional C ∗ -algebras, so that the unit is C . The tensor product of linear maps is defined in the usualway, while the tensor product of conjugate-linear maps can be defined similarly [28, Section 9.2.1].However, note that it does not make sense to define the tensor product of a linear map with a conjugate-linear one. The ∗ operation is the involution on a C ∗ -algebra, which is conjugate-linear. The copymap ∆ A from A to A ⊗ A in fdC *- AlgU op Y is the multiplication map A ⊗ A µ A −−→ A determined onelementary tensors by A ⊗ A A A . The discard map from A to C in fdC *- AlgU op Y is defined to bethe unit inclusion map ! A : C → A sending λ ∈ C to λ A . A linear map B F −→ A is ∗ -preserving if andonly if it sends self-adjoint elements in B to self-adjoint elements in A . For convenience, we will dropthe op and work directly with the unital maps on the algebras from now on.Although we have introduced the categories FinStoch and fdC *- AlgU Y , we will be more explicitand work mainly with matrix algebras, linear maps, and positive maps in our main results. The abstractsetting will mainly be used in the next two sections to provide the general context. This means that there is a functor M → BZ (where BZ is the one object category whose set of morphisms equals Z = { , } and whose composition is defined by addition modulo 2 in Z ) and a tensor product is defined for all objects andall morphisms of the same degree. Morphisms sent to 0/1 are called even / odd . Note that the tensor product of morphisms ofdifferent degrees is not defined, but the collection of even morphisms is a symmetric monoidal category. Unitality is defined differently for odd morphisms. We exclude the details because we will not need this definition here. Every such finite-dimensional C ∗ -algebra is ∗ -isomorphic to a finite direct sum of (square) matrix algebras [11, Theo-rem 5.5]. Conditional distributions
Here, we review two formulations of Bayes’ theorem, which we express categorically. Throughout thissection, M will denote either a classical or quantum Markov category and C will denote some (notnecessarily monoidal) subcategory of M . Furthermore, all morphisms will be even from now on. Definition 3.1.
Given states I p −→ X and I q −→ Y , a state-preserving morphism X f −→ Y (i.e. q : = f ◦ p ) iswritten as a triple ( f , p , q ) . A left/right Bayes map for ( f , p , q ) is a morphism f L / f R : Y → X in M suchthat qf L X Y = p f X Y , q f R XY = pf XY (3.2)If all morphisms are in C , then f L / f R are said to be left/right Bayesian inverses of ( f , p , q ) (in C ).Bayes maps are automatically state-preserving. If all morphisms are ∗ -preserving, then there is nodistinction between left and right concepts (this is always the case in classical Markov categories [5, 21]). Definition 3.3.
Let I s −→ X ⊗ Y be a state and let p and q denote its marginals I s −→ X ⊗ Y π X −→ X and I s −→ X ⊗ Y π Y −→ Y , respectively. Here, π X and π Y are the projections , which are defined as π X : = (cid:0) X ⊗ Y id X × ! Y −−−−→ X ⊗ I ∼ = X (cid:1) and π Y : = (cid:0) X ⊗ Y ! X ⊗ id Y −−−−→ I ⊗ Y ∼ = Y (cid:1) . A conditional distribution of s given Y / X (or Y / X conditional for short) is a morphism Y s | Y −→ X / X s | X −→ Y such that qs | Y X Y = s X Y , s X Y = p s | X X Y . (3.4) Definition 3.5.
Let X and Y be objects, let I p −→ X be a state and let f , g : X → Y be morphisms. Themorphism f is said to be left/right p -a.e. equivalent to g iff pf Y X = pg Y X , p f X Y = p g X Y . (3.6)All of these definitions are quite similar. Indeed, if f L and f R are left and right Bayes maps forsome ( f , p , q ) , then they are automatically left and right a.e. unique, respectively. Furthermore, the Y / X conditionals are also left/right a.e. unique. A.e. equivalence agrees with the standard measure-theoreticnotion [5, Proposition 5.3 ,5.4] (as well as the C ∗ -algebraic one [23, Section 3.1], [21, Theorem 5.12]).With these preliminaries, Bayes’ theorem can now be expressed in two different ways. Theorem 3.7. [Bayes’ theorem via Bayesian inversion] Every triple ( f , p , q ) in FinStoch (a state-preserving ( X , p ) f −→ ( Y , q ) ) admits a (necessarily a.e. unique) Bayesian inverse (in FinStoch ). Theorem 3.8. [Bayes’ theorem via conditional distributions] Every joint state {•} s −→ X × Y in
FinStoch admits both (necessarily a.e. unique) X and Y conditionals. .J.Parzygnat 5
Example 3.9.
These two versions of Bayes’ theorem are often expressed as the equations p ( x | y ) p ( y ) = p ( y | x ) p ( x ) and p ( x | y ) p ( y ) = p ( x , y ) = p ( y | x ) p ( x ) , (3.10)respectively. Although it seems as though the former is a special case of the latter, notice that theinput data for each definition is different. The first version has input data a morphism X f −→ Y and astate {•} p −→ X (the state q on Y is obtained via composition). Meanwhile, the second version has inputdatum a state {•} s −→ X × Y . This distinction may seem pedantic, but it is crucial for generalizing to thenon-commutative setting [24]. Notation 3.11.
In what follows, if A is a matrix, then A † denotes is conjugate transpose. A matrix A ∈ M m ( C ) is positive iff it is self-adjoint ( A † = A ) and its eigenvalues are non-negative, equivalently A = C † C for some C ∈ M m ( C ) . If P ∈ M m ( C ) is an orthogonal projection (i.e. P † P = P ), then P ⊥ : = m − P denotes its complement projection. The standard matrix units of M m ( C ) will be denoted by E ( m ) i j with i , j ∈ { , . . . , m } . They satisfy E ( m ) i j E ( m ) kl = δ jk E ( m ) il , where δ jk is the Kronecker delta taking value1 when j = k and 0 otherwise. A linear map F : M n ( C ) → M m ( C ) is positive ( completely positive ) iffit ( F ⊗ id M k ( C ) ) sends positive matrices to positive matrices (for all k ∈ N ). In terms of the notation atthe beginning of Section 3, M = fdC *- AlgU Y and C = fdC *- AlgPU is the subcategory consisting of(linear) positive unital maps. If F : M n ( C ) → M m ( C ) is linear, then F ∗ denotes its adjoint with respectto the Hilbert–Schmidt inner product on matrices, i.e. F ∗ : M m ( C ) → M n ( C ) is the unique linear mapsatisfying tr ( F ∗ ( A ) B ) = tr ( AF ( B )) for all A ∈ M m ( C ) and B ∈ M n ( C ) . An example that appears oftenis the Hilbert–Schmidt dual of the inclusion ι M m ( C ) : M m ( C ) → M m ( C ) ⊗ M n ( C ) sending A to A ⊗ n ,and is given by the CPU map tr M n ( C ) , which is called the partial trace . Explicitly, tr M n ( C ) is determinedby its action on simple tensors, namely tr M n ( C ) ( A ⊗ B ) = tr ( B ) A , and it satisfies a partial form of cyclicitygiven by tr M n ( C ) (cid:0) ( A ⊗ B )( m ⊗ C ) (cid:1) = tr M n ( C ) (cid:0) ( m ⊗ C )( A ⊗ B ) (cid:1) (3.12)for all inputs A , B , C . Example 3.13.
Let A : = M m ( C ) and B : = M n ( C ) be two matrix algebras. Let ω = tr ( ρ · ) and ξ = tr ( σ · ) be states on A and B , respectively, with respective density matrices. Let B F −→ A be aunital linear map. If σ is positive definite (so that the state ξ is faithful), then there are unique left andright Bayes maps for ( F , ω , ξ ) . They are respectively given by F L ( A ) : = σ − F ∗ ( ρ A ) and F R ( A ) : = F ∗ ( A ρ ) σ − (3.14)for all A ∈ A . If F is ∗ -preserving, demanding that these two functions be equal is equivalent todemanding that there is a ∗ -preserving Bayes map F . In this case, its explicit formula is given by(see [24] for a proof) F ( A ) = √ σ − F ∗ ( √ ρ A √ ρ ) √ σ − . (3.15)Hence, if F is positive unital (PU) or completely positive unital (CPU), then so is F . The reader willnotice that (3.15) is the formula for the Petz recovery map [1, 3, 16, 25, 26]. However, we will later seethat the Petz recovery map is distinct from the Bayes map in general. The difference between the Petzrecovery map and the Bayes map is more pronounced in the case that σ is not positive definite (so that ξ is not faithful), though the details of this will not be discussed here (but see [22, 24]).Before using this example, we first need to explain how conditionals can be constructed using Bayesmaps more abstractly. Afterwards, we will look at several examples by combining the two results. This is especially due to the abusive notation of using p for all mathematical objects. Note that a.e. equivalence now reduces to equality since ξ is faithful. Conditional distributions
Theorem 4.1.
Let I s −→ X ⊗ Y be a joint state with marginals I p −→ X and I q −→ Y . Let Y L −→ X ⊗ Y andX R −→ X ⊗ Y be left and right Bayes maps for ( π Y , s , q ) and ( π X , s , p ) , respectively. Then the compositess | X : = (cid:0) X R −→ X ⊗ Y π Y −→ Y (cid:1) and s | Y : = (cid:0) Y L −→ X ⊗ Y π X −→ X (cid:1) are X and Y conditionals of s, respectively.Proof. By assumption qL X Y Y = s = s and p R YXX = s = s . (4.2)The definitions of s | X and s | Y are drawn as s | X XY : = R and s | Y YX : = L . (4.3)From this, we immediately obtain qs | Y X Y = qL = s = s X Y = s = p R = p s | X X Y , (4.4)which is the desired conclusion. (cid:4) This theorem, together with the left/right a.e. uniqueness of left/right Bayes maps, is useful becauseit allows us to write down explicit formulas for conditionals in the quantum setting, at least up to thesupports of the states. For the remainder of this work, we will focus on applying this to matrix algebras,rather than arbitrary finite-dimensional C ∗ -algebras. Corollary 4.5.
Set A : = M m ( C ) and B : = M n ( C ) . Let ζ ≡ tr ( ν · ) be a state on A ⊗ B (with densitymatrix ν ) whose marginals on A and B are given by ζ ◦ ι A = : ω ≡ tr ( ρ · ) and ζ ◦ ι B = : ξ ≡ tr ( σ · ) ,respectively. Suppose that ρ and σ are invertible. Then there are unique conditionals B F : = ζ | A −−−−→ A and A G : = ζ | B −−−−→ B given byF ( B ) = tr B (cid:0) ( m ⊗ B ) ν (cid:1) ρ − and G ( A ) = σ − tr A (cid:0) ν ( A ⊗ n ) (cid:1) . (4.6) The Hilbert–Schmidt duals of these maps are given byF ∗ ( A ) = tr A (cid:0) ν ( ρ − A ⊗ n ) (cid:1) and G ∗ ( B ) = tr B (cid:0) ( m ⊗ B σ − ) ν (cid:1) . (4.7) Proof.
The first claim follows from Theorem 4.1 and Example 3.13. For instance, F ( B ) = ι ∗ A (cid:0) ( A ⊗ B ) ν (cid:1) ρ − = tr B (cid:0) ( A ⊗ B ) ν (cid:1) ρ − . The second claim follows from the definition of the Hilbert–Schmidtinner product and the cyclic properties of the trace. (cid:4) .J.Parzygnat 7Are the conditionals F and G in Corollary 4.5 positive maps? Let’s look at some examples. Example 4.8.
In the notation of Corollary 4.5, take m = n = ν to be Bohm’s EPR densitymatrix ν : = (cid:20) − − (cid:21) corresponding to the pure state √ (cid:0) | ↑i ⊗ | ↓i − | ↓i ⊗ | ↑i (cid:1) , where | ↑i and | ↓i are just e = (cid:2) (cid:3) and e = (cid:2) (cid:3) expressed in Dirac notation, [4, 10]. Then the marginal density matrices ρ and σ both equal . Since this is invertible, the conclusions of Corollary 4.5 apply. Hence, F (cid:18)(cid:20) a bc d (cid:21)(cid:19) = tr B a b c d a b c d − − = (cid:20) d − b − c a (cid:21) = (cid:20) − (cid:21)(cid:20) a bc d (cid:21) T (cid:20) −
11 0 (cid:21) , (4.9)which shows that F is PU, but not CPU. The same formula is obtained for G . It is worth comparingthis expression to the one obtained by using the Petz recovery map instead of the Bayes map. The Petzrecovery map R : A ⊗ B → A associated to the inclusion ι A : A → A ⊗ B and the state ζ on A ⊗ B is given by R ( A ⊗ B ) = p ρ − tr B (cid:0) √ ν ( A ⊗ B ) √ ν (cid:1) p ρ − = B ( ν ( A ⊗ B ) ν ) , (4.10)where we have used the fact that ρ = and ν = ν (because ν is a rank 1 density matrix), so that √ ν = ν . Precomposing R with the inclusion gives F ′ : = R ◦ ι B , which acts as F ′ (cid:18)(cid:20) a bc d (cid:21)(cid:19) =
12 tr B a + d − a − d − a − d a + d
00 0 0 0 = (cid:20) a + d a + d (cid:21) = tr (cid:0)(cid:2) a bc d (cid:3)(cid:1) . (4.11)Notice that the map F ′ , obtained using the Petz recovery map, is actually CPU, unlike the conditional F we obtained in (4.9). However, which one of these two maps recovers the standard EPR correlations?Suppose that Alice (represented by A ) obtains new evidence in the form of a state ϕ = h↑ | · | ↑i (for example, suppose that she set up an apparatus to measure the spin and obtained the result spin up).Then by applying the maps F and F ′ to these states via pullback, Alice infers that Bob (represented by B ) would obtain the state on B given by ( ϕ ◦ F ) (cid:18)(cid:20) a bc d (cid:21)(cid:19) = d = (cid:28) ↓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) a bc d (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ↓ (cid:29) and ( ϕ ◦ F ′ ) (cid:18)(cid:20) a bc d (cid:21)(cid:19) = tr (cid:0)(cid:2) a bc d (cid:3)(cid:1) . (4.12)The first map shows that the spin up state for Alice changes to the spin down state for Bob once themap F is applied. On the other hand, the map F ′ always gives the totally mixed state for Bob. Thisindicates that F is a more suitable inference map describing the EPR correlations, since F ′ loses all theentanglement (more precisely, it is an entanglement breaking channel). Note that analogous conclusionshold if the evidence Alice has is the spin in any direction: F will predict the opposite spin for Bob while F ′ still predicts the totally mixed state. More details relating this to Bayesian updating will be presentedelsewhere [22].The fact that F , and not F ′ , reproduced the EPR correlations suggests that it has its merits anddeserves further study (an alternative derivation of the EPR correlations is done via conditional densitymatrices in [17, Section V.A.3]). Example 4.8 also shows that a joint state can have positive conditionalsthat are not necessarily CPU. But do the conditionals always need to be positive? The next exampleshows that the answers to this question is no. One could equivalently obtain the Hilbert–Schmidt duals and act on the associated density matrices in the Schr¨odingerpicture.
Conditional distributions
Example 4.13.
Set A : = M ( C ) and B : = M ( C ) . A general pure state in C ⊗ C ∼ = C is of theform | Ψ i = c ↑↑ | ↑↑i + c ↑↓ | ↑↓i + c ↓↑ | ↓↑i + c ↓↓ | ↓↓i , where c ↑↑ , c ↑↓ , c ↓↑ , c ↓↓ ∈ C satisfy | c ↑↑ | + | c ↑↓ | + | c ↓↑ | + | c ↓↓ | = | ↑↓i = | ↑i ⊗ | ↓i and similarly for the other vectors). Given p ∈ ( , ) , set q : = − p and let ν : = | Ψ ih Ψ | be the density matrix in A ⊗ B ∼ = M ( C ) associated to the pure statewith c ↑↑ = q p , c ↑↓ = q q , c ↓↑ = − q p , and c ↓↓ = q q . Then ν = p √ pq − p √ pq √ pq q −√ pq q − p −√ pq p −√ pq √ pq q −√ pq q , ρ = (cid:20) q − pq − p (cid:21) , and σ = (cid:20) p q (cid:21) . (4.14)Thus, ρ − = pq h p − qp − q i and σ − = h p − q − i . Using Corollary 4.5, one obtains the explicit formulas F (cid:0)(cid:2) a bc d (cid:3)(cid:1) = " a + d + pc + qb √ pq d − a + pc − qb √ pq d − a + qb − pc √ pq a + d − pc + qb √ pq , G ∗ (cid:0)(cid:2) a bc d (cid:3)(cid:1) = " a + d + pb + qc √ pq d − a + qc − pb √ pq d − a + pb − qc √ pq a + d − pb + qc √ pq , (4.15) G (cid:0)(cid:2) a bc d (cid:3)(cid:1) = a − b − c + d q qp ( a − b + c − d ) q pq ( a + b − c − d ) a + b + c + d , and (4.16) F ∗ (cid:0)(cid:2) a bc d (cid:3)(cid:1) = a − b − c + d q pq ( a − b + c − d ) q qp ( a + b − c − d ) a + b + c + d . (4.17)If p = q = , then all of these maps are positive. Indeed, given a positive matrix of the form C : =[ ab ] [ a b ] = h aa abba bb i , F and G send this matrix to F ( C ) = (cid:20) a + bb − a (cid:21) (cid:2) a + b b − a (cid:3) and G ( C ) = (cid:20) a − ba + b (cid:21) (cid:2) a − b a + b (cid:3) when p = q = . (4.18)However, when p = , then neither F nor G are positive. In fact, neither F nor G are ∗ -preserving, whichis a necessary condition for positivity. We will come back to this in the next section. The expressions for conditionals in Corollary 4.5 have two disadvantages. First, they are only partiallydefined on the supports. Second, they need not be positive maps. A necessary condition for positivityis ∗ -preservation, so we will first analyze when conditionals are ∗ -preserving. All positive matrices are non-negative sums of matrices of this type. Hence, proving F ( C ) and G ( C ) are positive is sufficientto proving that F and G are positive, respectively. Proving F and G are positive is also equivalent to proving F ∗ and G ∗ arepositive. We have not discussed this aspect here because we assumed the marginals are invertible. See [24] for more details regardingsupports and their role in Bayesian inversion. The analogous situation for conditionals is part of ongoing work. .J.Parzygnat 9
Lemma 5.1.
Let A : = M m ( C ) , B : = M n ( C ) be the algebras and ζ = tr ( ν · ) , ω = tr ( ρ · ) , and ξ = tr ( σ · ) the states as defined in Corollary 4.5. Then ∗ -preserving conditionals B F : = ζ | A −−−−→ A and A G : = ζ | B −−−−→ B respectively exist if and only if (cid:2) ρ , tr B (cid:0) ν ( m ⊗ B ) (cid:1)(cid:3) = ∀ B ∈ B , i.e. h ρ , tr B (cid:0) ν ( m ⊗ E ( n ) kl ) (cid:1)i = ∀ k , l ∈ { , . . . , n } , (5.2) and (cid:2) σ , tr A (cid:0) ν ( A ⊗ n ) (cid:1)(cid:3) = ∀ A ∈ A , i.e. h σ , tr A (cid:0) ν ( E ( m ) i j ⊗ n ) (cid:1)i = ∀ i , j ∈ { , . . . , m } , (5.3) respectively.Proof. By Corollary 4.5, the formulas for F and G are uniquely determined. These linear maps are ∗ -preserving if and only if F ( B † ) = F ( B ) † and G ( A † ) = G ( A ) † for all inputs, or equivalently F ( B † ) † = F ( B ) and G ( A † ) † = G ( A ) for all inputs. We begin with F and assume F is ∗ -preserving. Then,tr B (cid:0) ( m ⊗ B ) ν (cid:1) ρ − = F ( B ) = F ( B † ) † = (cid:16) tr B (cid:16)(cid:0) m ⊗ B † (cid:1) ν (cid:17) ρ − (cid:17) † = ρ − tr B (cid:0) ν ( m ⊗ B ) (cid:1) . (5.4)Multiplying both sides by ρ and using the properties of the partial trace, this is equivalent totr B (cid:0) ν ( m ⊗ B ) (cid:1) ρ = ρ tr B (cid:0) ν ( m ⊗ B ) (cid:1) . (5.5)Since every B can be expressed as a linear combination B = ∑ k , l B kl E ( n ) kl , this is equivalent to the secondcondition in (5.2). By a similar calculation, if (5.2) holds, then F is ∗ -preserving. An analogous argumentshows that G is ∗ -preserving if and only if (5.3) holds. (cid:4) Example 5.6.
One can also use Lemma 5.1 to prove that the maps F and G in Example 4.13 are not ∗ -preserving if p = .Lemma 5.1 provides a necessary condition for positive conditionals to exist. Are they sufficient?Namely, if a conditional is ∗ -preserving, is it necessarily positive? The reason we ask this question isbecause this is (perhaps surprisingly) true for Bayes maps on matrix algebras (when one of the densitymatrices has full support). In fact, the ∗ -preserving condition is strong enough to imply complete positiv-ity for Bayes maps [24]. Based on Example 4.8, we so far know that ∗ -preservation is not strong enoughto imply complete positivity (or even Schwarz-positivity) for conditionals, so it is natural to ask aboutpositivity alone. In the following theorem, we settle this question. Theorem 5.7.
In the notation of Lemma 5.1, positive conditionals B F : = ζ | A −−−−→ A and A G : = ζ | B −−−−→ B existif and only if Equations (5.2) and (5.3) hold, respectively (i.e. if and only if ∗ -preserving conditionalsexist).Proof. It suffices to prove the claim for F . If F is positive, then it is automatically ∗ -preserving, whichis where Lemma 5.1 applies. Conversely, suppose that F is ∗ -preserving. Then by Corollary 4.5, Equa-tion (5.2), the properties of the partial trace, and the functional calculus F ( B † B ) = tr B (cid:0) ( m ⊗ B † B ) ν (cid:1) ρ − = p ρ − tr B (cid:0) ( m ⊗ B ) √ ν √ ν ( m ⊗ B † ) (cid:1)p ρ − . (5.8)Since the right-hand-side of this expression is manifestly positive, F is positive. (cid:4) Since ρ commutes with tr B (cid:0) ( m ⊗ B ) ν (cid:1) for every B , any function of ρ commutes with tr B (cid:0) ( m ⊗ B ) ν (cid:1) as well (see “TheFunctional Calculus” series in [20]). ∗ -preserving conditions do not hold for all elements in the domain algebras, we canalways find maximal subspaces on which F and G are positive.
Definition 5.9.
In the notation of Lemma 5.1, set A ρ c : = (cid:8) A ∈ A : [ ρ , A ] = (cid:9) and B σ c : = (cid:8) B ∈ B : [ σ , B ] = (cid:9) (5.10)to be the commutants of { ρ } and { σ } inside A and B , respectively. Set B ν : = n B ∈ B : tr B (cid:0) ν ( m ⊗ B ) (cid:1) ∈ A ρ c o and A ν : = n A ∈ A : tr A (cid:0) ( A ⊗ n ) ν (cid:1) ∈ B σ c o (5.11)to be the conditional domains of ν inside B and A , respectively. A (concrete) operator system inside M k ( C ) is a (complex) vector subspace O ⊆ M k ( C ) such that k ∈ O and A ∈ O implies A † ∈ O . Lemma 5.12.
In the notation of Definition 5.9, B ν and A ν are operator systems.Proof. It suffices to prove this for B ν . First, B ν is a subspace by linearity. Second, tr B ( ν ) = ρ and ρ ∈ A ρ c imply n ∈ B ν . Third, if B ∈ B ν , thentr B (cid:0) ν ( m ⊗ B † ) (cid:1) = (cid:0) tr B (cid:0) ( m ⊗ B ) ν (cid:1)(cid:1) † = (cid:0) tr B (cid:0) ν ( m ⊗ B ) (cid:1)(cid:1) † . (5.13)Since A ρ c is a ∗ -algebra tr B (cid:0) ν ( m ⊗ B ) (cid:1) ∈ A ρ c implies (cid:0) tr B (cid:0) ν ( m ⊗ B ) (cid:1)(cid:1) † ∈ A ρ c . Hence, B † ∈ B ν by (5.13). Thus B ν is an operator system. (cid:4) In this way, although one might not be able to condition on the full algebra to obtain a positive map,one might be able to condition on an operator system inside that algebra.
Example 5.14.
In terms of Example 4.13 and assuming p = , one can show A ρ c = (cid:26)(cid:20) a bb a (cid:21) : a , b ∈ C (cid:27) ⊂ A and B σ c = (cid:26)(cid:20) a d (cid:21) : a , d ∈ C (cid:27) ⊂ B (5.15)are the commutants. The conditional domains are given by B ν = (cid:26)(cid:20) a d (cid:21) : a , d ∈ C (cid:27) ⊂ B and A ν = (cid:26)(cid:20) a bb a (cid:21) : a , b ∈ C (cid:27) ⊂ A . (5.16)Example 5.14 suggests that B ν and A ν are not only operator systems, but they might even be C ∗ -subalgebras. Is this always the case? Example 5.17.
The answer to this question is no, though the simplest counterexample I could currentlyfind involves a 9 × A = M ( C ) and B = M ( C ) . Its expression is notparticularly enlightening, so I have chosen to not record it here. Corollary 5.18.
In the notation of Lemma 5.1 and Definition 5.9, there exist conditionals B F −→ A and A G −→ B such that the restrictions B ν ֒ → B F −→ A and A ν ֒ → A G −→ B are positive unital maps fromoperator systems to C ∗ -algebras. In terms of the Hilbert–Schmidt duals (the Schr¨odinger picture), therestrictions A ρ c ֒ → A F ∗ −→ B and B σ c ֒ → B G ∗ −→ A are positive trace-preserving maps between C ∗ -algebras. Also, I could not find a 4 × ν for which the conditional domains are not C ∗ -subalgebras, and I suspect thatthis may always be the case. I hope to resolve this in future work. .J.Parzygnat 11Positivity of the Hilbert–Schmidt duals guarantees that density matrices living in the respective com-mutants always get sent to density matrices under the conditionals. Example 5.19.
In terms of Example 4.13 (see also Example 5.14), suppose that Bob (represented by B ) obtains new evidence in the form of a state ϕ = h↑ | · | ↑i . This is represented by the density matrix | ↑ih↑ | , which is in B σ c . The conditional G ∗ sends this density matrix to (cid:2) − − (cid:3) . In other words,with evidence | ↑ih↑ | , Bob will infer that the state Alice receives is given by √ ( | ↑i − | ↓i ) . However, ifBob obtains new evidence that is represented by a density matrix not in B ρ c , such as (cid:2) − − (cid:3) , then theimage of this under G ∗ is given by + √ pq h − p − qq − p i . Although this is a matrix with nonnegativeeigenvalues (they are 0 and 1), it is not a density matrix because it is not self-adjoint, unless p = q = . The work presented here includes preliminary investigations on the structure of conditioning in the set-ting of hybrid classical-quantum systems (finite-dimensional C ∗ -algebras) from the Markov categoryperspective. We have focused only on matrix algebras and joint states for which the marginal densitymatrices are invertible. Our definitions are distinct from those of [15], which defines conditioning interms of predicates and uses operations analogous to those used to define the Petz recovery map (simi-lar constructions are done using the Q / calculus in [7, 17]). The root of the distinction between thesetwo approaches comes from the fact that we use the multiplication map to formulate Bayes maps, eventhough it is not a positive map. By using quantum Markov categories [21], we have been able to constructconditioning in a way completely analogous to what is done in the classical theory, while still using themultiplication map, and then finding conditions for which the resulting maps are positive.Some work in progress includes the extension of the results presented here to the case where themarginal density matrices are not invertible. Although this seems like an innocent generalization, thisis where most of the intricate details occur when analyzing the case of disintegrations and Bayesian in-version in [23, 24]. It is also what accentuates the difference between Bayesian inverses and the Petzrecovery map. Another crucial generalization is to the case of general finite-dimensional C ∗ -algebras,i.e. direct sums of matrix algebras, to include hybrid classical-quantum systems. Upon obtaining suitablenecessary and sufficient conditions for positive conditionals to exist, one should show that these condi-tions are automatically satisfied for commutative algebras in such a way so that the conditional versionof Bayes’ theorem (Theorem 3.8) is reproduced (Theorem 3.7 was already reproduced in [24]).I see many interesting future directions based on the ideas presented here. For example, what doesthe set of joint states admitting positive conditionals look like? What is the structure of conditionalsfor multi-partite (as opposed to bi-partite) states on quantum systems? For example, one can take three(classical) random variables and construct multiple conditionals for other purposes, such as defining con-ditional independence or constructing Markov chains. There are also theorems describing the consistencyof successive conditioning in classical probability theory (for a viewpoint similar to the one presentedhere, see [13, Section 11], particularly Lemma 11.11, and the references therein). One wonders if suchresults still hold in the quantum setting from this perspective, at least on the conditional domains definedhere. Another direction for future investigations is to obtain approximate versions of these results usingdistance measures between states, such as the fidelity or statistical distance. Acknowledgements.
This research has received funding from the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation program (QUASIFT grant agreement677368).2 Conditional distributions
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