A geometric view on quantum incompatibility

A geometric view on quantum incompatibility Anna Jenčová Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia Genoa, June 2018

Outline Introduction GPT: basic definitions and examples Incompatibility: - characterization - incompatibility witnesses and degree - maximal incompatibility Incompatibility and Bell non-locality Steering

General probabilistic theories: basic notions states: preparation procedures of a given system convex structure: probabilistic mixtures of states Assumption: Any state space is a compact convex subset K R m. effects: yes/no experiments determined by outcome probabilities in each state respect the convex structure of states: affine maps K [0, 1] Assumption: All affine maps K [0, 1] correspond to effects. For the more general framework, see e.g (G. Chiribella, G. D Ariano, P. Perinotti, PRA 2010)

General probabilistic theories: basic notions measurements: (with finite number of outcomes) described by outcome statistics in each state affine maps K n n : simplex of probabilities over {0,..., n} given by effects: f i (x) = f (x) i, i = 0,..., n, f i = 1 i Assumption: All affine maps K n correspond to measurements.

General probabilistic theories: basic examples Classical systems: state spaces: m effects: vectors in R m+1 with entries in [0, 1] measurements: classical channels T : m n The measurements are identified with (m + 1) (n + 1) stochastic matrices (conditional probabilities) {T (j i)} i,j : T (j i) = f (δ m i ) j, δ m i = vertices of m

General probabilistic theories: basic examples Quantum systems state spaces: S(H) = density operators on a Hilbert space H, dim(h) < effects: E(H) = quantum effects, 0 E I, E B(H) measurements: POVMs on H M 0,..., M n E(H), M i = I i

General probabilistic theories: basic examples Spaces of quantum channels state spaces: C A,A = set of all quantum channels (CPTP maps) B(H A ) B(H A ) effects: f E(C A,A ), f (Φ) = Tr M(Φ id R )(ρ AR ), Φ C A,A, for some state ρ AR S(H AR ) and effect M E(H A R) measurements: f 0,..., f n, f i (Φ) = Tr M i (Φ id R )(ρ AR ), Φ C A,A, for some ρ AR S(H AR ) and a POVM {M 0,..., M n } on H A R.

GPT and ordered vector spaces Ordered vector space: (V, V + ) a real vector space V (dim(v ) < ) a closed convex cone V + V, generating in V, V + V + = {0} Dual OVP: an ordered vector space (V, (V + ) ) vector space dual V dual cone (V + ) = {ϕ V, ϕ, x 0, x V } We have V = V, (V + ) = V +.

GPT and ordered vector spaces Any state space K determines an OVP: A(K) = all affine functions K R A(K) + = positive affine functions E(K) = {f A(K), 0 f 1 K }, 1 K is the constant unit function Then (A(K), A(K) + ) is an OVP, E(K) is the set of all effects. A norm in A(K): f max = max f (x) x K

GPT and ordered vector spaces Let (V (K), V (K) + ) be the dual OVP. K {ϕ V (K) +, ϕ, 1 K = 1} a base of V (K) + V (K) + λ 0 λk the cone generated by K V (K) the vector space generated by K Base norm: ψ K = inf{a + b, ψ = ax by, a, b 0, x, y K}, ψ V (K) - the dual norm to max.

GPT and ordered vector spaces: self-duality We say that the cone V + is (weakly) self-dual if V + (V + ) classical: V ( n ) + A( n ) + ( (R n+1 ) + ) quantum: V (S(H)) + A(S(H)) + ( B(H) + ) not true for spaces of quantum channels not true for all spaces of classical channels

Composition of state spaces: tensor products Assumption: For state spaces K A and K B, the joint state space K A K B is a subset in V (K A ) V (K B ). We have: K A min K B K A K B K A max K B minimal tensor product: separable states K A min K B = co{x A x B, x A K A, x B K B } maximal tensor product: no-signalling K A max K B := {y V (K A ) V (K B ), f A f B, y 0, 1 A 1 B, y = 1}

Composition of state spaces: tensor products classical: na min nb = na max nb = nab the probability simplex on {0,..., n A } {0,..., n B } quantum: S(H A ) S(H B ) = S(H AB ) S(H A ) min S(H B ) separable states S(H A ) max S(H B ) normalized entanglement witnesses quantum channels: C A,A C B,B = CAB,A caus B causal bipartite channels C A,A min C B,B = CAB,A loc B local bipartite channels C A,A max C B,B causal, not necessarily CP

Channels and positive maps Channels: transformations of the systems allowed in the theory affine maps between state spaces K K affine maps K V (K ) + extend to positive maps of the ordered vector spaces (V (K), V (K) + ) (V (K ), V (K ) + ) not all affine maps are allowed in general: n m : all classical channels S(H) S(H ): must be completely positive

Entanglement breaking maps A positive map T A : K A V (K A )+ is entanglement breaking (ETB) if (T A id B )(K A max K B ) V (K A min K B ) + for all state spaces K B. T A is ETB iff it factorizes through a simplex: T A : K g T 0 n V (K ) + (measure (g) and prepare (T 0 ))

Duality The space of all linear maps V (K) V (K ), with the cone of positive maps is an ordered vector space. Its dual is the space of linear maps V (K ) V (K), with the cone of positive ETB maps, duality: T, T = Tr TT

Polysimplices A polysimplex is a Cartesian product of simplices S lo,...,l k := l0 lk with pointwise defined convex structure. states of a device specified by inputs and allowed outputs theories exhibiting super-quantum correlations (S. Popescu, D. Rohrlich, Found. Phys. 1994; J. Barrett, PRA 2007; P. Janotta, R. Lal, PRA 2013)

Polysimplices S = S lo,...,l k : convex polytope, with vertices s n0,...,n k = (δ l 0 n0,..., δ l k nk ) δj i is the j-th vertex of li A(S) + : generated by effects of the projections m i : S l0,...,l k li, m i 0,..., m i l i E(S), The base of A(S) + is the dual polytope.

Polysimplices: examples Square (gbit, square-bit): = 1 1 V ( ) + A( ) + - weakly self-dual the only polysimplex with this property m 1 1 m 0 1 m 0 0 m 1 0 s 0,1 s 1,1 s 0,0 s 1,0 the cone A( ) +

Polysimplices: examples Hypercube: n = 1 1 base of A( n ) + : a cross-polytope e 0 s 0,1,1 s 1,1,1 m 2 1 s 0,0,1 s 1,0,1 e 1 e 2 m 1 1 m 0 1 s s 0,1,0 s 1,1,0 m 0 m 1 0 0 s s 1,0,0 0,0,0 m 2 0 the cube 3 octahedron

Polysimplices: examples Prism: s 0,1 s 2,1 1 2 m1 1 m 0 2 s 1,1 m 0 0 m 0 1 s 0,0 s 2,0 s 1,0 m 1 0 S 2,1 = 2 1 a base of A(S 2,1 ) +

Polysimplices and classical channels Let S = k+1 n. A correspondence between s k+1 n and stochastic matrices T : T (j i) = m i j(s), s = (T ( 0),..., T ( k)) k+1 n is isomorphic to the set of all classical channels k n. Any polysimplex is isomorphic to a face in a set of classical channels.

Polysimplices and quantum channels There are channels R : C A,A m n and R : m n C A,A, such that RR = id. The maps are determined by ONBs { i A }, { j A } as R(Φ)(j i) = j, Φ( i i A ) j A, i, j; Φ C A,A R (s)(ρ) = i,j m i j(s) i, ρ i A j j A, ρ S(H A ); s m n. Such maps are called: R - retraction, R - section. Note that R R is a projection (onto a set of c-c channels).

Incompatible measurements in GPT A collection of measurements f 0,..., f k, f i : K li, is the same as a channel F = (f 0,..., f k ) : K S l0,...,l k : F (x) = (f 0 (x),..., f k (x)), f i = m i F, i = 0,..., k compatible: marginals of a single joint measurement g : K L = l0 lk that is, (f 0,..., f k ) : K g L J S f 0,..., f k are compatible if and only if (f 0,..., f k ) is ETB.

Incompatibility witnesses By duality of the spaces of maps: F = (f 0,..., f k ) : K S is incompatible if and only if there is an incompatibility witness: a map W : S V (K) + such that Tr FW < 0

Incompatibility witnesses Any W : S V (K) + is determined by images of vertices: w n0,...,n k = W (s n0,...,n k ) W is ETB iff there are ψ i j V (K)+ such that w n0,...,n k = i ψ i n i

Incompatibility witnesses A witness must be non-etb, but this is not enough Characterization of witnesses: W : S V (K) + is a witness iff no translation of W along K is ETB. Translation along K: W : S V (K) +, such that W (s) = W (s) + v, for some 1 K, v = 0.

Incompatibility witnesses in Bloch ball z 0 W w 1,1 w 0,1 v W w 1,1 w 0,1 W x + w 1,0 w 1,0 y w 0,0 1 w 0,0

Incompatibility witnesses for two-outcome measurements We have another characterization if S is a hypercube k+1 : Let W : k+1 V (K) + pick a vertex: s n0,...,n k all adjacent edges: e 0,..., e k e 0 s 0,1,1 s 1,1,1 s 0,0,1 s 1,0,1 s e 1 e 2 s 0,1,0 s 1,1,0 k W (e i ) K > 2 1 K, W ( s) i=0 s s 1,0,0 0,0,0

Examples of extremal witnesses for pairs of effects It is enough to use extremal incompatibility witnesses: extremal as maps S V (K) + Some examples for S = : Square-bit: K = extremal non-etb maps = symmetries of the square r s dihedral group D 4 : group of order 8 2 generators: r, s non-etb, no nontrivial translations: witnesses

Examples of extremal witnesses for pairs of effects Quantum states: K = S(H) extremal non-etb maps: parallelograms in B(H) + with rank one vertices: x 00 x 00 + x 11 x 11 = ρ = x 01 x 01 + x 10 x 10, incompatibility witness if perimeter (in trace norm) > 2Tr ρ for compatibility of pairs of effects, it is enough to consider restrictions to 2-dimensional subspaces

Incompatibility degree Can we quantify incompatibility? (M.M. Wolf et al., PRL 2009; P. Busch et al., EPL 2013; T. Heinosaari et al., J. Phys. A 2016; D. Cavalcanti, P. Szkrzypczyk, PRA 2016) Incompatibility degree: the least amount of noise that has to be added to obtain a compatible collection. different definitions by the choice of noise we choose coin-toss measurements = constant maps f p (x) p Collection of coin-tosses = constant map always compatible (ETB) F s : K s = (p 0,..., p k ) S

Incompatibility degree Let F, F s : K S, s S. all channels F ETB channels F s coin-tosses We put ID s (F ) = min{λ, (1 λ)f + λf s is ETB}, (T. Heinosaari et al. PLA 2014) ID(F ) := inf s S ID s(f )

Incompatibility degree by incompatibility witnesses For s int(s), let us denote W s := {W : S V (K) +, W (s) K} and Then ID s (F ) = q s (F ) := min W W s Tr FW. 0 if q s (F ) > 0 q s(f ) 1 q s(f ) otherwise. This expression is related to (dual) linear programs for incompatibility degree e.g. (M. Wolf, D. Perez-Garcia, C. Fernandez, PRL 2009)

ID attainable for pairs of quantum effects Using extremal witnesses B(H) +, we can prove: For quantum state spaces, we have max ID(F ) = 1 1 F :S(H) 2 - for ID s, s the barycenter of, proved already in (M. Banik et al., PRA 2013)

Maximal incompatibility in GPT For any s S, it is known that ID s (F ) k k + 1 The joint measurement for 1 k+1 F + k k+1 F s: choose one measurement in F uniformly at random replace all others by coin-tosses We say that F is maximally incompatible if ID(F ) = k k+1.

Maximal incompatibility for effects For two-outcome measurements, we have a nice characterization: Let F : K k : F is maximally incompatible if and only if F is a retraction. The corresponding section is the witness W : k K such that ID(F ) is attained. There exist k maximally incompatible effects on K if and only if there exists a projection K K whose range is affinely isomorphic to the hypercube k.

Maximal incompatibility: examples Polysimplices: Let M : S k+1, M = (m 0 n 0,..., m k n k ), n i {0,..., l i } Then M is maximally incompatible. Quantum channels: There are m = dim(h A ) maximally incompatible effects on C A,A compose the retraction R : C A,A m n with M as above. (cf. M. Sedlák et al., PRA 2016; AJ, M. Plávala, PRA 2017)

Bell non-locality in GPT Bell scenario: f 0 A,..., f k A A i A A y B f 0 B,..., f k B B i B j A p(j A, j B i A, i B ) j B The conditional probabilities satisfy the no-signalling conditions: j A p(j A, j B i A, i B ) = p A (j A i A ), i B p(j A, j B i A, i B ) = p B (j B i B ), i A j B

Bell non-locality in GPT In our setting: F A = (fa 0,..., f k A A ), F B = (fb 0,..., f k B B ), y K A K B (F A F B )(y) S A max S B There is a correspondence S A max S B no-signalling conditional probabilities: - the no-signalling polytope s p(j A, j B i A, i B ) := (m i A ja m i B jb )(s)

Bell non-locality in GPT Local hidden variable model: Λ q(λ) λ f 0 A,..., f k A A i A A y B f 0 B,..., f k B B i B q A (j A i A, λ) q B (j B i B, λ) j A j B p(j A, j B i A, i B ) = λ q(λ)q A (j A i A, λ)q B (j B i B, λ) (H. M. Wiseman, S. J. Jones, A. C. Doherty, PRL 2007)

Bell witnesses and Bell inequalities p(j A, j B i A, i B ) admit LHV iff s S A min S B : - the local polytope Bell witnesses: entangled elements in A(S A min S B ) + Extremal: finitely many µ 1,..., µ N Bell inequalities: s S A min S B µ i, s 0, i = 1,..., N µ i M i extremal affine maps S A A(S B ) + Let s = (F A F B )(y). If F A or F B is compatible or y separable, then s S A min S B.

The CHSH inequality If S A = S B = : the CHSH witnesses: µ isomorphisms the CHSH inequality: M : V ( ) + A( ) + 0 µ, (F A F B )(y) = 1 ( 1 1 ) 2 2 a 0 (b 0 + b 1 ) + a 1 (b 0 b 1 ), y a i = 1 2(f i A ) 0, b i = 1 2(f i B ) 0

Bell inequalities and the incompatibility degree Relation of violation of Bell inequalities to incompatibility degree: If F A is incompatible, then for any y K A K B, any Bell witness µ and s int(s A ), we have µ, F A F B (y) µ max q s (F A ).

Bell inequalities and the incompatibility degree Maximal violation of CHSH inequality: CHSH bound Quantum case: Tsirelson bound Equality case for the CHSH bound: If K = S(H) and S A =, then there is some H B H A, F B : S(H B ) and y S(H AB ) such that s is the barycenter of µ, F A F B (y) = 1 2 q s(f A ) (cf. M. M. Wolf et al., PRL 2009; P. Busch, N. Stevens, PRA 2014)

Bell inequalities and the incompatibility degree Sketch of a proof using incompatibility witnesses: µ, F A F B (y) = Tr F A W (Tr F s W )q s (F A ), with S A M A(S B ) + F B A(K B ) + T V (K A ) + W W is an incompatibility witness if Bell inequality is violated. (M is a map related to µ and T to y.) Bell inequalities are obtained from special incompatibility witnesses.

Bell inequalities and the incompatibility degree For the equality: A M A( B ) + F B B(H B ) + B(H A ) + W All incompatibility witnesses are obtained from CHSH inequalities.

Bell inequalities and the incompatibility degree In general: S A M A(S B ) + F B A(K B ) + T V (K A ) + if S A of S B, M : V (S A ) + A(S B ) + is never an isomorphism: weaker witnesses = there exists incompatible collections that do not violate Bell inequalities W (M. T. Quintino, T. Vértesi, N. Brunner, PRL 2014)

Steering in GPT Quantum steering: (E. Schrödinger, Proc. Camb. Phil. Soc. 1936) Rigorous definition (in GPT setting): f 0,..., f k i A y B p(j i) j x j i assemblage: {p(j i), x j i }, x j i K B, p(j i) probabilities p(j i)x(j i) = y B K B, i j

Steering in GPT Local hidden state (LHS) model: f 0,..., f k i Λ q(λ) λ x λ A y B q(j i, λ) j x j i p(j i)x j i = λ q(λ)q(j i, λ)x λ. (cf. H. M. Wiseman, S. J. Jones, A. C. Doherty, PRL 2007)

Assemblages and tensor products Let F A = (f 0,..., f k ). (F A id B )(y) S max K B assemblages elements β S max K B : p(j i)x j i = m i j id B, β admits LHS model if and only if β is separable for β = (F A id B )(y): no steering if y is separable or F A are compatible. steering witnesses: all entangled elements in A(S min K B ) + steering degree