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Which graph states are useful for quantum information processing? Mehdi Mhalla, Mio Murao , ∗ †‡ Simon Perdrix, Masato Someya, and Peter S. Turner ∗ † † 1 1 0 Abstract 2 n Graphstates[5]areanelegantandpowerfulquantumresourceformeasurement a based quantum computation (MBQC). They are also used for many quantum pro- J tocols (error correction, secret sharing, etc.). The main focus of this paper is to 1 provide a structural characterisationof the graphstates that can be used for quan- 3 tuminformationprocessing. Theexistenceofagflow(generalizedflow)[8]isknown to be a requirement for open graphs (graph, input set and output set) to perform ] h uniformlyandstronglydeterministiccomputations. Weweakenthegflowconditions p to define two new more general kinds of MBQC: uniform equiprobability and cons- - t tant probability. These classes can be useful from a cryptographic and information n pointofviewbecauseeventhoughwecannotdoadeterministiccomputationingen- a u eralwecanpreservethe informationandtransferitperfectly fromthe inputs to the q outputs. We derive simple graph characterisations for these classes and prove that [ thedeterministicanduniformequiprobabilityclassescollapsewhenthecardinalities 2 of inputs and outputs are the same. We also prove the reversibility of gflow in that v case. Thenewgraphicalcharacterisationsallowustogofromopengraphstographs 6 in general and to consider this question: given a graph with no inputs or outputs 1 fixed, which vertices can be chosen as input and output for quantum information 6 processing? We present a characterisationof the sets of possible inputs and ouputs 2 . forthe equiprobabilityclass,whichisalsovalidfordeterministiccomputationswith 6 inputs and ouputs of the same cardinality. 0 0 1 1 Introduction : v i X The graph state formalism [5] is an elegant and powerful formalism for quantum in- r formation processing. Graph states form a subfamily of the stabiliser states [4]. They a provide a graphical description of entangled states and they have multiple applications in quantum information processing, in particular in measurement-based quantum computation (MBQC) [9], but also in quantum error correcting codes [4] and in quantum ∗CNRS,LIG, Université deGrenoble, France †Graduate School of Science, The University of Tokyo, Japan ‡NanoQuine, The Universityof Tokyo, Japan 1 protocols like secret sharing [7, 6]. They offer a combinatorial approach to the characterisation of the fundamental properties of entangled states in quantum information processing. The invariance of the entanglement by local complementation of a graph [10]; the use of measure of entanglement based on the rank-width of a graph [11]; and the combinatorial flow characterisation [1] of deterministic evolutions in measurement- based quantum computation witness the import role of the graph state formalism in quantum information processing. In this paper, we focus on the application of graph states in MBQC and in particular on the characterisation of graphs that can be used to perform quantum information processinginthiscontext. Theexistenceofagraphicalconditionwhichguaranteesthata deterministicMBQCevolution canbedrivendespiteoftheprobabilisticbehaviourofthe measurements is a central pointin MBQC. Ithas already been proven that the existence of a certain kind of flow called glfow characterises uniformly stepwise determinism [1]. In section 3, we introduce a simpler but equivalent combinatorial characterisation using focused gflow and we provide a simple condition of existence of such a flow as the existenceofarightinversetotheadjacencymatrixofthegraph. Wealsoproveadditional properties in the case where the number of input and output qubits of the computation are the same: the gflow is then reversible and the stepwise condition [1] on determinism is not required to guarantee the existence of a gflow. The main contribution of this paper is the weakening of the determinism condition in order to consider the more general class of information preserving evolutions. Being information preserving is one of the most fundamental property that can be required for a MBQC computation. Indeed, some non-deterministic evolutions can be information preserving when one knows the classical outcomes of the measurements produced by the computation. Such evolutions are called equi-probabilistic – when each classical outcome occurs with probability 1/2 – or constant-probabilistic in the general case. In section 4, we introduce simple combinatorial conditions for equi-probabilistic and constant-probabilistic MBQC by means of excluded violating sets of vertices. We show, in the particular case where the number of input and output qubits are the same, that graphs guaranteeing equi-probabilism and determinism are the same. In section 6, using thisgraphicalcharacterisation, weaddressthefundamentalquestionoffindinginputand output vertices in an arbitrary graph for guaranteeing an equi-probabilistic (or deterministic) evolution. To this end, we show that the input and output vertices of a graph must form transversals of the violating sets induced by the equi-probabilistic characterisation. Finally, in the last section, we investigate several properties of the most general and less understood class of constant probabilistic evolutions. 2 Measurement-based quantum computation Inthissection,themainingredientsofmeasurementbasedquantumcomputation(MBQC) are described. More detailed introductions can be found in [2, 3]. An MBQC is described by (i) an open graph (G,I,O) (G is a simple undirected graph, I,O V(G) ⊆ are called resp. input and output vertices); (ii) a map α : OC [0,2π), where → 2 OC := V O, which associates with every non ouputvertex an angle; and (iii) two maps \ x,z : OC 0,1 V(G) called corrective maps. A vertex v supp(x(u)) supp(z(u)) is → { } ∈ ∪ called a corrector of u, where supp(y)= u y = 1 . Themaps x,z should beextensive u { | } in the sense that there exists a partial order over the vertices of the graph s.t. any ≺ corrector v of a vertex u is larger than u, i.e. v supp(x(u)) supp(z(u)) implies u v. ∈ ∪ ≺ Let N : C 0,1 I C 0,1 V(G) be the preparation map which associates with any { } { } → arbitraryinputstatelocatedontheinputqubitstheinitialentangledstateoftheMBQC: 1 N = ( 1)q(x,y) x,y x √2IC X − | ih | | | x 0,1 I,y 0,1 IC ∈{ } ∈{ } whereq : 0,1 V(G) N = x E(G) (supp(x) supp(x)) associates withevery x { } → 7→ | ∩ × | thenumberof edges of thesubgraphG = (V(G) supp(x),E(G) (supp(x) supp(x))) x ∩ ∩ × induced by x. Theone-qubitmeasurements,parametrizedbyanangleα ,ofeverynon-outputqubit u u are inducing the following projection P (α) : C 0,1 V(G) C 0,1 O of the entangled s { } { } → state onto the subspace of the output qubits, where s 0,1 OC stands for the classical ∈ { } outcomes of the one-qubit measurements: 1 P (α) = eαx·s y xy s √2IC X | ih | | | x 0,1 OC,y 0,1 O ∈{ } ∈{ } with α = α(u) and x s is the bitwise conjonction of x and s. MoxreovPer,u∈asduappp(txa)tive Pauli co·rrections depending on the classical outcomes of the measurements and on the corrective maps, are applied during the computation leading, for any possible classical outcomes s 0,1 OC, to the following overall (postselected) ∈ { } evolution χ :C 0,1 I C 0,1 O: s { } { } →   χs = Ps(α) Y Xsx(u)Zsz(u) N · · u V(G)  ∈ where X and Z are Pauli operators: X = X and Z = Z . s s s u supp(s) u s u supp(s) u An MBQC is implementing the quantumNop∈eration χ .NTh∈e evolution is { s}s 0,1 OC as follows: a classical outcome (also called branch) s 0,1∈{OC}is produced and the ∈ { } input state φ C 0,1 I is mapped to the state χ φ C 0,1 O (up to a normalisation). { } s { } | i ∈ | i∈ The probability for an outcome s 0,1 OC to occur is p = χ φ 2. s s ∈ { } || | i|| Theoverall evolution can bedecomposedinto severalsteps, correspondingtoapossi- bleimplementationoftheMBQCmodel: firsttheinputstate φ isencodedintotheopen | i graph state φ = N φ , then the local measurements (qubit u is measured according G | i | i the observable cos(α(u))X +sin(α(u))Y) and the local Pauli corrections are performed. This sequence of local operations is done according to the partial order induced by the correction maps x,z. 3 3 Determinism Definition 3.1. An MBQC (G,I,O,α,x,z) is strongly deterministic if all the branches are implementing the same map, i.e. U s.t. s 0,1 OC, χ = 1 U. s ∃ ∀ ∈ { } √2|OC| Lemma 3.1. If an MBQC is strongly deterministic then it implements an isometry. Proof. Since Ps 0,1 OC χ†sχs = I, U†U = I so U is an isometry and the MBQC imple- ∈{ } ments the super operator ρ UρU . † 7→ In order to point out the combinatorial properties of MBQC, the angles of measurements and the corrective maps can be abstracted away in the following way in order to keep only the influence of the initial open graph. Definition 3.2. An open graph (G,I,O) guarantees uniformly strong determinism if x,z s.t. α, (G,I,O,α,x,z) is strongly deterministic. ∃ ∀ An MBQC is said to guarantee stepwise strong determinism if any partial computation is also strongly deterministic. The gflow of an open graph is defined as follows, based on the use of the odd neighborhood of a set of vertices: for a given subset S of vertices in a graph G, Odd(S) := v V(G) s.t. N(v) S = 1 [2] . { ∈ | ∩ | } Definition 3.3. (g, ) is a gflow of (G,I,O), where g :OC 2Ic, if for any u, ≺ → — if v g(u), then u v; ∈ ≺ — u Odd(g(u)); ∈ — if v Odd(g(u)) and u= v then u v. ∈ 6 ≺ Theorem 3.1 ([1]). An open graph (G,I,O) guarantees uniform stepwise strong determinism iff (G,I,O) has a gflow. 3.1 Focused gflow Since the gflow is not uniquewe introduce a stronger version called focused gflow, which is unique if the number of inputs and outputs are the same. The focused gflow gives rise to a simpler characterisation of uniform stepwise strong determinism. The focused gflow is based on the use of extensive maps. Definition 3.4. g : OC 2IC is a focused gflow of (G,I,O) if g is extensive – i.e. the → transitive closure of the relation (u,v) s.t. v g(u) is a partial order over V(G) – and { ∈ } u OC, Odd(g(u)) OC = u ∀ ∈ ∩ { } Theorem 3.2. Anopen graph (G,I,O) guarantees uniform stepwise strong determinism iff (G,I,O) has a focused gflow. 4 Proof. We prove that (G,I,O) has a gflow iff it has a focused gflow. First, assume g is a focused gflow, and let be the transitive closure of (u,v) s.t. v g(u) . is a ≺ { ∈ } ≺ partial order and by definition, if v g(u) then u v. Moreover u Odd(g(u)) = u . ∈ ≺ ∈ { } Finally, if v Odd(g(u)) and v = u then v O, so there is no element larger than ∈ 6 ∈ v by definition of . Thus (g, ) is a gflow. Now, assume (g, ) is a gflow. We call ≺ ≺ ≺ the depth of a vertex u its distance to the output, i.e. the length of longest strictly increasing sequence u u .. u s.t. u O. We construct a focus gflow g 1 k k f ≺ ≺ ≺ ∈ by induction on the depth of the vertices. If u is of depth 1 then g (u) := g(u). If u f is of depth larger than 2, let g (u) := g(u) g (v), where is the symmetricdifference: A B = (fA B) (A BL).v∈SOinddc(eg(Ou)d)∩dO(AC,v6=Bu)f= Odd(A) O⊕dd(B), ⊕ ∪ \ ∩ ⊕ ⊕ Odd(g (u)) OC = (Odd(g(u)) Odd(g (v))) OC = (Odd(g(u)) OC) f(Odd∩(g(u)) u ) OC)L=v∈Oudd.(gM(u)o)∩reOoCv,evr6=ug is exftensive∩since the relation R∩ f ⊕ \{ } ∩ { } induced by g is s.t. uRv = u v so thetransitive closure of R is a partial order. f ⇒ ≺ 3.2 Induced adjacency matrix and reversibility We introduce the notion of induced adjacency matrix of an open graph and show that an open graph has a gflow if and only if its induced matrix has a DAG as right inverse. Definition 3.5. The induced adjacency matrix of an open graph (G,I,O) is the subma- trix A OCof the adjacency matrix A = m ,(u,v) V(G) of G removing the rows G|IC G { u,v ∈ } of O and column of I, i.e. A OC = m ,(u,v) OC IC . G|IC { u,v ∈ × } The induced matrix A OC is the matrix representation of the linear map W G|IC 7→ Odd(W) IC which domain is 2OC and codomain is 2IC. ∩ Theorem 3.3. (G,I,O) has a gflow iff there exists a DAG1 F = (V(G),E) s.t. A OC.A IC = I G|IC F|OC Proof. (only if) Assume (G,I,O) has a gflow. Thanks to lemma 3.2 w.l.o.g. (G,I,O) has a focused gflow g . Let F = (V(G),E) be a directed graph s.t. (u,v) E(F) f ∈ ⇐⇒ v g (u). Notice that u OC, A IC 1 = 1 where 1 is a binary vector ∈ f ∀ ∈ F|OC {u} gf(u) X s.t. (1 ) = 1 u X. Moreover, since g is extensive, F is a DAG. Thus X u f ⇐⇒ ∈ A OCA IC 1 = A OC1 = 1 = 1 . (if) Assume F = (V(G),E) G|IC F|OC {u} G|IC g(u) Odd(gf(u))∩OC {u} be a DAG s.t. A OC.A IC = I, then let g : OC 2IC = u N+(u). Since F is a G|IC F|OC → 7→ F DAG, g is extensive, and 1 = A IC (1 ) = A OCA IC 1 = 1 , so Odd(g(u)) OC = u . Odd(g(u))∩OC F|OC g(u) G|IC F|OC {u} {u} ∩ { } Thus, according to theorem 3.3, an open graph has a gflow if and only if it has a DAG as right inverse. Notice that this DAG is nothing but the graphical description of the focused gflow function: the set of successors of a vertex u is the image of u by the focused gflow function. 1DAG:Directed Acyclic Graph 5 As a corollary of Theorem 3.3, (G,I,O) has no gflow if I > O . Indeed, for | | | | dimension reasons, if I > O the matrix A OC has no right inverse. When I = O | | | | G|IC | | | | the focused gflow is reversible in the following sense: Theorem 3.4. When I = O , (G,I,O) has a gflow iff (G,O,I) has a gflow. | | | | Proof. Notice that the induced adjacency matrix of (G,O,I) is the transpose tA OC G|IC of the one of (G,I,O). Moreover, since A OC is squared, A IC is both right and left G|IC F|OC inverse of A OC. Thus, A IC .tA IC = t(A IC .A OC) = I. As a consequence G|IC G|OC F|OC F|OC G|IC A OC has a right inverse which is a DAG since the transpose of a DAG is a DAG. G|IC 4 Relaxing Uniform Determinism Focused gflow guarantees uniformly stepwise strong determinism. We consider here two more general classes of MBQC evolutions: the equi-probabilistic case where all the branches occur with the same probability, independent of the input state; and the constant probability case where all the branches occur with a probability independentof the input state. We show that both equi-probabilitic and constant probabilistic evolutions are information preserving and admit a simple graphical characterisation by means of violating sets. Definition 4.1. An MBQC (G,I,O,α,x,z) is: — equi-probabilistic if for any input state φ C2I and any branch s 0,1 OC, | i ∈ ∈ { } p = χ φ = 1 . s || s| i|| 2|OC| — constant-probabilistic if for any branch s 0,1 OC the probability p = χ φ s s ∈ { } || | i|| that the branch s occurs does not depend on the input state φ . | i Constant probabilistic (and hence equi-probabilistic) evolutions are information preserving in the sense that if one knows the branch s of the computation (i.e. the classical outcome) then he can recover the initial input state of the computation. Indeed, if an MBQC is constant probabilistic then the map φ χ φ is constant, thus s | i 7→ || | i|| χ†sχs = ps.I. If ps = 0 then the branch never occurs, otherwise the branch s is implementing an isometry. Remark: Notice that the knowledge of the branch s, which is necessary the case in the MBQCmodelbecauseofthecorrectivestrategy,isessentialtomakeanequi-probabilistic evolution information preserving. Indeed, consider the quantum one-time pad example with s 0,1 2, χ = σ /2 where σ is a Pauli operator (σ = I, σ = X,σ = s s s 00 01 10 ∀ ∈ { } Y,σ = Z). This evolution is equi-probabilistic but if the information of the branch is 11 nwohticthakisenclienatrolyacncootuinntf,ortmheatcioornrepspreosnedrvininggs.uper operator is ρ 7→ Ps∈{0,1}2σsρσs† = I/2 We prove that uniform equi- and constant probabilities have simple graph characterisations by violating sets, where uniformity is defined similarly to the determinism 6 case: (G,I,O) guarantees uniformly constant (resp. equi-) probabilisty if x,z s.t. α, ∃ ∀ (G,I,O,α,x,z) has a constant (resp. equi-) probabilistic evolution. Theorem 4.1. An open graph (G,I,O) guarantees uniform equiprobability iff W OC,Odd(W) W I = W = ∀ ⊆ ⊆ ∪ ⇒ ∅ A nonempty set W OC such that Odd(W) W I is called an internal set. ⊆ ⊆ ∪ Theorem 4.1 says that an open graph (G,I,O) guarantees uniform equi-probability if and only if it has no internal set. Proof. (if) First we assume that there is no internal set and we show that every branch occurs with the same probability 1/2OC , independently of the input state and the | | set of measurement angles. For a given open graph (G,I,O), a given input state φ and a given set of measurement angles α , we consider w.l.o.g. the 0- v v OC |brianch, i.e. the branch where all outcomes ar{e 0}2.∈The probability of this branch is p = + φ 2 = 1 eiαx x φ 2 where α = α .x ||Qv∈Och αv | Gi|| 2|OC|||Px∈{0,1}OC h | Gi|| x Pv∈OC v v and φ = E + φ . As a consequence, | Gi G| iIC | iI p = 1 ei(αy αx) φ x y φ 2|OC| Px,y 0,1 OC − h G| ih | Gi = 1 ∈{ } eiαu φ x y φ 2|OC| Pu 1,0,1 OC Px,y 0,1 OC s.t. x y=uh G| ih | Gi = 1 ∈{− } eiαu ∈{ } φ x− 1+u x 1 u φ 2|OC| Pu 1,0,1 OC Px 0,1 VuCh G| iVuC (cid:12) 2 (cid:11)Vuh |VuC (cid:10) −2 (cid:12)Vu| Gi = 1 ∈{− } eiαu φ ∈{1+}u (cid:12) x x 1 u (cid:12) φ 2|OC| Pu 1,0,1 OC h G|(cid:12) 2 (cid:11)Vu(cid:16)Px 0,1 VuC | ih |(cid:17)(cid:10) −2 (cid:12)Vu| Gi = 1 ∈{− } eiαu φ (cid:12)1+u 1 u∈{ φ} (cid:12) 2|OC| Pu 1,0,1 OC h G|(cid:12) 2 (cid:11)Vu(cid:10) −2 (cid:12)Vu| Gi = 1 ∈{− } eiαup (cid:12) (cid:12) 2|OC| Pu 1,0,1 OC u whereV = i O∈{C− u }= 0 , 1 u = 1 ui ,andp = φ 1+u 1 u φ . Notice tuhat{fo∈r any|v i 6 IC}, (cid:12)(cid:12)φ±2 (cid:11)V=u 1Ni∈Vu(cid:12)(cid:12) ±2Za(cid:11)i φ u h Ga|(cid:12)(cid:12).2T(cid:11)hVuus(cid:10)fo−2r (cid:12)(cid:12)aVnuy| Gi ∈ | Gi √2Pa∈{0,1} NG(v)(cid:12) G\v(cid:11) ⊗ | iv u 1,0,1 OC s.t. V = , there exists v IC VC O(cid:12) dd(V ) (which is not empty u u u ∈ {− } 6 ∅ ∈ ∩ ∩ by hypothesis) such that: p = φ 1+u 1+u X φ u h G|(cid:12) 2 (cid:11)Vu(cid:10) 2 (cid:12)Vu Vu| Gi = 1 (cid:12) φ (cid:12) a Za 1+u 1+u X Zb φ b = 21Pa,b∈{0,1}φ(cid:10) G\vZ(cid:12)(cid:12)ha |v N1G+(vu)(cid:12)(cid:12) 2 1(cid:11)+Vuu(cid:10) 2X(cid:12)(cid:12)VuZaVu NφG(v)(cid:12)(cid:12) G\v(cid:11)| iv = 12Pa∈{0,1}((cid:10) 1G)\av(cid:12)(cid:12)φNG(vZ)(cid:12)(cid:12)a 2 (cid:11)Vu1+(cid:10)u 2 (cid:12)(cid:12)Vu1+uVu NZGa(v)(cid:12)(cid:12)XG\v(cid:11)φ = 12Pa∈{0,1}(−1)a(cid:10)φG\v(cid:12)(cid:12) 1N+Gu(v)(cid:12)(cid:12) 21+(cid:11)uVu(cid:10)X2 (cid:12)(cid:12)Vφu NG(=v)0Vu(cid:12)(cid:12) G\v(cid:11) where the2fPacato∈r{0(,1}1−)a co(cid:10)meGs\vfr(cid:12)(cid:12)o(cid:12)(cid:12)m2th(cid:11)eVufa(cid:10)ct2th(cid:12)(cid:12)Vaut XVu(cid:12)(cid:12)anGd\vZ(cid:11)a are commuting when − Vu NG(v) a = 0 and anticommuting when a = 1 since v Odd(V ). As a consequence, it remains u ∈ in p only the case where V = , so p = 1 φ φ = 1 . u ∅ 2|OC|h G| Gi 2|OC| (only if) Now we prove that the existence of a violating set implies that there exists a particular input state and a particular set of measurement angles such that some branches occur with probability 0. Let W OC s.t. Odd(W ) WC IC = and 0 ⊆ 0 ∩ 0 ∩ ∅ 2The other branches are taken into account by considering a different set of measurement angles e.g. the branch where all outcomes are 1 corresponds to the 0-branch when the set of measurements is {αv+π}v∈OC. 7 P = P be a Pauli operator defined as follows: v V v N ∈ X if v W and v / Odd(W ) 0 0  ∈ ∈ v V,Pv = Y if v W0 Odd(W0) ∀ ∈ ∈ ∩ I otherwise   Let φ = + 0 be an input state. Notice that | 0i | iW0∩I ⊗| iW0C∩I PEG|+iIC |φ0i === (((−−−111)))|||EEE(((WWW000)))|||EEEGGGXX|+WWi00IZC|+O|φidI0dC(iW∪,W0)∩0WZ0OCd|d+(WiI0C)∩|Wφ00Ci|0iW0C∩I where E(W) = E (W W) is the set of the internal edges of W. Thus E + φ ∩ × G| iIC | i0 is an the eigenvector of P associated with the eigenvalue ( 1)E(W0), implying that if | | − each qubit v W is individually measured according to the observable P producing 0 v ∈ the classical outcome s 0,1 , then s = E(W ) [2]. As a consequence, for the input φ and any svet∈o{f me}asuremePntvs∈Wα0 v | s.t. 0α| = 0 if v W Odd(W )C 0 v v OC v 0 0 and α =|π/2i if v W Odd(W ), all the{bra}nc∈hes s s.t. s ∈= 1+∩E(W ) [2] occurvwith probabi∈lity00.∩ 0 Pv∈W0 v | 0 | Theorem 4.2. An open graph (G,I,O) guarantees uniform constant probability iff W OC,Odd(W) W I = L(W) I = ∀ ⊆ ⊆ ∪ ⇒ ∩ ∅ where L(W):= Odd(W) W denotes a local set. ∪ A nonempty set W OC such that Odd(W) W I and L(W) I = is called a ⊆ ⊆ ∪ ∩ 6 ∅ strongly internal set. Theorem 4.2 says that an open graph (G,I,O) guarantees uniform constant probability if and only if it has no strongly internal set, or equivalently if and only if all internal sets are ‘far enough’ from the inputs. Proof. (if) Firstwe assumethat thereis nostrongly internal set andwe show that every branch occurs with a probability independent of the input. Using the notations of the proof of theorem 4.1, it only remains to prove thatp is independentof theinputfor any u u= 0suchthatIC VC Odd(V ) = andL(V ) I = . NotethatOdd(V ) V IC u u u u u 6 ∩ ∩ ∅ ∩ ∅ ⊆ ⊆ so p = φ 1+u 1+u X φ u h G|(cid:12) 2 (cid:11)Vu(cid:10) 2 (cid:12)Vu Vu| Gi = ( 1)(cid:12)E(Vu) φ 1+(cid:12)u 1+u E Z X + φ − | |h G|(cid:12) 2 (cid:11)Vu(cid:10) 2 (cid:12)Vu G Odd(Vu) Vu| iIC | iI = ( 1)E(Vu) φ (cid:12)1+u 1+u(cid:12) E Z + φ − | |h G|(cid:12) 2 (cid:11)Vu(cid:10) 2 (cid:12)Vu G Odd(Vu)| iIC | iI = ( 1)E(Vu)+Vu O(cid:12)dd(Vu) φ (cid:12)1+u 1+u φ − | | | ∩ |h G|(cid:12) 2 (cid:11)Vu(cid:10) 2 (cid:12)Vu| Gi Moreover, for any v V , since v IC(cid:12), φ 1+u (cid:12) 1+u φ ∈ u ∈ h G|(cid:12) 2 (cid:11)Vu(cid:10) 2 (cid:12)Vu| Gi = 1 φ a Za 1+u (cid:12) 1+u Zb (cid:12) b φ 2 Pa,b∈{0,11}+(cid:10)uvG\v(cid:12)(cid:12)h |v NG(v)(cid:12)(cid:12) 2 (cid:11)Vu1(cid:10)+uv2 (cid:12)(cid:12)Vu NG(v)| iv(cid:12)(cid:12) G\v(cid:11) = 1 φ Z 2 1+u 1+u Z 2 φ = 21 (cid:10)φG\v(cid:12)(cid:12) 1N+Gu(v)(cid:12)(cid:12) 2 1(cid:11)+Vuu\v(cid:10) 2φ(cid:12)(cid:12)Vu\v NG(v)(cid:12)(cid:12) G\v(cid:11) So, by i2n(cid:10)duGct\ivo(cid:12)(cid:12)n(cid:12)(cid:12), 2φ(cid:11)Vu\1v+(cid:10)u 2 (cid:12)(cid:12)V1u+\vu(cid:12)(cid:12) G\φv(cid:11) = 1 φ φ = 1 . This shows h G|(cid:12) 2 (cid:11)Vu(cid:10) 2 (cid:12)Vu| Gi 2|Vu| (cid:10) G\Vu(cid:12)(cid:12) G\Vu(cid:11) 2|Vu| that pu does not depen(cid:12)d on the inpu(cid:12)t state. (cid:12)(cid:12) 8 (only if) Now we prove that the existence of a strongly internal set implies that there exists a particular set of measurement angles such that some branches occur with probability zero for some input state and with nonzero probability for other inputs. Let W OC s.t. Odd(W ) WC IC = , u L(W ) I, and P = P be a Pa0ul⊆i operator defined l0ike∩in t0he∩proof of∅theo0re∈m 4.1.0W∩e consider the fNollvo∈wVingvinput states: φ = + 0 a for a 0,1 . Notice that PE + φ = | ai | iW0∩I⊗| iW0C∩I\u0⊗| iu ∈ { } G| iIC | aiI (−1)a+|E(W0)|EG|+iIC |φaiI. Let αv = π/2 if v ∈ W0∩Odd(W0) and αv = 0 otherwise. We consider a branch s of measurement which occurs with a nonzero probability if the icnopnusetqsuteantcee,isif|φth0ei.inNpuotticsteattehaist tφhis,bthraisnbchrasnacthissfieosccPurvs∈wWi0tshvp=rob(a−b1il)i|tEy(W0.0)|. As a 1 | i 5 Uniform Equiprobability versus Gflow Existence Since the existence of a gflow implies strongly uniform determinism it also implies uniform equiprobability. In general uniform equiprobability does not imply gflow: Lemma 5.1. When I = O , there exists an open graph that satisfies uniform equiprob- | |6 | | ability but that has no gflow. Proof. Consider the graph depicted in Figure 1. It is easy to see that it has no gflow, as no subset of the outputs has a single vertex as its odd neighorhood. On the other hand, all the subsets of OC have a nonempty external odd neighborhood in IC. v v 2 5 v v 1 3 v v 6 4 Figure 1: Open graph (G,I,O) with I = v and O = v ,v satisfying the uniform 1 5 6 { } { } equiprobability condition but having not gflow. However, in the particular case where I = O , the existence of a gflow implies | | | | uniform equiprobability. Theorem 5.1. When I = O , (G,I,O) guarantees uniform equiprobability iff it has a | | | | gflow. Proof. We only have to prove that uniformequiprobability implies the existence of gflow (the other direction is obvious). We prove the existence of a gflow for (G,O,I) which, according to theorem 3.4, implies the existence of a gflow for (G,I,O). Since (G,I,O) is uniformly equiprobable, the matrix A IC is injective, so reversible. Indeed, for G|OC any W OC, A IC .1 = 1 = 0 = Odd(W) I W I ⊆ G|OC W ∅ ⇐⇒1 Odd(W)∩IC ⇒ ⊆ ⊆ ∪ so W = . The matrix A IC − is the induced matrix of a directed open graph ∅ (cid:16) G|OC(cid:17) 9 (H,O,I), where H is chosen s.t. vertices in O have no successor. In the following we show that H is a DAG. By contradiction, let S V(H) be the shortest cycle in H. ⊆ Notice that S OC since vertices in O have no successor. A IC .(A IC ) 1.1 = ⊆ G|OC G|OC − S 1 A IC .1 = 1 Odd (Odd (S) OC) IC = S. Let WS :=⇐O⇒dd (SG)|OCOOCd.dHS(iSn)c∩eOSC is theSsh⇐or⇒test cycleG, S HOdd ∩(S). M∩oreover S OC H H ∩ ⊆ ⊆ so S W. Thus Odd (W) W IC which implies W = , so S = . Thus H is a G ⊆ ⊆ ∪ ∅ ∅ DAG. Notice that thanks to Theorem 5.1 the stepwise condition in the characterisation of gflow can be removed, improving Theorem 3.1: Corollary 5.1. When I = O , if (G,I,O) guarantees uniform strong determinism iff | | | | it has a gflow. Proof. Uniform strong determinism implies equiprobability which ensures the existence of gflow when I = O . | | | | 6 Choosing Inputs and Outputs The fact that the characterisation of uniform probability is by internal subsets allows us to have abetter view of thefollowing general problem: given a graph, which vertices can bechosenasoutputsandinputsformeasurementbasedquantuminformationprocessing. Definition 6.1. Given a graph G, for any A V(G), let be the collection of internal A ⊆ E sets outside A: := S V,S = Odd(S) SC AC = A E { ⊆ 6 ∅∧ ∩ ∩ ∅} A transversal of a collection C of sets is a set that intersects all the elements of C. The set of all transversals of is T( ) := S V, S S S = . A A ′ ∀ A ′ E E { ⊆ ∈ E ∩ 6 ∅} Lemma 6.1. If an open graph (G,I,O) guarantees uniform equiprobability then O ∈ T( ). E∅ Proof. By contradiction if W and W O = , then Odd(W) WC = , so Odd(W) W IC which implie∈s WE∅ = . It c∩ontradic∅ts the fact that W∩ . ∅ ⊆ ∪ ∅ ∈E∅ Theorem 6.1. An open graph (G,I,O) guarantees uniform equiprobability if and only if O T( ). I ∈ E Proof. O T( ) W ,W O = W OC,W / W I I I ∈ E ⇐⇒ ∀ ∈ E ∩ 6 ∅ ⇐⇒ ∀ ⊆ ∈ E ⇐⇒ ∀ ⊆ OC, (Odd(W) WC IC W = ) W OC,(Odd(W) W I W = ). ¬ ∩ ∩ ∧ 6 ∅ ⇐⇒ ∀ ⊆ ⊆ ∪ ⇒ ∅ Theorem 6.2. Given a graph G and two subsets of vertices I and O with I = O , the | | | | open graph (G,I,O) guarantees equiprobability iff I T( ) and O T( ). I ∈ E∅ ∈ E Proof. When I = O , if (G,I,O) guarantees equiprobability then (G,I,O) has a gflow | | | | (Theorem 5.1) and thus (G,O,I) has a gflow (Theorem 3.4) as well. As a consequence (G,I,O) guarantees uniform equiprobability so I T( ). ∈ E∅ 10