aa r X i v : . [ m a t h - ph ] J un AREA LAW FOR RANDOM GRAPH STATES
BENOˆIT COLLINS, ION NECHITA, AND KAROL ˙ZYCZKOWSKI
Abstract.
Random pure states of multi-partite quantum systems, associated witharbitrary graphs, are investigated. Each vertex of the graph represents a genericinteraction between subsystems, described by a random unitary matrix distributedaccording to the Haar measure, while each edge of the graph represents a bi-partite,maximally entangled state. For any splitting of the graph into two parts we considerthe corresponding partition of the quantum system and compute the average entropyof entanglement. First, in the special case where the partition does not “cross” anyvertex of the graph, we show that the area law is satisfied exactly. In the generalcase, we show that the entropy of entanglement obeys an area law on average, thistime with a correction term that depends on the topologies of the graph and of thepartition. The results obtained are applied to the problem of distribution of quantumentanglement in a quantum network with prescribed topology.
Contents
1. Introduction 12. Random graph states 33. Exact area law for adapted partitions 64. One-vertex marginals 85. Defining the boundary surface for a general partition 96. A general area law for graph states 127. Rank of random graph states and a transport problem 138. Some results for different subsystem dimensions 159. Perspectives and open questions 19References 201.
Introduction
Entanglement in many-body quantum systems is a subject of a considerable recentinterest [1–3]. In several physical problems it is important to describe correlationsbetween two selected parts of a composed quantum system. For any pure state de-scribing the entire system, such correlations can be characterized quantitatively by the entanglement entropy H , equal to the von Neumann entropy of the reduced mixedstate obtained by the partial trace over a selected subsystem. Note that this quantitydepends explicitly on the partition of the entire system into subsystems. Entanglement entropy H does not usually scale proportionally to the size of theselected region S . As described in [3, 4], for different systems the entanglement entropyis approximately proportional to the boundary of this set, denoted by ∂S . Consideringa three dimensional body S , the size of its boundary is proportional to the area of S and not to its volume. Thus, for any set S of subsystems, the size of its boundary | ∂S | will be called the “area” of S . If for a sufficiently large system the leading contributionto the entanglement entropy of a given state | Ψ i with respect to the partition { S, T } isproportional to the area separating both subsystems, we will say that the area law issatisfied. In the case of one dimensional systems, the area law implies that the entropysaturates asymptotically to a constant, as the area of the boundary consists of twoisolated points.Area laws are studied in context of black hole physics and holographic principle [5,6],and for ground states of quantum lattice systems with local interactions [7–10]. Undersome technical assumptions is it possible not only to show that the area law holdsfor the ground state of a given two–dimensional model system, but also to derive thenegative correction term called topological entanglement entropy [11,12], and show thatit is universal and depends only on the topology of the interaction. Investigations ofthe area law were performed also for subsets with fractal boundaries [13].For any quantum state of a system of interacting particles, described by a lattice or agraph, it is interesting to analyze the degree of entanglement between two given nodesof the graph [14]. This issue is relevant in studies on generating entanglement betweengiven nodes of a graph [15], entanglement swapping and quantum repeater systems [1]and quantum communications in noisy networks [16].The aim of this work is to study the area law for the ensembles of random statesassociated to a graph. The authors have previously introduced [17] an ensemble of purequantum states for which the structure of interactions and entanglement are encodedin the graph. In the present work, we study the average entropy of entanglement forelements of this ensemble, and we show that it obeys asymptotically and on averagearea laws of the form(1) E (Ψ) = H ( ρ S ) = | ∂S | log N − h Γ ,S + o (1) , where Ψ is a graph state, ρ S is the reduced density operator, | ∂S | is the “area” ofthe boundary of the partition { S, T } , N is the dimension of the subsystems and h Γ ,S is a numerical constant depending on the graph Γ and on the partition { S, T } . Thefunctionals E and H denote, respectively, the entropy of entanglement for pure statesand the von Neumann entropy of mixed density matrices.In previous work, the entanglement entropy for a concrete quantum state of a givencomposite quantum system (see e.g. [3]) has been investigated. Here however, we discussthe statistical properties of ensembles of quantum states. We will show under whichassumptions the area law holds exactly in our model, while in the opposite case thecorrections to the area law are derived.The paper is organized as follows. The model of graph random states from [17] isrecalled in Section 2, and a detailed example is worked out. In Section 3, the notion of REA LAW FOR RANDOM GRAPH STATES 3 adapted marginals of graph states is introduced and it is shown that for such partitionsof the graph the area law holds exactly. A more general case of the problem, for whichthe area law for random graph states holds only asymptotically is treated in Section 4.In Section 5, a definition of the boundary is proposed in the most general setting. Thegeneral area law is stated in Section 6, while an application to a transport problem ispresented in Section 7. Section 8 provides an outline of the adjustments to be madeto the main results in the case where the dimensions of the relevant Hilbert spaces aredifferent. 2.
Random graph states
We recall here the model of random graph states introduced in [17]. The most generaldefinition of this model will not be recalled here, but for completeness we sketch belowthe main ingredients of the construction. For any undirected graph Γ consisting of m edges B , . . . , B m and k vertices V , . . . V k , we associate an ensemble of random purestates | Ψ i . They describe a quantum system consisting of n = 2 m particles describedin the composed Hilbert space H ⊗ mN . The dimension N of any subspace is arbitraryand in particular we will analyze the asymptotic limit N → ∞ .Each edge of the graph, which connects subsystems labeled by i and j , representsthe maximally entangled state, | Φ i + ij = √ N P Nx =1 | x i i ⊗ | x i j , between two Hilbert spaces H i and H j . Each vertex V of the graph, of degree b , represents a generic interactionbetween b subsystems, described by a random unitary matrix of size N b , distributedaccording to the Haar measure . In this way, independent random unitary matricesdescribe unknown interactions in each node, which are assumed to be generic. Theabove discussion can be summarized in the following formula, describing an elementfrom the ensemble:(2) | Ψ i = " O V vertex U V { i,j } edge | Φ i + i,j , where | Φ i + i,j are maximally entangled states and U V are independent Haar unitaryoperators at each vertex.Our assumptions differ therefore from the model analyzed in [18], in which edges ofthe graph denote maximally entangled states of two qubits, while the vertices repre-sent deterministic local unitary gates or local measurements. A more general graphmodel of quantum networks was investigated in [1, 14] and later studied in context ofentanglement percolation [19]. In this version of the model any edge represents a givenbipartite state of two qubits, while a vertex denotes a deterministic unitary swap gateor a local measurement. Note that in the system investigated here each dot at the endof an edge of a graph represents a quantum subsystem described by a N dimensionalHilbert space. The parameter N is arbitrary, but our results are obtained under theassumption that the dimension is large. BENOˆIT COLLINS, ION NECHITA, AND KAROL ˙ZYCZKOWSKI
Let us divide the set of n = 2 m particles into two disjoint subsets, labeled by S and T . The subset T will correspond to subsystems over which the averaging is performed:a marginal of the graph state | Ψ ih Ψ | defined by the partial trace over these subsystems,(3) ρ S = Tr T | Ψ ih Ψ | , forms a mixed state ρ S supported on the remaining subspaces S = { , . . . , n } \ T .Note that any graph Γ and its partition P trace = { S, T } determines an ensemble ofrandom mixed states ρ S . In [17] we studied statistical properties of such an ensem-ble, which depends on the topology of the graph and on its partition. Ensembleof random mixed states can be characterized by the average von Neumann entropy E H ( ρ S ) = − E Tr ρ S log ρ S . By definition this quantity is equal to the average entropyof entanglement of the random pure state | Ψ i with respect to the prescribed partition P trace = { S, T } .As a first example for this graphical notation, consider the graph shown in Fig. 1(a),which consists of m = 10 edges (loops and multi-edges are allowed) and k = 5 vertices.Fig. 1(b) shows all n = 2 m = 20 subsystems denoted by small black dots and therandom interaction between some of them represented by the gray circles at the verticesof the graph. The partition { S, T } is indicated graphically by placing diagonal crosseson the small black dots corresponding to elements in T (“traced subsystems”), whileelements in S have no crosses on top of them (“surviving subsystems”). V V V V V (a) V V V V V (b) Figure 1.
An example of a graph state (a) and one of its marginals (b).Let us now consider another example, the graph consisting of a single edge, m = 1presented in Fig. 2. The only edge forms a loop here, so there is a single vertex V only, k = 1. The partition splits the n = 2 partite system into two subsystems S and T . Inthis case we will relax for a moment the assumption that the sizes of both subsystemsare equal and will denote them by N and N ′ , respectively. In such a case the onlyparameter of the model is the ratio between the sizes of both subsystems c = N/N ′ .As both subsystems are coupled at the vertex V by a random unitary matrix U ∈ U ( N N ′ ) the assumption on the existence of the maximally entangled state becomes REA LAW FOR RANDOM GRAPH STATES 5 V (a) V (b) Figure 2.
A single loop graph in the standard notation (a) and simpli-fied notation (b). This degenerate graph represents bipartite system ( S - the black dot and T - crossed dot) and leads to random mixed stateswith level density described by Marchenko–Pastur distributions.irrelevant here. The partial trace over the subsystem T leads to a random mixed state ρ S of size N [20]. It can be represented by a normalized Wishart matrix ρ S = GG † / Tr GG † ,where G is a rectangular non-hermitian random Ginibre matrix of size N × N ′ (recallthat a Ginibre matrix has i.i.d. complex Gaussian entries). The spectral density of themixed state ρ S of size N is known to be described asymptotically by the Marchenko–Pastur distribution π c ( x ), in the following sense (see [21] for details):(4) almost surely, lim N →∞ µ N = π c , where the limit corresponds to the weak convergence of probability measures, µ N is theempirical distribution of the rescaled eigenvalues of ρ S (5) µ N = 1 N N X i =1 δ cNλ i ( ρ S ) , and the parameter c is the asymptotic ratio of dimensions c = lim N →∞ N ′ /N . TheMarchenko-Pastur distribution is given by(6) π c = max(1 − c, δ + p c − ( x − − c ) πx [1+ c − √ c, c +2 √ c ] ( x ) dx. The average von Neumann entropy of a random mixed state ρ S is given by the integral,(7) E H ( ρ S ) = ln N − Z x log x dπ c ( x ) + o (1) = ln N − h c + o (1) , where h c is the entropic correction(8) h c = ( + c log c if c > c if 0 < c < . Depending on the desired amount of generality, we are sometimes going to work onthe model in which all subsystems are described by Hilbert spaces of the same size N .We will also consider the general version of the model [17], in which only the pairs ofsubsystems connected by an edge, which describes a maximally entangled state, have thesame dimensions. Hilbert spaces of different dimensions, as in the Marchenko–Pasturcase treated above, will be allowed. However, we shall ask that these dimensions grow BENOˆIT COLLINS, ION NECHITA, AND KAROL ˙ZYCZKOWSKI at fixed ratios, imposing the asymptotic regime dim H i = d i N , for some fixed positiveconstants d i . 3. Exact area law for adapted partitions
In this section we show that the area law holds exactly for graph states, provided thatthe marginal under consideration satisfies a particular condition, called adaptability .Recall that to any graph state we associate two partitions of the set of n = 2 m subspaces: a vertex partition P vertex which encodes the vertices of the graph, and a pairpartition P edge which encodes the edges (corresponding to maximally entangled states).More precisely, two subsystems H i and H j belong to the same block of P vertex if they areattached to the same vertex of the initial graph. Each edge ( i, j ) of the graph contributesa block of size two { i, j } to the edge partition P edge . Recall that a marginal (3) of arandom graph state | Ψ ih Ψ | is specified by a 2-set partition P trace = { S, T } . Definition 3.1.
A marginal ρ S is called adapted if (9) P trace > P vertex for the usual refinement order on partitions. In other words, a marginal is adapted ifand only if the number of traced out systems in each vertex is either zero or maximal.If this is the case, then the partition boundary, which splits the graph into parts { S, T } ,does not cross any vertices of the graph. Because of the above property, for adapted marginals, we can speak about traced outvertices , because if one subsystem of a vertex is traced out, then all the other systemsof that vertex are also traced out. For the graph state in Figure 1(a) we consider theadapted marginal obtained by partial tracing vertices V and V , see Figure 3.We now define precisely what we mean by area laws [3]. The partition { S, T } definesa boundary between the set of vertices that are traced out and vertices that survive. Inthe subsequent sections, the following definition will be generalized to take into accountnon-adapted marginals. Definition 3.2.
The boundary of the adapted partition { S, T } is defined as the set ofall (unoriented) edges e = { i S , j T } in the graph state with the property that i S ∈ S and j T ∈ T . Equivalently, it is the set of edges of the type . The boundary of apartition shall be denoted by ∂S .The area of this boundary is its cardinality | ∂S | , i.e. the number of edges between S and T . In the example of Figure 3, the boundary is represented by a dashed (green) line.The area of the boundary in this case is 5: in Figure 3, the boundary line intersects 5edges.The main result of this section is that the area law holds exactly for adapted marginalsof graph states, where we allow arbitrary dimensions of subsystem. Note that, for agiven (boundary) edge { i, j } , we have d i = d j , the common dimension of the maximallyentangled state corresponding to the edge { i, j } . REA LAW FOR RANDOM GRAPH STATES 7
Theorem 3.3.
Let ρ S be an adapted marginal of a graph state | Ψ ih Ψ | . Then, theentropy of ρ S has the following exact, deterministic value: (10) H ( ρ S ) = log Y { i,j }∈ ∂S d i N = | ∂S | log N + log Y { i,j }∈ ∂S d i for each value of the size parameter N . In the particular case where all the Hilbertspaces have dimension N (i.e. d i = 1 for all i ), the area law takes the form (11) H ( ρ S ) = | ∂S | log N. Proof.
At fixed N , the random density matrix ρ S has the following simple expression(12) ρ S = O C ∈ Π Svertex U C O { i,j }∈ Π inedge | Φ + i,j ih Φ + i,j | ⊗ O { i,j }∈ Π outedge I d i N d i N O C ∈ Π Svertex U C ∗ , where Π Svertex is the set of surviving vertices, Π inedge is the set of edges connecting survivingvertices and Π outedge = ∂S is the set of edges connecting surviving with traced vertices,i.e. crossed edges. Since the entropy is unitarily invariant and additive with respect totensor products, it is immediate that(13) H ( ρ S ) = X { i,j }∈ ∂S log( d i N ) = | ∂S | log N + log Y { i,j }∈ ∂S d i . (cid:3) V V V V V ST Figure 3.
An adapted marginal for the graph state in Figure 1(a). Thedashed (green) line represents the boundary between the traced–out sub-systems T and the surviving subsystems S . BENOˆIT COLLINS, ION NECHITA, AND KAROL ˙ZYCZKOWSKI
For the system corresponding to the graph shown in Figure 3 with all subsystems ofsize N the von Neumann entropy reads(14) H ( ρ S ) = 5 log N. This follows from the fact that ρ S is in this case a unitary conjugation of a maximallymixed state of size N with an arbitrary pure state of size N .4. One-vertex marginals
We shall look now at the simplest situation for which an approximate area law holds.We are considering marginals with a unique surviving vertex which may contain tracedout subsystems. We refer to the results in [17], Section 6.3, where the exact same situa-tion was studied. In this particular setting, let us introduce some appropriate notationfor the subsystems of the surviving vertex V . The subsystems of V are partitioned, onone hand, into surviving S subsystems and traced-out subsystems T ′ (note that T ′ isa subset of the set of all traced systems in the original graph). Moreover, the edgesof the graph introduce a different partition of V , into subsystems attached by loopsof V (denoted by F ) and subsystems connected to other vertices, which form a set G .Hence, three important parameters with respect to the only surviving vertex: | G | , thenumber of edges connecting this vertex to other traced out vertices, | T ′ | the number oftraced out Hilbert spaces in this vertex and | S | , the number of surviving Hilbert spaces.For the example presented in Figure 4, one has G = { , , , } , T ′ = { , , , } and S = { , , , } . H H H H H H H H V Figure 4.
Example of a graph state marginal with only one surviving vertex.Except for some trivial, degenerate situations ( S = V , T = V or G = V ), the entropyof the reduced density matrix is genuinely random, and its asymptotic behavior can beinferred from Theorem 6.4 of [17]. Theorem 4.1.
In the limit N → ∞ , the average von Neumann entropy of the reduceddensity matrix ρ S has the following behavior: (15) E H ( ρ S ) = o (1) + log( d S N | S | ) if | S | < | T ′ | + | G | ;log( d T ′ d G N | T ′ | + | G | ) − h d T ′ d G /d S if | S | = | T ′ | + | G | ;log( d T ′ d G N | T ′ | + | G | ) if | S | > | T ′ | + | G | . where h c is the entropic correction for a Marchenko-Pastur distribution, defined inequation (8) . In the particular case where d i = 1 for all i , one has (16) E H ( ρ S ) = min {| S | , | T ′ | + | G |} log N − δ | S | , | T ′ | + | G | + o (1) . REA LAW FOR RANDOM GRAPH STATES 9
At this point, it is not obvious how to give an interpretation of Theorem 4.1 and ofequation (16) in particular, as an area law. Recall that because of the random unitarymatrix acting on the vertex V , the exact indices of the traced subsystems inside thevertex V are irrelevant (at least in the case of trivial relative dimensions). Hence, thetopological notion of boundary separating the sets S and T becomes ambiguous. In thefollowing section, a more general definition of the boundary of a partition will be givenvia a combinatorial optimization problem.5. Defining the boundary surface for a general partition
Before stating and proving the area law in the most general setting, we have tointroduce a well-defined notion of boundary surface for a general, possibly non-adapted,partition { S, T } . In this section, we shall restrict our attention to the case where allthe subsystems have the same dimension N . We shall come back to the general case ofarbitrary ratios in Section 8.A naive attempt at a definition is to consider the topological notion of boundarythat was used in Section 3. In Figure 5, two a priori different marginals of the samegraph state are represented. Given the fact that independent random unitary matricesact on vertices V and V , one can swap the two Hilbert spaces (say of V ) and leavethe distribution of the reduced state invariant. Hence, the two states should have thesame statistical properties, although they have different boundary areas: note that theboundary line does not cross any edge on the left hand side picture (null boundary),whereas the right hand side picture has a boundary area of 2 (two intersections). Hence,it is obvious that such a definition of the boundary area is not suitable. The rest of thissection is devoted to introducing a new definition for the boundary area of a partitionmotivated by a strong connection to the maximal flow problem. Note that we restrictourselves to the case of identical systems, d i = 1, so that dim H i = N . V V (a) V V (b) Figure 5.
Different assignment of “crosses” yield different boundaries.The partition on the left has zero boundary (no cuts) and the partitionon the right has a boundary of 2: two edges of the graph are cut by thedashed (green) lines.Consider, as before, an unoriented graph Γ = (
V, E ) having possibly multiple edgesand loops. Consider also a counting function s : V → N , such that, for all vertex v ∈ V ,0 s ( v ) deg( v ) and put t ( v ) = deg( v ) − s ( v ). These functions will represent thenumber of surviving (resp. traced) subsystems inside a given vertex. To the pair (Γ , s )we shall associate two combinatorial structures: a network N Γ ,s and a set of markedfattened graphs F Γ ,s . In Theorem 5.2 we show that the maximal flow in the network N Γ ,s is equal to maximal number of crossings in F Γ ,s , providing the connection betweenstatistical properties of the reduced state ρ S and a combinatorial object related to thepartition { S, T } . This will be the main ingredient in the proof of the general formulationof the area law, Theorem 6.1.Let us first describe the network N Γ ,s . Start with the natural network associated toΓ, with the same vertex set as Γ and with capacities C given by the formula(17) C ( v, w ) = ( number of edges between v and w in Γ if v = w ;0 if v = w. To this network, add two distinguished vertices, which were called in [17] id and γ . Theremaining capacities are defined by C (id , v ) = t ( v ) = deg( v ) − s ( v ) and C ( v, γ ) = s ( v ).The network N Γ ,s defined in this way has been shown in [17, Section 5.2] to be intimatelyconnected to the statistical properties of random graph states.We move now to the second combinatorial object associated to the pair (Γ , s ). First,starting from Γ, construct the fattened graph Γ fat with vertex set(18) V fat = G v ∈ V { v , v , . . . , v deg( v ) } and with edge set E fat corresponding to the edges of Γ in such a way that every twoedges are now disjoint. We keep track of the fattening operation by a projection map f : V fat → V defined by f ( v i ) = v . The fattening operation is depicted in Figure 6.A marking of fattened graph Γ fat compatible with the counting function s is a subset M ⊂ V fat such that, for all v ∈ V ,(19) | M ∩ f − ( v ) | = s ( v ) . We write M s to denote the fact that the marking M is compatible with the countingfunction s . Note that there always exist compatible markings and that the markingis unique if and only if s ( v ) is either 0 or maximal for every vertex v . This extremalsituation corresponds to adapted marginals, which were studied in Section 3. Thenumber of crossings of a marking M , denoted cr( M ), is the number of edges in E fat which connect a marked vertex with an unmarked one(20) cr(Γ fat , M ) = |{ i, j } ∈ E fat | ( i ∈ M, j / ∈ M ) or ( i / ∈ M, j ∈ M ) }| . For the first marking in Fig. 6b, one has cr(Γ fat , M ) = 6, while for the one shown inFig. 6c, the number of crossings reads cr(Γ fat , M ) = 8.The main result of this section is the graph-theoretical Theorem 5.2, which makesuse of the following important definition. Definition 5.1.
For a graph Γ and a partition { S, T } of its associated nodes, definethe area of the boundary (or, simply, area ) of the partition { S, T } as (21) | ∂S | = max M s cr(Γ fat , M ) , where the maximum is taken over all the markings M of the set of vertices of the fattenedgraph Γ fat compatible with the counting function s . REA LAW FOR RANDOM GRAPH STATES 11 V V V V V (a) V V V V V (b) V V V V V (c) Figure 6.
The fattening of the graph in Figure 1 (a) and two markings, M in panel (b) and M in (c), compatible with the count function (num-ber of marked subsystems in each vertex), s ( V ) = s ( V ) = s ( V ) = 1, s ( V ) = 3, and s ( V ) = 2. Marked subsystems are represented by emptydots, and the number of crossings cr is equal to the number of edges withone vertex filled and the other one empty. Theorem 5.2.
For any graph Γ , the maximal flow X Γ ,S in the network N Γ ,S is equalto the area of the boundary of the partition { S, T } (22) X Γ ,S = | ∂S | . Proof.
We shall prove inequalities in both directions. First, consider a compatiblemarking M s . We shall construct a set of augmenting paths in the network havinga total flow of cr(Γ fat , M ). For every crossing edge attached to a single vertex v of Γ,consider the augmenting path id → v → γ . For a crossing edge e = ( v, w ), where theblack dot is in vertex v and the empty dot is in vertex w , consider the augmenting pathid → v → w → γ . In this way, to each crossing edge, we associate a unit of flow from id to γ , proving thus cr(Γ fat , M ) X Γ ,s . Maximizing over all compatible markings M proves the first inequality.Let us move now to proving the other direction. To this end, consider a set ofaugmenting paths in the network N Γ ,s , achieving the maximal flow X Γ ,s . Letid → v → v → · · · → v k → γ be such an augmenting path of length k >
1. If k = 1, choose an edge ( v , v ) in Γ fat and mark one of its vertices as filled and the other one as empty. Otherwise, one canfind the edges ( v , v ), ( v , v ), . . . , ( v k − , v k ) in the fattened graph Γ fat . Color theseedges in the following way: • ( v , v ) : v filled, v empty; • ( v , v ) : v empty, v empty; • · · ·• ( v k − , v k ) : v k − empty, v k empty.In this way, for each augmenting path of unit flow (one can always assume this, at thecost of repeating edges), one assigns a unique crossing in the fattened graph. It followsthat, for this marking M , one has cr(Γ fat , M ) = X Γ ,s . This proves the theorem. (cid:3) A general area law for graph states
This section contains the proof of the main result of the paper, Theorem 6.1.
Theorem 6.1 (Area law for random graph states) . Let ρ S be the marginal { S, T } of agraph state Γ . Then, as N → ∞ , the area law holds, in the following sense (23) E H ( ρ S ) = | ∂S | log N − h Γ ,S + o (1) , where | ∂S | is the area of the boundary of the partition { S, T } defined in 5.1 and h Γ ,S isa positive constant, depending on the topology of the network N Γ ,S (and independent of N ).Proof. The idea is to combine moment computations from [17] for the random matrix ρ S with the combinatorial identity proved in Theorem 5.2. Recall the following momentformula from [17, Theorem 5.5]:(24) ∀ p > , E Tr( ρ pS ) = N − X Γ ,S ( p − ( | B p | + o (1)) , where B p ⊂ N C ( p ) k is the subset of non-crossing partitions (or, equivalently, geodesicpermutations) corresponding to the augmenting paths leading to the maximal flow X Γ ,S in the network N Γ ,S (for details, see [17]). We can restate the above asymptoticexpression as a limit: (we write simply X = X Γ ,S )(25) ∀ p > , lim N →∞ E N X Tr (cid:2) ( N X ρ S ) p (cid:3) = | B p | . In other words, the measures(26) µ N = E N X N X X i =1 δ N X λ i ( ρ S ) REA LAW FOR RANDOM GRAPH STATES 13 have limiting moments given by | B p | . Note that there is a unique probability measurehaving moments | B p | , since one has the bound | B p | Cat kp (4 k ) p which is exponen-tial, and thus, by Carleman’s condition, these moments uniquely define the probabilitymeasure. Since the limit points of the tight sequence of measures µ N are uniquelydetermined by their moments, it follows that [22, Theorem C.9] µ N converges weaklytowards a measure µ satisfying ∀ p > , Z x p dµ ( x ) = | B p | . Finally, one has(27) E H ( ρ S ) = X log N − h Γ ,S + o (1) , with a correction term equal to the entropy of the asymptotic measure µ ,(28) h Γ ,S = Z x log xdµ ( x ) . (cid:3) If the measure µ N converges to the Marchenko-Pastur distribution with parameter c , the above term coincides with the entropy h c defined in (8).7. Rank of random graph states and a transport problem
We start by looking at a linear algebra problem, the maximum rank of a marginal ofa graph state, over the set of unitary operations on vertices.
Theorem 7.1.
The maximum rank of a graph state marginal ρ S is the area of theboundary of the partition { S, T } (29) max U ,...,U k rk ρ S = | ∂S | . Moreover, this maximum can be achieved by choosing the U i to be permutation matricesand the state ρ ∗ S which achieves the maximum can be taken maximally mixed. The above result shows that the maximum achievable rank of a graph state is equal,asymptotically, to the rank of a random marginal. We shall investigate further thisproperty by presenting an alternative formulation to our problem, bridging togetherthe max-flow, max-crossings and the rank aspects. It also has the advantage of being“operational”. The notation below mirrors the one in the rest of the paper and in [17].Note that there is no randomness in the problem below, and N can be arbitrary. Weconsider below the simplest one–qubit case, N = 2.Consider the following problem (see Figure 7 for an example). Problem.
The company
EntanglementFactory has k research facilities around theworld, let us call them V , . . . , V k . As a result of past experiments, entangled statesare shared between pairs of these laboratories, as follows : labs V i and V j share E ij singlet states | Φ + i ∈ C N ⊗ C N . Each facility V i has an unlimited supply of extra N -ditswhich are not entangled with anything else. One day, company EntanglementFactory V (2 , V (1 , V (2 , (a) V (2 , V (1 , V (2 , (b) V (2 , V (1 , V (2 , (c) Figure 7.
An instance of the problem, where S i and T i are noted inparenthesis after each V i . In (b), we create locally singlet states. Thestates sent to A are circled in red in (c), the other ones being sent to B .Note that at each site we consider S i + T i particles.receives an order from the company WantEntanglement for a supply of entangled states.Company
WantEntanglement has two research facilities A and B and offers to ship N -dits from each factory V i to A and/or B , as follows: S i N -dits can be shipped from V i to A and T i N -dits can be shipped from V i to B . Company WantEntanglement pays$10 for each unit of entanglement it will have between stations A and B . What isthe maximal profit the company EntanglementFactory can make, just by using localunitary matrices at each site V i ?We shall answer this question in the following three scenarios, each situation imposingsome physical restrictions or liberties on the system. Scenario 1 : No initial entanglement.
Suppose that all the entangled particlesshared between pairs of V i ’s are lost. The best we can do is to create locally, at each V i , maximally entangled (singlet) states and to ship one half to A and the other half to B . The number of entangled pairs between A and B will then be Y = X i min( S i , T i ) . Scenario 2 : Global operations are allowed.
Suppose that company
Entangle-mentFactory has the ability of performing global operations on all of its facilities V i .Then it is easy to produce a maximally entangled state between A and B : Y = min( X i S i , X i T i ) . REA LAW FOR RANDOM GRAPH STATES 15
Scenario 3 : Entanglement with local operations.
Without any assumptions, weshow that Y = X Γ ,S . First, notice that the problem in this case is just as a restatement of the rank theoremdiscussed earlier, before taking the partial trace of the state. The facilities A and B define the partition with respect to which the partial trace is considered. The entropyof entanglement of the pure state shared between A and B is just the von Neumannentropy of the reduced density matrix ρ S . Note that the following inequality concerningthe number of entangled pairs in different scenarios holds Y Y Y . One can characterize entanglement by using the generalized R´enyi entropy H q ( ρ ) := − q ln Tr ρ q . Note that in the limit the R´enyi parameter q tends to unity this expressionreduces to the von Neumann entropy, lim q → H q ( ρ ) = H ( ρ ). Furthermore, the rank rkof a matrix is given by the generalized entropy of order zero, log rk ρ = H ( ρ ).To summarize, the results on the rank and on the R´enyi entropy H q have the followingtranslation Theorem 7.2.
There exist local unitary operations U i (which can be taken to be per-mutation matrices, i.e. at each site V i we just have to say where each particle goes, to A or B ) such that H ( ρ ) = H ( ρ ) = H p ( ρ ) = X Γ ,S log N, for any q ≥ . In other words, ρ is essentially a maximally mixed state. Moreover, thepermutation matrices involved can be computed efficiently, using a flow algorithm. In the random case (suppose the engineers of company
EntanglementFactory areon vacation, so the staff decides to implement at each site V i random, independentlocal unitary transformations), we have the following result, which is a restatement ofTheorem 6.1. Theorem 7.3.
For independent random Haar unitary matrices U i , we have, almostsurely as N → ∞ , H ( ρ ) = X Γ ,S log N and E H ( ρ ) = X Γ ,S log N − h Γ ,S + o (1) . Comparing the two results above, one concludes that the random choice is nearlyoptimal. 8.
Some results for different subsystem dimensions
In this final section, we analyze simple graphs in the general setting, where we allowsubsystems to have different dimensions. Although we can not state a general arealaw for marginals of such graph states, we perform direct computations in some simplesettings using the full machinery developed in [17]. Our main tool is the followingresult, valid for random graph states with subsystems of dimensions d i N . Theorem 8.1 (see [17, Theorem 5.4]) . The asymptotic moments of a graph state mar-ginal ρ S are given by the formula E Tr( ρ pS ) = (1 + o (1)) N − X ( p − X ( β ,...,β k ) ∈ B k Y i =1 ( d S i ) γ − β i ) k Y i =1 ( d T i ) β i (30) · Y i We are going to discuss here a simple graph described by twoedges joined in one vertex, used to model the trans–horizon entanglement during theprocess of evaporation of a black hole [23, 24]. The entire system is thus composedout of four subsystems, two of which have the same dimension equal to d N , while theother two have the dimensions d N . The ratios d , are treated as parameters of themodel. We shall consider two marginals of this graph state, see Figure 8. In both cases,the network associated to the marginal is the same, and has a maximum flow X = 2.Moreover, the set B of permutations achieving this maximum flow is(31) B = { ( β , β , β ) ∈ S p : id = β β β = γ } . First case : subsystems of size d N are traced. Applying Theorem 8.1 to this setting,we obtain E Tr ρ pS = (1 + o (1)) N − p − X id= β β β = γ d γ − β )2 d γ − β )2 d β )1 d β )1 (32) · d β − β ) − p d β − β ) − p ( d d ) − p = (1 + o (1)) N − p − d − p d X σ ∈ NC ( p ) (cid:18) d d (cid:19) σ . In other words(33) lim N →∞ d N E Tr( d N ρ S ) p = X σ ∈ NC ( p ) (cid:18) d d (cid:19) σ . REA LAW FOR RANDOM GRAPH STATES 17 d N d NV V V (a) d N d NV V V (b) d N d NV V V (c) d N d NV V V (d) id V V V γ (e) Figure 8. The black hole graph, in the simplified notation (a) and thediagram with 4 subsystems (b). In the middle row, two of its marginals:on the left (c), d N -sized subsystems are traced out and on the right (d),subsystem of both dimensions are traced out. In the bottom row, thenetwork associated to both marginals (e).Thus, the rescaled random matrix d N ρ S ∈ M d N ( C ) converges in moments to theMarchenko–Pastur distribution of parameter d /d . Let us now compute the averageentropy of this random matrix. Use(34) lim N →∞ d N E H ( d N ρ S ) = − h d /d to show that E H ( ρ S ) = o (1) + ( log( d N ) − d d if d > d log( d N ) − d d if d < d , (35) = log( d N ) − d D + o (1) , where d = min( d , d ) and D = max( d , d ). Note that the formula above is symmetricin d and d , a consequence of the fact that the non-zero spectra of the two reduceddensity operators of a pure state are identical. Second case : subsystems of both sizes are traced. In this case, the moment formulareads(36) E Tr ρ pS = (1 + o (1)) N − p − ( d d ) − p +1 X id= β β β = γ . Thus, the rescaled random matrix d d N ρ ∈ M d d N ( C ) converges in moments to theMarchenko–Pastur distribution of parameter 1. The entropy computation in this caseis easier:(37) E H ( ρ S ) = log( d d N ) − 12 + o (1) . Note that the two entropy formulas agree in the case d = d . Double line graph - the oxygen molecule O . The graph presented in Figure 9,which can be symbolically represented by O = O , might be interpreted as the oxygenmolecule. We will discuss here the general version of the model in which there aretwo pairs of subsystems of size d N and d N respectively, and look at two differentmarginals. The two marginals correspond to the same network, which has a maximalflow X = 2. The set B of permutations achieving this maximum flow is B = { ( β , β ) ∈ S p : id β β γ and id β β γ } (38) = { ( β , β ) ∈ S p : id β = β γ } , which is in bijection with the set found for the “black-hole” graph. First case : subsystems of size d N are traced. Proceeding as in the case of the “black-hole” graph, we obtain the same result as in the first case above. Thus, as before(39) E H ( ρ S ) = log( d N ) − d D + o (1) , where d = min( d , d ) and D = max( d , d ). Second case : subsystems of both sizes are traced. Again, we obtain the same result asin the corresponding case of the “black-hole” graph(40) E H ( ρ S ) = log( d d N ) − 12 + o (1) . To conclude, note that in general, the relative size of the traced out systems matters(unless d = d ). At this point, we have no interpretation for the surprising fact thatthe two graphs studied here yield the same output entropies. REA LAW FOR RANDOM GRAPH STATES 19 d Nd NV V (a) d Nd NV V (b) d Nd NV V (c) d Nd NV V (d) id V V γ (e) Figure 9. The oxygen graph, in the simplified notation (a) and thediagram with 4 subsystems (b). In the middle row, two of its marginals:on the left (c), d N -sized subsystems are traced out and on the right (d),subsystem of both dimensions are traced out. In the bottom row, thenetwork associated to both marginals (e).9. Perspectives and open questions In this work, we study the structured model of random pure quantum states intro-duced in [17] from the perspective of area laws. We showed, in the situation wherethe vertex size is constant, that the entropy of entanglement satisfies, on average, anarea law, for a suitable definition of surface area. Indeed, since we are dealing withunitary mixing at each vertex, the usual notion of area does not make sense, so onedefines surface area via a combinatorial optimization procedure. In the final section ofthe paper, we studied some situations where Hilbert space dimension varies, in the caseof very simple graphs. Unfortunately, our current methods (area defined via combina-torial optimization) are not adapted anymore, and some further work is necessary toestablish an area law in this more general setting.A mathematical improvement over the current results would be to obtain estimatesfor the probabilities of failure of the announced area laws. Indeed, our results focus onaverage quantities and it would be interesting for to derive large deviations bounds forthe entropy of entanglement at fixed (but large) Hilbert space dimension N . Another direction for future work would be to continue the project started in [17] andto analyze different models of structured entanglement, motivated by solid state physics.Indeed, our starting assumption is that the initial entanglement between vertices isencoded by maximally entangled states. It would be natural to drop this assumptionand to work with generic entanglement , that could be generated, say, by associating tograph edges an independent set of unitary matrices. This would render the graph statemodel more symmetric and make it more realistic. Acknowledgments. It is a pleasure to thank Pawe l Kondratiuk for fruitful discus-sions on random graph states, and the anonymous referees for several useful remarks andcomments. Financial support by the Polish National Centre of Science under the grantnumber DEC-2011/02/A/ST1/00119 and by the Deutsche Forschungsgemeinschaft un-der the project SFB Transregio–12 is gratefully acknowledged. I. N. acknowledgesfinancial support from the ANR project OSvsQPI and the PEPS-ICQ CNRS projectCogit. The research of B. C. was partly supported by an NSERC discovery grant, anOntario’s ERA and AIMR. References [1] Acın A, Cirac J I and Lewenstein M 2007 Entanglement percolation in quantum networks Nat.Phys. Phys. Rev. Lett. Rev.Mod. Phys. 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B.C.: Department of Mathematics, University of Ottawa, Canada; AIMR, TohokuUniversity, Sendai; CNRS, Lyon 1, France. [email protected] I.N.: CNRS, Laboratoire de Physique Th´eorique, IRSAMC, Universit´e de Toulouse,UPS, 31062 Toulouse, France. [email protected]