Cooperation and dependencies in multipartite systems

We propose an information-theoretic quantifier for the advantage gained from cooperation that captures the degree of dependency between subsystems of a global system. The quantifier is distinct from measures of multipartite correlations despite sharing many properties with them. It is directly computable for classical as well as quantum systems and reduces to comparing the respective conditional mutual information between any two subsystems. Exemplarily we show the benefits of using the new quantifier for symmetric quantum secret sharing. We also prove an inequality characterizing the lack of monotonicity of conditional mutual information under local operations and provide intuitive understanding for it. This underlines the distinction between the multipartite dependence measure introduced here and multipartite correlations.


I. INTRODUCTION
Identifying and quantifying dependencies in multipartite systems enables their analysis and provides a better understanding of complex phenomena.The problem has been addressed by several communities, considering both classical and quantum systems.For example, in neuroscience and genetics measures of multipartite synergy were put forward [1][2][3][4][5][6], in quantitative sociology quantifiers of coordination were introduced [7], redundancy was quantified in complex systems [8], and in physics and information processing quantities aimed at characterizing genuine multiparty correlations were studied in depth [9][10][11][12][13][14].The former quantifiers are motivated mathematically, keeping the combinatorial aspects of complex systems in mind, e.g., the synergy is the difference in the information all subsystems have about an extra system as compared to the total information contained in any subset of the systems.Many of the latter quantifiers involve difficult optimizations and are therefore hard to compute, e.g. in order to compute the extractable work used to define genuinely multipartite correlations in [12] one has to optimise over all protocols with local unitary operations, local dephasings and classical communication, which does not admit any computer-friendly parameterisation.Here, we introduce an operationally defined, simple and computable quantifier of multipartite dependency in terms of information gain from cooperation when some parties meet and try to deduce the variables of some of the remaining parties.We show how it differs from multipartite correlations often discussed in this context, prove its essential properties and discuss examples and applications.
It turns out that, in order to compute the quantity introduced here, it is sufficient to consider the respective conditional mutual information between only two subsystems.The conditional mutual information is a well established quantity in quantum information theory and broader physics.It captures the communication cost of quantum state redistribution [15][16][17][18], topological entanglement entropy [19][20][21][22] and squashed entanglement [23].It is related to quantum discord [24,25], intrinsic steerability [26], conditional erasure cost of a tripartite state [27,28] and the conditional quantum onetime pad [29].Additionally, conditional quantum mutual information appears in thermodynamics [30], in relation to high-energy physics [31][32][33] and Markov chains [34][35][36][37][38][39].Therefore, the dependence introduced here is a relevant quantity in all these problems if we symmetrize them and look for the worst case scenario.Concrete examples are given below.Furthermore, all these different studies will benefit from the inequality we prove here, which characterizes the lack of monotonicity of quantum conditional mutual information under general local operations.

II. MULTIPARTITE DEPENDENCE
Let us begin by briefly recalling fundamental relationships, e.g., that two classical variables X 1 and X 2 are statistically independent if their probabilities satisfy P (X 1 |X 2 ) = P (X 1 ).Alternatively, the statistical independence can be stated in terms of entropies with the help of both the Shannon entropy [40] H(X) = − d i=1 P (x i ) log d P (x i ), where d is the number of outcomes, and the conditional entropy H(X|Y ) = − i,j P (x i , y j ) log d P (xi,yj) P (yj ) .Note that throughout this paper logarithms are base d.As a measure of dependence of two variables X 1 and X 2 one introduces the corresponding entropic difference H(X 1 ) − H(X 1 |X 2 ), the so-called mutual information I(X 1 : X 2 ) [41].Similarly, the quantum mutual information captures the dependence between quantum subsystems [42], characterized by the state ρ, when we use the von Neumann entropy S(ρ) = −Tr(ρ log d ρ) in place of the Shannon entropy.However, already in the case of three variables there are two levels of independence.The variable X 1 can be independent of all other variables, i.e., P (X 1 |X 2 X 3 ) = P (X 1 ), or it can be conditionally independent of one of them, e.g., P (X 1 |X 2 X 3 ) = P (X 1 |X 2 ).The former dependence is again captured by the mutual information I(X 1 : X 2 X 3 ), while the so-called conditional mutual information I(X 1 : considers the latter.It is thus natural to define the tripartite dependence as the situation where any variable depends on all the other variables.This can be quantified as the worst case conditional mutual information Due to strong subadditivity the conditional mutual information is non-negative and hence D 3 ≥ 0 [40].D 3 vanishes if and only if there exists a variable such that already a subset of the remaining parties contains all available information about it.Note that this condition is also satisfied if a variable is not correlated with the rest of the system at all.The value of D 3 can be interpreted using an alternative expression for conditional mutual information, e.g., I(X 1 : X 3 |X 2 ) = I(X 1 : X 2 X 3 ) − I(X 1 : X 2 ).Accordingly, one recognizes from Eq. (1) that D 3 expresses the gain in information about the first subsystem that the second party has from cooperating with the third party.Non-zero value of D 3 ensures that any two parties always gain through cooperation when accessing the knowledge about the remaining subsystem.The minimal gain over the choice of parties is an alternative way to compute D 3 .Of course by taking this minimum some information about the underlying quantum state is inevitably lost, but our quantity is designed to be sensitive to genuinely multipartite properties of the state only.
In the context of quantum subsystems we can rewrite the conditional mutual information as I(X 1 : is the entanglement cost of merging a state X 1 with X 2 , see Ref. [43], we can interpret the conditional mutual information as the extra cost of merging states one by one (X 1 with X 2 and X 3 with X 2 ) instead of altogether (X 1 X 3 with X 2 ).D 3 is the minimum extra cost of this merging.
As another example, quantum conditional mutual information prominently appears in the definition of Markov chains.For a tripartite system in state ρ 123 , vanishing I(X 1 : X 3 |X 2 ) = 0 means that there exists a recovery map R X2→X2X3 , such that ρ 123 = R X2→X2X3 (ρ 12 ), i.e. the global state can be recovered by applying map from X 2 to X 2 X 3 on marginal X 1 X 2 .The global state is then called a quantum Markov chain.More generally, it turns out that the state R X2→X2X3 (ρ 12 ) has a high fidelity with ρ 123 for all states with small I(X 1 : X 3 |X 2 ) [34].A small values of D 3 therefore indicates that there exists a subsystem from which the global state can be recovered, approximating a quantum Markov chain, and a large D 3 shows that there is no such subsystem.
Moving on to larger systems, we note that there are more conditions to be considered already in order to define the four-partite dependence.In analogy to the tripartite case the first condition is to require that cooperation of any triple of parties provides more information about the remaining subsystem, e.g., I(X 1 : X 2 X 3 X 4 ) − I(X 1 : X 2 X 3 ) must be positive.But one should also impose that cooperation between any pair brings information gain about the two remaining variables, e.g., I(X 1 X 2 : X 3 X 4 ) − I(X 1 X 2 : X 3 ) must be positive.The former condition demands a positive conditional mutual information, I(X 1 : X 4 |X 2 X 3 ) > 0, while the latter one requires I(X 1 X 2 : X 4 |X 3 ) > 0. In order to compute D 4 one takes the minimum of these two conditional mutual informations over all permutations of subsystems.Note, however, that from the chain rule for mutual information and its non-negativity we have, e.g., ) and therefore it is sufficient to minimize over the conditional mutual information between two variables only.We emphasize that this step simplifies the computation significantly.The same argument applies for arbitrary N and leads to the definition of Npartite dependence where the minimum is taken over all permutations of the subsystems.In the case of a quantum system in state ρ we obtain where i, j = 1 . . .N and i = j.Tr i ρ denotes a partial trace over the subsystem i.In general, calculating the Npartite dependence requires computation and comparison of N 2 values, i.e., scales polynomially as N 2 , whereas for permutationally invariant systems it is straightforward.
One may also like to study k-partite dependencies within an N -partite system.To this aim we propose to apply the definitions above to any k-partite subsystem and take the minimum over the resulting values.

III. PROPERTIES
We now prove essential properties of the introduced dependence measure and explain why it is distinct from the multipartite correlations.
A. Pure states First of all, for pure quantum states |Ψ , the dependence can be simplified as where ρ i is the state of the system after removing all but the i-th particle, i.e., D N (|Ψ ) is given by the smallest quantum mutual information in two-partite subsystems, without any conditioning.Here, we made use of the fact that both subsystems of a pure state have the same entropy: S(Tr i ρ) = S(ρ i ) for ρ = |Ψ Ψ|.In Appendix G we prove the following upper bound on D N for pure states It is a consequence of the trade-off relation between the quantum mutual information for different two-particle subsystems of a pure global state and the definition of D N where the smallest conditional mutual information is chosen.In particular, the bound is achieved by the . This can be seen from Eq. ( 4).The two-particle subsystems of the GHZ state are of the form d |jj jj| and their mutual information equals 1.Additionally, the quantum mutual information is bounded by 1 whenever the state ρ ij is separable.This follows from the fact that separable states have non-negative quantum conditional entropy S i|j (ρ ij ) ≥ 0 [75], and accordingly their mutual information is bounded as where the last inequality is the consequence of using logarithms to base d.A comprehensive list of dependencies within standard classes of quantum states is given in Tab.I.The analytical formula for the Nqubit Dicke states with e excitations, |D e N , is presented in Appendix F. In short, if one fixes e and takes the limit N → ∞, the dependence D N vanishes.For e being a function of N , e.g., e = N/2, the dependence D N tends to 1/2.

B. Maximum dependence
The maximal N -partite dependence over classical distributions of d-valued variables is given by 1 (recall that our logarithms are base d) and follows from the fact that classical mutual information cannot exceed the entropy of each variable.On the other hand, quantum mutual information is bounded by 2 and this is the bound on D N optimized over quantum states (see Appendix E).As seen, pure quantum states satisfy the classical bound of 1, but there exist mixed states belonging to the class of k-uniform states, in particular for k = N − 1 [46], achieving the bound of 2. In the case of N qubits (for N even)  [11]; the GHZ state is described in the text; L k and R k stands for the linear cluster and the ring cluster states of k qubits (in general, the graph states are defined by the elements of the stabilizer group for a particular linear or ring graphs, as shown in [81]) and Ψ4 is discussed in [45].k-uniform states are states where all k-partite marginals are maximally mixed, whereas AME(n,d), so-called absolutely maximally entangled states, refer to ⌊n/2⌋-uniform states of d dimensions [74].
the optimal states have the following form where σ j are the Pauli matrices and σ 0 denotes the 2 × 2 identity matrix.Note that ρ max is permutationally invariant and gives rise to perfect correlations or anticorrelations when all observers measure locally the same Pauli observable.These states are known as the generalized bound entangled Smolin states [47,48].They are a useful quantum resource for multiparty communication schemes [49] and were experimentally demonstrated in Refs.[50][51][52][53][54][55].Per definition for (N − 1)-uniform states all reduced density matrices are maximally mixed, with vanishing mutual information, whereas the whole system is correlated.In Appendix E we provide examples of states which maximize D N for arbitrary d and show in general that the only states achieving the maximal quantum value of 2 are (N − 1)-uniform.
Let us also offer an intuition for values of D N above the classical bound of one.As shown in Appendix G this can only happen for mixed quantum states.One could then consider an auxiliary system which purifies the mixed state.High values of D N correspond to learning simultaneously the variables of the subsystems and the auxiliary system.Note that making this statement mathematically precise may be difficult as the problem is equivalent to the interpretation of negative values of conditional entropy [43,56,57].
A practical implication of the fact that mixed states achieve maximal dependence is that coupling an initially closed system to an environment can improve multipartite dependencies within the system.Furthermore, this is the only way of generating non-classical values of D N as dynamics within a closed system initialised in a pure state cannot beat the classical bound.

C. Comparison with multipartite correlations
Let us begin with a simple example that illustrates the difference between multipartite correlations and multipartite dependence.Consider three classical binary random variables described by the joint probability distribution P (000) = P (111) = 1 2 .All three variables are clearly correlated as confirmed, e.g., by quantifiers introduced in Refs.[13,14].However, the knowledge of, say, the first party about the third party does not increase if the first observer is allowed to cooperate with the second one.By examining her data, the first observer knows the variables of both remaining parties and any cooperation with one of them does not change this.There is no information gain and hence this distribution has vanishing tripartite dependence.
On the other hand, let us consider the joint probability distribution with P (000) = P (011) = P (101) = P (110) = 1  4 , which can describe also a classical system.Any two variables in this distribution are completely uncorrelated, but any two parties can perfectly decode the value of the remaining variable.Hence the gain from cooperation is 1 and so is the value of D 3 .
Nevertheless, D N does satisfy a number of properties that are expected from measures of genuine multipartite correlations.Any such quantifier should satisfy a set of postulates put forward in Refs.[12,14].We now show that most of them also hold for D N and we precisely characterize the deviation from one of the postulates.In Appendices A-C we prove the following properties of the dependence: (i) If D N = 0 and one adds a party in a product state then the resulting (N + 1)-party state has D N = 0.
(ii) If D N = 0 and one subsystem is split with two of its parts placed in different laboratories then the resulting (N + 1)-party state has D N +1 = 0.
(iii) D N can increase under local operations.Let us denote with the bar the quantities computed after local operations.We have the following inequality: where systems X 1 and X 2 are the ones minimizing D N , i.e., before the operations were applied.
The properties (i) and (ii) hold for all quantifiers of multipartite correlations.It is expected that measures of multipartite correlations are also monotonic under local operations (though note that often this condition is relaxed in practice, see e.g.quantum discord [42,58,59]).In the present case, the monotonicity property does not hold in general for D N , however, property (iii) puts a bound on its maximal violation (see Appendix D for a concrete example).Moreover, it has a clear interpretation: local operations that uncorrelate a given subsystem from the others may lead to information gain when the less correlated party cooperates with other parties.
Let us explain this more quantitatively for the conditional mutual information between variables X 1 and X 2 .While it is well-known that this quantity is monotonic under local operations on subsystems not in the condition [60], we prove in Appendix C that the following inequality is satisfied under local operations on arbitrary subsystem (being the origin of property (iii)): The second line is non-negative due to the data processing inequality and it quantifies how much the local operations have uncorrelated the variables in the condition X 3 . . .X N from the variables X 1 X 2 .This sets the upper bound to the lack of monotonicity of the conditional mutual information.
Let us also mention that the lack of monotonicity under local operations has been discussed in the context of complexity measures for multipartite systems [61][62][63].Those measures involve a different concept of correlation, in terms of the number of particles that have to be coupled in the Hamiltonian for which the discussed state is a thermal state.However, these measures a similar spirit to the dependence in that they capture the improvement in approximating a given distribution when more and more particles are coupled in the Hamiltonian.

D. Partial extension of classical interpretations
As shown, in classical information theory D N has a clear interpretation as information gain from cooperation.Agents could measure their subsystems and the information gain from the outcomes would match the computed value of D N .In the following we partially extend this interpretation to quantum systems, when the values of D N do not exceed unity.Let us proceed by comparing examples of classical and quantum distributions.As shown in the previous section a classical mixture of P (000) = P (111) = 1  2 admits D 3 = 0. To the contrary, a superposition 1 ) has a rather non-trivial value of dependence given by D 3 = 0.9183.However, this does not imply that there exists a set of local measurements on |W which yields a classical distribution with higher dependence.In fact, by optimising the dependence of local measurement results over all projective quantum measurements one obtains the classical 2  3 .Note, however, that D 3 compares the information two systems have about the third one with the pairwise information.It is therefore natural to also consider joint measurements on two parties.We have therefore computed the postmeasurement state i,j Π 12 j ⊗Π 3 k |W W |Π 12 j ⊗Π 3 k , where Π 12 j are the rank-one projectors on the first two particles and correspondingly Π 3 k are for the last qubit.Indeed, when we optimise over these projectors the dependence of the post-measurement state precisely matches the value computed for the W state, i.e.D 3 = 0.9183.
The examples given suggest that perhaps the interpretation of the quantum value of D N (whenever not exceeding unity) can be given as the highest dependence of the classical dataset that can be measured on the quantum state (including joint measurements).It turns out that this is in general not the case.We have found examples of pure four-qubit states for which there exist local measurements with outcomes producing D 4 higher than that of the corresponding quantum states.This further demonstrates property (iii) listed above.

IV. APPLICATIONS AND EXAMPLES
Multipartite dependence can be computed for both classical and quantum systems and is a generic quantifier of information gain from cooperation that can be used across science.Here we discuss examples of applications of D N in quantum information and briefly mention its role in data science.

A. Quantum secret sharing
An intuitive application of D N is secret sharing [64][65][66][67] with the additional constraint that the secret could be shared by any party.We refer to this problem as symmetric secret sharing.In the tripartite setting it requires collaboration of two parties in order to read out the secret of the remaining party.In the classical version of this problem the secret is a random variable, e.g., the measurement outcome of, say, the first observer.It is thus required that both, the second as well as the third party alone has only little or no information about the secret, i.e., I(X 1 : X 2 ) and I(X 1 : X 3 ) are small, while both of them together can reveal the result of the first observer, i.e., I(X 1 : X 2 X 3 ) is large or unity.Clearly, D 3 is the relevant figure of merit and due to the minimization in (1), the secret can be generated at any party.The states ρ max derived above appear well suited for this task and since they admit perfect correlations along complementary local measurements, by following the protocol in [66], the quantum solution to the secret sharing problem offers additionally security against eavesdropping.
This security has also been recently quantified with the conditional quantum mutual information in Ref. [29].The dependence introduced here is in contrast taking into account that any participant could be sharing the secret.
Let us also define the quantum secret sharing task and demonstrate the relevance of the dependence in this context.Suppose Alice has a quantum state ρ, called the secret, which she wants to split into n shares such that the secret is recoverable only when a party has all n shares.A quantum secret sharing scheme [67] is a map E n : A → X ⊗n such that, where Tr k is the partial trace over an arbitrary set of subsystems and C Q (Λ) is the quantum capacity of the channel Λ.This condition encodes the requirement that from any subset of shares one is not capable of recovering the initial quantum information in ρ.The rate of a quantum secret sharing scheme is given by the quantum capacity of the channel E n .The concrete protocol utilizing ρ max is as follows.Consider a quantum secret ρ of a single qubit.Using the teleportation protocol with state ρ and one subsystem of ρ max in Alice's possession, she performs the encoding map E N −1 to her qubit: (10) In this way Alice prepares n = N − 1 shares of the secret.We denote the state above as ρ N −1 .Since for any ρ we have (Tr , no subset of observers can recover the quantum secret.All of them together, however, can recover it perfectly with the following "reverse" teleportation scheme.Again take the resource (N − 1)uniform state and conduct a measurement in the basis |Ψ }, where µ n = 0, 1, 2, 3 and is the maximally entangled state.This measurement is conducted on the shares ρ N −1 and N − 1 subsystems of ρ max .If the measurement result corresponds to |Ψ the remaining qubit is in the state ρ, otherwise there exist unitary operations depending on the result that transform the single qubit output to ρ.
We now show that any N -partite state ρ c with maximally mixed marginals and non-classical dependence D N (ρ c ) > 1 is useful for the quantum secret sharing.Consider the encoding map E c : A → X ⊗N −1 with the Choi state given by ρ c [68], i.e., ( where |Φ is the maximally entangled state.The rate of quantum secret sharing admits the lower bound The steps are justified as follows.The first line follows from definition.Ineq.(12b) is the result of computing the quantum capacity of a channel [16,[69][70][71][72][73] with system A ′ being of the same dimension as system A and where S A ′ |X1...XN−1 (ρ) is the quantum conditional entropy of state ρ, (12c) follows because the maximally entangled state is a particular choice of φ A ′ A , and the Choi state of E c is ρ c .Eq. (12d) follows from the properties of entropy recalling that our logarithms are base d.
Eq. (12e) follows from the assumption that the state has maximally mixed marginals.Finally, the dependence is the worst case conditional mutual information.
Since the marginals of ρ c are maximally mixed, the same holds for the encoded state ρ N −1 = E c (ρ), i.e., no subset of parties can recover the quantum secret alone, yet for all of them together R > 0 holds for D N (ρ c ) > 1.
This lower bound is in general not tight, although it is achieved, e.g., by the (N − 1)-uniform states [74].In fact, all degradable channels give rise to the equality in (12b) and all symmetric states admit the equality in (12f).Note that for all pure states the lower bound on the rate is zero, whereas, e.g., [66] gives a quantum secret sharing scheme using a GHZ state with a unit rate.In this particular case it is easy to generalize the proof above.Since the GHZ state has classically correlated marginals (not maximally mixed), the conditional entropy in Eq. (12d) vanishes and the rate is lower bounded by the dependence alone, which is 1 for the GHZ state.
Finally, note that also situations where a subset of k parties is required to read the secret are of practical interest.Analogical arguments to the ones just given show that the dependence D k is the figure of merit for sharing the secret among any k-partite subsystem of N -party state where anyone could be the secret sharer.

B. Witnessing entanglement
As derived above, the values of D N exceeding 1 indicate that quantum systems are being measured.Furthermore, the quantum state of the systems is not pure.We now show that such values witness quantum entanglement.
By our assumption D N > 1. Per definition of dependence, it is the smallest difference of mutual informations.Let us label the subsystems such that this minimum is D N = I(X 1 : X 2 X 3 . . .X N ) − I(X 1 : X 3 . . .X N ), Since the second term is never positive, it is the first mutual information that has to be bigger than one, i.e.I(X 1 : X 2 X 3 . . .X N ) > 1. Writing the mutual information in terms of the quantum conditional entropy then gives For subsystems with the same dimension S(ρ 1 ) ≤ 1 and accordingly the conditional mutual information is negative.As shown by Cerf and Adami, this is only possible for entangled states [75].Note that this entanglement does not have to be genuinely multipartite.An example of a particular state violating the bound of 1 is given in Eq. (6).It also nicely demonstrates the point just given as it can be written as a mixture of correlated Bell states [47] and therefore is biseparable.Furthermore, the proof can be repeated for any number of subsystems k, i.e. if D k > 1 any k-party subsystem is entangled.A concrete example where this is the case is given by the four-party subsystems of absolutely maximally entangled state of 6 qubits (see Tab. I).

C. Entanglement without dependence
An intriguing question in the theory of multipartite entanglement is whether entanglement can exist without classical multipartite correlations [11].The examples of N -party entangled states with vanishing N -party classical correlations are known in the literature [76][77][78][79][80], though the corresponding notions of classical correlations do not satisfy all the postulates of Refs.[12,14].Here we ask whether there are genuinely multipartite entangled states with no multipartite dependence.
It turns out there exist even pure genuinely multipartite entangled states without multipartite dependence.
Consider any N -qudit cluster state (including linear, ring, 2D, etc.) for N ≥ 4. It was shown in Ref. [81] that all single-particle subsystems are completely mixed and there exists at least one pair of subsystems in the bipartite completely mixed state.The corresponding entropies are equal to S(ρ i ) = 1 and S(ρ ij ) = 2, and lead to D N = 0, due to Eq. ( 4).Therefore, the information about a particular subsystem cannot be increased when other subsystems are brought together which explains the impossibility of the corresponding secret sharing task [82][83][84].Note that there exist other subsets of observers who can successfully run secret sharing using a cluster state.

D. Dependence without correlations
Similarly we ask whether multipartite dependence can exist without multipartite correlations and vice versa.It turns out that all combinations are possible.The cluster states discussed in the previous subsection give rise to multipartite correlations and therefore show that multipartite correlations can exist without multipartite dependence.
Conversely, the dependence can be non-zero even in states with no correlations whatsoever.To this end consider the state It is N -party entangled and has vanishing all N -partite correlation functions [11].Yet, its D N is finite as shown in Tab.I.This again shows that multipartite dependence is distinct from multipartite correlations and captures other properties of genuinely multi-partite entangled systems.

E. Experimental states
We move to multipartite dependence in quantum optics experiments.Table II gathers quantum states prepared with photonic qubits.Details of the experimental preparation of the states and the evaluation of the dependence are given in Appendix H.We have chosen to present the states illustrating the properties discussed above.
The experimental data is in good agreement with the theoretical calculations.Deviations for the six qubit state D 3  6 result from reduced fidelities due to contributions of higher order noise in the state preparation.The same applies to the five qubit state ρ nc,5 derived from D 3 6 .Indeed, the states denoted as ρ nc , which have vanishing correlation functions between all N observers [77], clearly show a non-vanishing value for D N .Hence, these states are examples for "entanglement without correlations" and "dependence without correlations".Similarly, the experimental data of the linear cluster state L 4 indicates "entanglement without dependence" and "correlations without dependence".In the experiment, the GHZ 4 state ∼ |0000 + |1111 achieves the highest dependence of all considered states and is close to the theoretical dependence D 4 = 1, which is maximal over all pure states.The small value of D 3 for the four-partite GHZ state re- flects its property of having vanishing dependence for all tripartite classically correlated subsystems.

F. Data science
The dependence is also expected to find applications outside physics and we briefly sketch how it can be useful in data science.The problem of feature selection is to reduce available data to a smaller subset that faithfully represents the whole dataset, so that the predictions made on the basis of the subset would be the same as based on the entire set.In other words, we would like to eliminate variables that are not important.
Such variables can be identified from the conditions of extremal dependence.Other functions of conditional mutual information were considered in Refs.[85,86].Let us first analyze the case of D N ≈ 0. For example, if the minimizing conditional information is I(X 1 : X 2 |X 3 . . .X N ) ≈ 0 then either variable X 1 or X 2 can be eliminated as it does not improve the information between the remaining variables, e.g., I(X 1 : X 2 X 3 . . .X N ) ≈ I(X 1 : X 3 . . .X N ).On the other hand, D N ≈ 1 corresponds to the situation where each variable is independent of the rest, e.g., I(X 1 : X 3 . . .X N ) ≈ 0, (and hence at first sight one would have to keep track of all of them), but from N − 1 variables one can predict the remaining one, e.g.I(X 1 : X 2 X 3 . . .X N ) ≈ 1. Accordingly, one variable can be eliminated.

V. CONCLUSIONS
We have introduced a quantity, the multipartite dependence, as new tool for the characterisation of quantum states.It is surely the method of choice to determine whether and by what amount cooperation between any subsystems brings additional information about the remaining subsystems.It offers an extension to the charac-terization of multipartite properties via correlations.The dependence is directly calculable and has a clear interpretation.As such, several applications are identfied here clearly indicating its relevance for future studies in quantum communication and elsewhere.whereas projection of one photon's polarization onto the horizontal direction and tracing this photon out provides us the Dicke state D 1  3 .The tomographic data of D 2 4 was originally taken for Ref. [77].The five photon state ρ nc,5 was deduced from D 3  6 .The four-qubit GHZ 4 state as well as the linear cluster state L 4 were prepared in a two-photon setup, where the four qubits were encoded in two degrees of freedom per photon, the polarization and the path.A type-I SPDC source (two crossed type-I BBO crystals pumped with a 402 nm continuous-wave laser at 60 mW) generated polarization-entangled photon pairs.Using polarizing beam splitters the two polarization qubits were coupled to the path degree of freedom inside two displaced Sagnac interferometers.Right after a polarising beamsplitter, the four qubit GHZ 4 state is created, and the combined manipulation of polarisation and path qubits in the Sganac interferometer enables the transformation of this state to other multi-qubit entangled states [89].
Genarally, the evaluation of the dependence is based on tomographic data.As the dependence is calculated from the eigenvalues of the reconstructed state and its marginals, a careful treatment of the eigenvalues of the reconstructed states is crucial.However, as the linearly reconstructed states feature negative eigenvalues [90], we instead resort to a simple tomographic reconstruction based on the findings in [88].We model the state by ̺ = i λ i |ψ i ψ i | + λ ⊥ 1 1 ⊥ , where λ i (|ψ i ) are the unaltered eigenvalues (eigenstates) from the direct state estimate which are well above a noise threshold, λ ⊥ is the sum of the noise eigenvalues where 1 1 ⊥ denotes the identity matrix in the space spanned by the noise eigenstates.

TABLE II .
Illustrative values of dependence for several experimental quantum states.In brackets we give theoretical predictions for ideal states.