Entangling power of multipartite unitary gates

We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an $n$-partite gate as an explicit function of the gate, linking the entangling power of gates acting on $n$-partite Hilbert space of dimension $d_1 \ldots d_n$ to the entanglement of pure states in the Hilbert space of dimension $(d_1 \ldots d_n)^2$. Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the maximal entangling power and relate it to the absolutely maximally entangled (AME) states of a system with $2n$ parties. Finally, we provide a detailed analysis of the entangling properties of three-qubit unitary and orthogonal gates.


Introduction
Quantum entanglement is among the most important resources of modern quantum technologies. Along with quantum superposition, it is the main tool needed for superdense coding [1], quantum teleportation [2], efficient quantum tomography, measurement precision beyond classical limit [3], and many other practical applications [4,5].
In quantum computation, the computation algorithms are realized with quantum gates -unitary transformations U acting on the state-space of the system, that describe non-trivial interactions between different subsystems. In this context, a crucial property of a given gate acting on a composed quantum system is its entangling power τ (U ), which is defined [6] as the average entanglement created by the gate when acting on a generic fully separable pure state |ψ sep : Here τ denotes the chosen measure of entanglement and the brakets · represent the average over the set of pure states in the Hilbert space H with respect to the unique, unitarily invariant measure. For instance, characterizing the entanglement of an initially separable state by its negativity [7] allows one to introduce a quantity accessible in an experiment [8].
In general, independently of the measure τ selected, there exists no closedform expression for the entangling power as a function of the gate. In other words, given a certain gate U , its entangling power has to be computed numerically by performing the average (1), which is time-consuming and yields only approximate results.
The key exception is the class of bipartite gates of dimension d × d. For these systems, Zanardi [9] derived an expression relating the entangling power of the gate (1) with its linear entropy: where S L is the linear entanglement entropy of the gate, determined by the purity of its Schmidt vector in the operator Schmidt decomposition, and S is the SWAP gate. Note that above, the linear entropy is used in two different meanings with two different arguments: on the left as the chosen measure of entanglement of the pure state S L |ψ , and on the right as a function of the unitary matrix S L (U ). Needless to say, for most applications the expression of Zanardi is superior to the original definition (1), allowing one to make precise, analytical statements regarding the entangling power of bipartite quantum gates.
In this contribution, we generalize the notion of entangling power to multipartite unitary gates. Investigation of the problem was started by Scott [10], who derived formulae for entangling power in the case of several subsystems of an equal dimension d. These results are particularly important in the context of quantum correction codes [11,12,13]. In this work, we study entangling power from a different perspective, not as a resource for a given quantum protocol, but as a physical property of a given gate, possibly acting on subsystems of different dimensions.
More recently, investigation of the problem of the entangling power of quantum gates acting on multipartite systems was pursued by Chen et al [14]. They analyzed the minimal entanglement created by a given unitary gate acting on a pure product state, described by Shannon entropy of entanglement. In this work we use the linear entropy of entanglement, which allows us to derive explicit analytical expressions for entangling power of an arbitrary unitary gate.
We start by following the strategy of Zanardi [9] for tripartite systems of dimension d 1 d 2 d 3 . We rewrite the formula (1) for the entangling power of a tripartite unitary gate with respect to the measure of entanglement τ 1 called onetangle [15] in terms of explicit functions of the unitary matrix. Besides its potential to be utilized in practical applications, such as looking for optimal entangling gates, the formula offers additional physical insight into the gates' entangling power, linking it to the entanglement of states in the extended Hilbert space of dimension (d 1 d 2 d 3 ) 2 . These results are then generalized to unitary gates acting on a system with an arbitrary number n of subsystems. A thorough analysis of the entangling properties of n-partite gates, in particular three-qubit gates, is also provided.
This work is organized as follows. In Section 2, we briefly characterize onetangle -the aforementioned measure of entanglement, as well as relevant related topics. In Section 3, we present and prove the analytical formula for the entangling power as an explicit function of the matrix U ∈ U (d 1 d 2 d 3 ) -see Theorem 2. In Section 4, we explore the entangling properties of general tripartite gates, including the mean entangling power averaged over ensemble of random orthogonal/unitary matrices of a fixed size with respect to the Haar measure on the corresponding groups. Furthermore, we investigate the maximal entangling power and its relation to AME states [16,17] of six-party systems. In Section 5, we extend all the previous results to the general case of unitary gates acting on an arbitrary number of parties. In Section 6, we characterize the entangling properties of several relevant classes of three-qubit unitary gates. Finally, in Section 7, we summarize our findings and propose some related open problems.

Tripartite entanglement
We begin with tripartite systems, H = H 1 ⊗ H 2 ⊗ H 3 , dim H i = d i . As the measure of entanglement we choose the one-tangle [15,18], defined as: where for pure states denotes the so-called generalized concurrence -a measure of entanglement with respect to the given splitting g|g of the Hilbert space. One-tangle can be thus interpreted as a measure of the total amount of entanglement in the state. The range of one-tangle is defined by the range of the generalized concurrence [19]: where d g denotes the dimension of the partition g, with the former attained by separable states and the latter by maximally entangled states with respect to the bipartition g|g .
Contrary to the bipartite case, in the case of tripartite systems, there is a number of locally inequivalent classes of entangled states, and even maximally entangled states [20] (which shows why tripartite entanglement is significantly more complex than in the bipartite case). In particular, in the case of three qubits, there are two such classes: the GHZ class, represented by the state as well as the W class, represented by the state Intuitively, the entanglement in GHZ states can be understood as "genuine" tripartite entanglement. If one qubit is traced out, the resulting state is separable. The entanglement in W states, on the other hand, is more akin to bipartite entanglement. If one qubit is traced out, the resulting state is still entangled. The amount of entanglement in the GHZ and the W states with respect to onetangle is equal to 1 and 8/9, respectively, the former of which is in this case maximal and equal to the right hand side of inequality (5).

Entangling power of a given tripartite gate
We state the main result of this work -an analytical formula for the entangling power of unitary matrices, in two steps. Firstly, in Lemma 1, we write and prove the formula in a basis-dependent form. Then, in Theorem 2, we rephrase it in geometric terms.
There are two reasons for such a choice: firstly, dividing the statement into two steps should make it more accessible for the reader. Secondly, and more importantly, for some purposes, including several results stated further in this work, the basis-dependent form is more practical than its geometric counterpart.
To this end, we observe that the matrix elements of any unitary operator U ∈ SU (d 1 d 2 d 3 ) acting in the Hilbert space H = H 1 ⊗ H 2 ⊗ H 3 , dim H i = d i , can be conveniently written in a six-index notation as Before we proceed, we note that the Einstein summation convention is used throughout the whole work, i.e. repeating indices are summed upon.
We can now state the following.
Lemma 1. The definition (1) of the entangling power for a tripartite system with one-tangle (3) as the entanglement measure is equivalent to where defines the entangling power of the gate U with respect to the bipartition ab|c.
We emphasize that implied summation over all possible four-component vectors r, s, t is taken in eq.(10), with vector elements spanned by {0, . . . , d i − 1}.
Proof. Instead of averaging over states, we can average over local unitaries acting on some chosen separable state |ψ 0 . In other words, we can rewrite definition (1) as where we have assumed a tripartite system. It is clear from the definition of one-tangle (3), that the entangling power (12) with one-tangle as the input is the sum of three terms of the form All we need to do is to show that the above quantity has the conjectured form (10). The proof consists of two steps. In the first step we compute τ ab|c . With no loss of generality, we choose the basis of the Hilbert space to be such that |ψ 0 ≡ |000 is its first element. Then, Using this notation, one can patiently calculate τ ab|c according to the definition (4): first performing the outer product |U U |, then the partial trace tr ab |U U |, next its square tr ab |U U | 2 . The final result can be written as where while the functions f ab|c r, s, t (U ) are defined in eq. (11).
The second step is to compute ab|c . By definition (13), where the integration is to be performed in accordance with the normalized Haar measure on the unitary group. Formula (15) implies that where we have moved the function f ab|c r, s, t (U ) in front of the integrals to emphasize that it does not depend on matrices U i , i = 1, 2, 3. Now, all three mutually independent integrals can be easily calculated using the handy result from [21,22] regarding the integration of the second moments of random unitary matrices over the relevant unitary group: In the case at hand, Substituting into eq. (18), we immediately obtain the conjectured formula (10).
In order to restate formula (10) in geometric terms, we observe that to every Given a basis {|j 1 j 2 j 3 } of the Hilbert space H, the coefficients of the state |U ∈ H ⊗ H are defined by the following relation: where the dimensional factor ensures proper normalization. Note that the equality (21) is valid in any basis, so the association U → |U may be viewed as a geometric statement.
We can now restate Lemma 1 in geometric terms: Theorem 2. Definition (1) of the entangling power for a tripartite system with one-tangle (3) as the entanglement measure is given by eq. (9) with the entangling power of the gate U on the bipartition ab|c (10) equal to In the above expression, the summation is over all ordered bipartitions x |y of H : including the two trivial bipartitions 1 2 3 |· and ·|1 2 3 , where the dot denotes an empty set, while |U is the state (21) associated with the matrix U .
Proof. The proof consists of a relatively lengthy but straightforward direct calculation of the expression (22) in a chosen basis and comparison with the basisdependent formula (10).

Statistical properties of ensembles of tripartite gates
After establishing an explicit formula (22) for the entangling power of tripartite gates, we would like to explore some of its consequences regarding the entangling properties of general tripartite unitary gates.
We begin with a formula for the upper bound for the entangling power.
is bounded from above by˜ where˜ ab|c is the upper bound for the entangling power of the gate U on the bipartition ab|c (22): where the summation is over x |y as in (23) and d abx , d cy denote the dimensions of the respective bipartitions of the extended Hilbert space H ⊗ H .
Proof. It is immediate to see from the definition of the entangling power (9) that if the quantity (25) is indeed the upper bound for the the entangling power (4) of the gate U on the bipartition ab|c (22), the theorem is true. All that remains is to prove this assumption. Let us add and substract the following quantity from the right hand side of eq. (22). After regrouping the terms, we arrive at Recall that in accordance with eq. (5), the quantity 2 is the upper bound for τ abx |cy , reachable only by states that are maximally entangled with respect to the bipartition abx |cy . This means that each and every term in the above sum, and thus the sum itself, is non-negative. Since the sum enters the equation with a minus sign, ab|c (U ) is at most equal to the quantity˜ ab|c . This completes the proof.
Besides bounding the maximum value of the entangling power of unitary gates U ∈ U (d 1 d 2 d 3 ) from above, the theorem provides us with an interpretation of the formula for the entangling power. Looking at eq. (27) we can see that the value ab|c (U ) is the highest when the values τ abx |cy (|U ) are the highest. The entangling power of the tripartite gate U is proportional to the total amount of entanglement in the sixpartite state |U .
Note that Theorem 3 provides only an upper bound for the maximum value of the entangling power of unitary gates U ∈ U (d 1 d 2 d 3 ). This bound may not be tight in general. It follows immediately from our discussion that the bound is tight if and only if the maximizing U fulfills for all bipartitions abx |cy of H ⊗ H entering the formula for the entangling power. In fact, further inspection reveals that for the bound to be tight the above relation must be true also for all the other bipartitions. This is either because of the symmetric property of generalized concurrence: τ abx |cy = τ cy |abx , or the origin of the state |U as a unitary matrix. In other words, U must be such that the state |U is a maximally entangled state with respect to each of the bipartitions. Such states are known in the literature as absolutely maximally entangled states (AME) [16,17,23]. They can only exist in multipartite quantum systems where each subsystem has the same dimension [17], although even then, several cases are known for which AME states do not exist: for instance, there are no such states for four-and seven-qubit systems [24,25]. The set of all such states in an n-partite Hilbert space of subsystem dimension d is denoted by AME(n,d).
Because of the property of the AME states called multiunitarity [23], utilizing the recipe (21), one can use the AME states to construct a unitary matrix U maximizing the entanglement power 1 for tripartite gates acting on H ⊗3 d . A link between large entangling power of unitary operators and strongly entangled multipartite states in an extended space was already discussed by Scott in [10]. We are now in position to establish a more precise relation and propose here the following result. Furthermore, given a state |ψ(6, d) ∈ AME(6, d) the matrix elements of the maximizing U can be recovered using the recipe (21), explicitly As our last general result, we compute the mean entangling power over the unitary group U (d 1 d 2 d 3 ) and the (real) orthogonal group O (d 1 d 2 d 3 ).
Proof. The proof relies on the basis-dependent expression (10). Since the functions f ab|c r, s, t (U ) are given (11) in terms of the second moments of the unitary matrix, one can integrate the basis-dependent expression using the previously utilized formula (19), which results in the conjectured expression (30) for the unitary group.
In the case of the orthogonal group, it is possible to find an analog of the formula (19) for orthogonal matrices -we refer the reader to Appendix A for details. The proof is then fully analogous to the unitary case.
We conclude the section by applying Theorems 3 and 5 to the special case of evenly divided tripartite systems d 1 = d 2 = d 3 = d called three-qudit systems, in which the expressions given therein simplify significantly. Corollary 6. The mean entangling power of three-qudit gates U ∈ U (d 3 ) reads in the case of the unitary group U (d 3 ), and in the case of the orthogonal group O(d 3 ). Furthermore, this entangling power is bounded from above by˜ denoted by green diamonds. For reference, the mean entangling power (32) of three-qudit gates has also been plotted as blue circles. As seen, while for large dimension d the three quantities practically converge, for smaller dimensions the improvement of our upper bound over the theoretical maximum is significant.

Multipartite unitary gates
In this section, we generalize the results from Sections 3-4 to an arbitrary number of partitions. With the exception of Lemma 1, which is skipped due to its auxiliary nature, each of the theorems provided therein is reiterated here in a version for n partitions. The proofs are omitted, as they follow the same lines of reasoning as their tripartite counterparts.
In full analogy to the tripartite case, to every unitary matrix U acting in the Hilbert space H = The 2n-index notation has been used here to denote matrix elements, U j1...jn j 1 ...j n := j 1 . . . j n |U |j 1 . . . j n .
Furthermore, we establish the following summation convention. Summation over p|q is understood as a summation over all non-trivial, unordered bipartitions of H. Summation over x |y , on the other hand, is understood as a summation over all ordered bipartitions of H , including trivial cases. For example, in the simplest case of two partitions, n = 2, we have where a single dot represents an empty set. As the measure of entanglement, we choose the natural generalization of one-tangle (3) to n parties, where 2 n−1 −1 is the number of bipartitions and τ p|q is defined in eq. (4). We are now ready to present the results concerning unitary gates acting on n-partite systems.
Theorem 7. Definition (1) of the entangling power for an n-partite system with the the entanglement measure given by the n-particle generalization of onetangle (39) is equivalent to where denotes the entangling power of the matrix U with respect to the bipartition p|q of H.
Theorem 8. The entangling power of n-partite unitary gates U ∈ U (d 1 . . . d n ) is bounded from above by˜ where˜ p|q is the upper bound for the entangling power of the matrix U on the bipartition p|q: In the above expression, d px , d qy denote the dimensions of the respective bipartitions of the extended Hilbert space H ⊗ H .
Observe that if we set n = 2, the expression (45) is equivalent to the result of Zanardi for bipartite systems [6], up to the multiplicative factor 1/2 due to different normalizations of generalized concurrence used.
Corollary 11. The mean entangling power of n-qudit gates U ∈ U (d n ) reads when averaged over the unitary group U (d n ), and when averaged over the orthogonal group O(d n ). Furthermore, this entangling power is bounded from above bỹ where · denotes the floor function.
Let us make three remarks regarding the above corollary. Firstly, we note that the result (47) regarding the mean over n-qudit unitary group could also be obtained by considering an appropriately weighted average of means of entangling power classes introduced in [10].  Secondly, we observe that due to the effect of concentration of measure investigated recently in the context of the entangling power of random unitary matrices [14], one can expect that for large dimension d the number of parties n the value of the entangling power for a given unitary gate U will typically be close to the averaged values derived above.
Finally, we note that aside from the domain where d, n are "small", the mean values (47), (48) over the unitary and orthogonal groups are nearly identical. Indeed, one can check that Moreover, numerical simulations suggest similar results regarding the upper bound (49) for the entangling power˜ 1 . In other words, in any n-qudit system with a sufficiently large dimension d or number of partitions n, the values of the quantities 1 O(d n ) , 1 U (d n ) and˜ 1 are practically indistinguishable. A visual comparison of the three quantities, which further supports our claim, has been provided in Figure 2.
The above results show that for higher dimensions generic unitary matrix acting on n-partite subsystems provides entangling power close to the maximal one. Hence, the corresponding random state determined in eq. (36) becomes asymptotically close the AME state of 2n parties.

Entangling properties of three-qubit unitary gates
As a special case, in this section we characterize the entangling properties of typical three-qubit gates. We consider the following four classes of such gates: • permutation matrices P(8), • diagonal unitary matrices D(8), • unitary matrices U (8), • orthogonal matrices O(8).
We mention that the latter three are also known in the literature [26,27] as the Circular Poissonian Ensemble (CPE), Circular Unitary Ensemble (CUE) and Circular Real Ensemble (CRE), respectively.
Permutations P(8). There are exactly 8! = 40 320 three-qubit permutation matrices P ∈ P (8). Since this number is finite and relatively small, it is possible to calculate the entangling power (9) of every permutation. This yields exactly 21 different entangling classes, ranging from 0 to max P(8) 1 = 64/81 ≈ 0.79, with mean 1 P(8) = 184 315 ≈ 0.58. The classification of permutations with respect to their entangling power is provided in Table B1 in Appendix B.
where ϕ i ∈ [0, 2π) are taken with respect to the flat measure on the eight-torus. In this parametrization, the entangling power (9) reduces to a relatively short expression where c ij kl := cos (ϕ i + ϕ j − ϕ k − ϕ l ). Since the average value of the cosine function over the whole period is 0, we can immediately state that the average entangling power of diagonal unitary matrices is equal to the constant term in the above expression, 1 D(8) = 10 27 ≈ 0.37. Finding the maximum value is a more complex task that requires optimization over ϕ i . To this end, we introduce new variables ω i , δ j and perform the following change of variables: Note that while this operation reduces the number of parameters from eight to seven, it comes with no loss of generality, as the global phase of unitary gates is irrelevant for physical considerations (including the entangling power). In other words, one of the eight parameters in equation (51) has been redundant from the start. In the new parametrization (53), formula for the entangling power (52) takes a particularly simple form, Notably, it depends only on the three parameters δ i . This should not be surprising: a three-qubit gate can be decomposed into a tensor product of three one-qubit gates if and only if it is possible to assign a definite phase differences between the three subgates. Thus, the entangling power of the gate is proportional to how "difficult" it is to assign these three phases, which is measured by the parameters δ i . With just three independent parameters, it is possible to find all the extremal points of the entangling power (54), i.e. points in which all its first derivatives vanish. Since the cube [0, 2π] 3 (δ 1 , δ 2 , δ 3 ) is a closed and bounded set, one of the extremal points has to be the global maximum of the function.
Using this method, we find that the maximum of the entangling power over diagonal unitary matrices is equal to max D ( Unitary matrices U (8). In the case of random unitary matrices, using Corollary 6 we immediately arrive at the mean value 1 U (8) = 2/3 ≈ 0.67, as well as the upper bound for the maximum˜ 1 = 8/9 ≈ 0.89. Since there do exist absolutely maximally entangled states of six qubits, or AME(6,2) [28,23], it follows from Corollary 4 that the aforementioned upper bound is tight, max U (8) 1 =˜ 1 = 8/9. Indeed, an example maximizing unitary which arises from direct application of the corollary to the AME(6,2) state provided in eq. (10) in [23].
To put the things into a wider perspective, we remind the reader that the amount of entanglement in the maximally entangled W states (7) is also equal to 8/9. This means that on average, the action of the unitary matrices maximizing 1 on separable states produces states characterized by entanglement precisely equal to that of the maximally entangled W states.
Orthogonal matrices O (8). Since the unitary matrix (57) maximizing the entangling power over the unitary group U (8) is orthogonal in addition to being unitary, it immediately follows that the maximum entangling power over the orthogonal group is equal to max O(8) 1 = max U (8) 1 = 8/9. As for the mean value, making use of Corollary 6 once again we find 1 O(8) = 208/315 ≈ 0.66.
The results of this section are summarized in Table 1 and further illustrated in Figure 3 showing a probability histogram of the entangling power for ensembles of 8! permutation matrices of size eight, the same number of random diagonal unitary matrices, random orthogonal matrices and random unitary matrices distributed according to the Haar measure.

Concluding remarks
In this work we have studied the entangling properties of multipartite unitary gates with respect to the chosen measure of entanglement τ 1 . We have derived an  These results were then generalized to unitary gates acting on n parties. In particular, we have found that a gate U acting on H ⊗n d which saturates the upper bound for entangling power corresponds to an AME state of 2n subsystems with d levels each.
Finally, we have employed our findings to analyze in detail the entangling properties of relevant classes of three-qubit unitaries. We have shown that a generic unitary gate of size 2 3 = 8 typically has a slightly larger entangling power than a generic orthogonal gate and much larger entangling power than typical permutation matrices or diagonal unitary matrices.
Based on this work, we propose two main directions for future research. Firstly, due to the existence of two inequivalent classes of maximally entangled three-qubit states, in addition to one-tangle studied in this contribution, there are two more valid measures of three-qubit entanglement. It would be especially interesting to find a formula analogous to ours for so-called three-tangle, which measures genuine tripartite entanglement in the state. We stress that due to the way these measures are defined, finding the expression for the entangling power with respect to one of them would immediately yield the expression for the other upon the use of our formula.
Secondly, the nonlocal properties of a given unitary gate U applied k-times, and the influence of the local interlacing dynamics V loc was recently studied for a bipartite setup [29,30,31], where the quantity e p (U V loc ) k , closely related to the entangling power, was investigated. It would be interesting to extend some of these results also for the multipartite case.
It is a pleasure to thank Arul Lakshminarayan for fruitful discussions and correspondence. Financial support by Narodowe Centrum Nauki under the grant number DEC-2015/18/A/ST2/00274 is gratefully acknowledged.

Appendix A Averages over orthogonal group
In order to find second moments of the orthogonal group O(d), it is necessary to find proper orthogonal Weingarten functions, W g O (q), r for two permutations q and r. These were first provided in [32] and further extended for certain setups in [33,34]. All the equations in this Appendix are based on these two extensions.
The desired integral can be expanded by these means to the form: The Weingarten functions W g O (q), r are calculated by joining the two permutations q and r into a new permutation qr. This new permutation can be uniquely broken into cycles, whose lengths are the only property relevant for our purposes. For each cycle with length l in the structure of qr, there is another cycle of length l -see Lemma 1.16 in [34]. By taking half of these cycles, the value of W g O (q), r is uniquely determined.
As an example, we consider W g O (p 1 ), p 1 with an involution p i defined above: so that the joint permutation p 1 p 1 consists of four cycles of length one. Taking half of these cycles yields two cycles of length 1, which we write as [1,1]. In [34] (App. B2) one can find this value: .
The same result holds for W g O (p i ), p i with any of the above p i . If the permutations p i and p j are different, then the permutation p i p j consists of two cycles of length 2 and so Inserting the above formulas into the desired integral (A1), we obtain the required formula used to derive expression (31).