Universal gates for transforming multipartite entangled Dicke states

We determine the minimal number of qubits that it is necessary to have access to in order to transform Dicke states into other Dicke states. In general, the number of qubits in Dicke states cannot be increased via transformation gates by accessing only a single qubit, in direct contrast to other multipartite entangled states such as GHZ, W and cluster states. We construct a universal optimal gate which adds spin-up qubits or spin-down qubits to any Dicke state by minimal access. We also show the existence of a universal gate which transforms any size of Dicke state as long as it has access to at least the required number of qubits. Our results have important consequences for the generation of Dicke states in physical systems such as ion traps, all-optical setups and cavity-QED settings where they can be used for a variety of quantum information processing tasks.


I. INTRODUCTION
Entanglement is a key resource facilitating a wide range of emerging quantum technologies, such as quantum computing [1], communication [2,3] and sensing [4]. It has been well established theoretically [5] and experimentally demonstrated between various particles, including photons, atoms and ions [6]. Entanglement between two particles [7] has been routinely prepared and used in different physical systems for a variety of tasks [1][2][3][4][5]. However, in order to make full use of the power of entanglement for quantum technologies and to probe deeper into the foundations of quantum mechanics, there has been an increasing push toward making larger numbers of particles entangled with each other. As the number of particles increases beyond two, different types of entangled states that cannot be converted into each other using local operations and classical communication (LOCC) [8] emerge. Greenberger-Horne-Zeilinger (GHZ) [9], cluster [10], Dicke [11] and W states [12], are examples of such inequivalent classes. This rich variety of structurally complex states among many particles holds great promise for a wide range of applications in quantum information. However, it is this same complexity that makes their preparation and manipulation difficult. Thus, understanding the limits for preparing and manipulating large multipartite entangled states are of great interest and urgently needed.
In this paper we derive the minimal number of qubits that are necessary to be accessed for expanding/reducing any given Dicke state. We show that, unlike W [13], GHZ and cluster states [14], Dicke states in general cannot be transformed by local access to only a single qubit. We consider gates for transforming Dicke states by minimal access. In the case of the expansion of W states, by accessing only one qubit there is a universal optimal gate which can expand any size of W states with maximum success probability. Similarly to such a case, we derive a universal optimal gate which adds either spin-up or spin-down qubits to Dicke states by minimal access. We then construct universal gates which add/subtract given numbers of spin-up and spin-down qubits with a nonzero success probability, regardless of the size of an initial Dicke state. This work has important implications for assessing the amount of control one needs in the preparation and manipulation of Dicke states for quantum information applications, such as quantum algorithms [15], quantum games [16] and multi-agent quantum networking [17].

II. NECESSARY CONDITION FOR TRANSFORMING A DICKE STATE
An N -qubit Dicke state with M 1 excitations is the equally weighted superposition of all permutations of Nqubit product states with M 1 spin-up (|1 ) and M 0 = N − M 1 spin-down (|0 ), and is written as , andP is a projector onto the symmetric subspace with respect to the permutation of any two particles. For example,P |2, 0 = |00 = |D 0 2 ,P |1, 1 = (|01 + |10 )/2 = √ 2|D 1 2 , and P |0, 2 = |11 = |D 2 2 . We assume that |D M1 N is shared between two subsystems A and B, and denote this as |D M1 N AB , with subsystem A holding a total of k qubits and subsystem B holding the remaining qubits, as shown in Fig. 1 (a). Here we derive the minimum number k of qubits that should be accessed in order to transform the state |D M1 N AB into a state |D M1+m1 N +n AB , where |n| is the total number of qubits added for n > 0 and deleted for n < 0, and similarly |m 1 | is the added/deleted number of qubits in |1 , while m 0 ≡ n − m 1 represents for the added/deleted number of qubits in |0 . For the trivial cases of M 0 = 0 (M 0 + m 0 = 0) and M 1 = 0 (M 1 + m 1 = 0), the states N+n AB by accessing only k qubits in A. In the case of reduction, n qubits are eliminated from A.
of system AB are product states |1 . . . 1 and |0 . . . 0 , respectively. In the following, we will study only the nontrivial cases where M 0 > 0, M 1 > 0, M 0 + m 0 > 0 and M 1 + m 1 > 0. We consider a local transformation scenario in which access to subsystem B is forbidden, and the transformation task is carried out by collectively manipulating the k qubits of subsystem A only [see Fig. 1 In this scenario, the whole system after the transformation is composed of N − k qubits in subsystem B and k + n qubits in subsystem A. Thus, we have We now derive a necessary condition for the transformation of Dicke states. The minimum number of spin-up qubits in subsystem B is given by α = max{M 1 − k, 0} for the initial Dicke state |D M1 N , and is given by . Since subsystem B is left untouched in the transformation, α ′ ≥ α should hold. Thus, k ≥ M 1 is necessary for the transformation with m 0 > 0. Since a similar argument holds for the transformation with m 1 > 0, we have as a necessary condition for transforming a Dicke state to another Dicke state. Note that for other cases, we trivially have k ≥ −n for m 0 ≤ 0 and m 1 ≤ 0. (3)

III. SUFFICIENT CONDITION FOR TRANSFORMING A DICKE STATE
Here we show that condition (2) is sufficient for transformation of a Dicke state to another Dicke state, and we derive the maximum probability for the transformation. We first decompose the Dicke state in Eq. (1) by using the symmetric bases in subsystems A and B. When we expand C M1 NP |M 0 , M 1 AB in the computational basis, it is given by the sum of C M1 N terms with unit amplitude. From these terms, select those that have j spin-up qubits in subsystem B. The sum of these selected terms should be given by where the range of the summation over j is given by Using this decomposition, we rewrite Eq. (1) as (5) For k > −n, decomposition of the desired state |D M1+m1 N +n AB obtained from the transformation is similarly given by with α ′ = max{M 1 − k − m 0 , 0} and β ′ = min{N − k, M 1 + m 1 }. Since access is allowed only to subsystem A, the marginal state in subsystem B does not change through the transformation process, which implies the following relation as tr with q min ≡ min α ′ ≤j≤β ′ q j (8) and Here it should be understood that C 0 0 ≡ 1, and C M1−j k = 0 for M 1 − j < 0 and M 1 − j > k.
When conditions (2a) and (2b) are satisfied, we have α ≤ α ′ and β ≥ β ′ , respectively, resulting in p max > 0. Under the conditions, we construct a gate M A which achieves the upper bound on the success probability in Eq. (7). The gate M A is composed of a success operator M s and a failure operatorM f satisfyingM † sM s +M † fM f = I. We defineM s bŷ f , we have p ≤ p max , where p max is given by Eq. (7) with α ′ = β ′ = M 1 − |m 1 | and k = −n = |m 0 |+ |m 1 |, and is strictly positive. In such a case, the success operator for the gate M A which achieves the upper bound on the success probability is defined bŷ which is a linear functional but we denote it as a linear operator for convenience here. From Eqs. (5) and (11), we obtainM s |D M1 As a result, we conclude that conditions (2) and (3) are necessary and sufficient for the transformation of the Dicke states.

IV. UNIVERSAL OPTIMAL GATES FOR TRANSFORMING DICKE STATES BY ADDING ONE TYPE OF SPIN WITH MINIMAL ACCESS
When we expand a W state, which is a special case of Dicke states with only one excitation M 1 = 1, i.e. |W N = |D 1 N , a universal optimal gate M A which achieves the expansion to |W N +m0 = |D 1 N +m0 (m 0 > 0) by accessing only one qubit is constructed asM s = |W n+1 AA 1| + (n + 1) −1 |0 ⊗n+1 A A 0|. The expansion can be done regardless of the size of the initial W state asM s |W N = √ p|W N +n with success probability p = (N + n)N −1 (n + 1) −1 , which coincides with p max calculated from Eqs. (7), (8) and (9) for any N .
Here we show that such a universal optimality is partially generalized to Dicke states under the following conditions: (a) the gate increases at most one type of spin, and (b) the gate accesses the minimum number of qubits to achieve the transformation.
For a gate with m 0 ≤ 0 and m 1 ≤ 0, the condition (b) means k = −n = |m 0 | + |m 1 |. Then the gate shown in Eq. (11) achieving p max only depends on m 1 and n. Thus it works as a universal optimal gate for any input with M 0 ≥ |m 0 | and M 1 ≥ |m 1 |.
We can also construct a universal optimal gate for m 1 > 0, m 0 ≤ 0 and k = M 0 by using the symmetry between |0 and |1 . Let us define a new operatorM s1 by interchanging the definition of |0 and |1 in Eq. (12), namely, replacing m 1 by m 0 and |D b a A by |D a−b a A . After rewriting the parameter j by k − j, we arrive at Here the positivity comes from α s ≤ j ≤ β s , implying that 0 ≤ k − j ≤ k and 0 ≤ k + m 1 − j ≤ k + n. In Eq. (16), by substituting j = k − M 1 + j ′ , and relabelling j ′ as j,M univ s is rewritten bŷ The success probability of the transformation is From q k min > 0, we see that the transformation succeeds with a nonzero probability whenever p max > 0.
For convenience, we classify the universal gates into four cases according to the signs of m 0 and m 1 , and show the range of applicable input Dicke states (M 0 , M 1 ) for each class in Fig. 2. The input states outside of the designated region are not transformable by any means (p max = 0), while those in the area are transformed with a nonzero success probability by the gate M univ A . Thus, the gates are universal gates for the Dicke-state transformation.

VI. CONCLUSION
Contrary to the expansion of GHZ, cluster and W states, Dicke states cannot be transformed by locally accessing only one qubit in general. We have derived the minimum number of qubits that should be accessed to transform a Dicke state to another Dicke state. Similarly to expansion of W states, when we access the minimum number of qubits for the transformation, one can construct universal optimal gates which add one type of spin to a given Dicke state. We have also constructed a universal optimal gate which deletes both types of spin from a given Dicke state with the minimum access of qubits. Finally, we have shown the existence of universal gates which transform any Dicke state satisfying the derived condition for the transformation with nonzero probabilities. We believe that the results are important for understanding the amount of control needed in the preparation and manipulation of Dicke states for future quantum information applications.