Distillation of supersinglet states

The gist: This paper introduces an entanglement distillation protocol for supersinglet states composed of N qubits, which uses local operations and classical communication (LOCC) to generate higher fidelity supersinglet states from multiple noisy copies

Protocol Overview

The distillation protocol is a LOCC protocol designed to prepare an entangled state with improved purity starting from multiple noisy copies of the same state. The process involves several key steps, including preparation, twirling, permutation symmetrization, and local measurements The procedure proceeds as follows: 1. Prepare an initial state ρ with total spin zero, with the dominant fidelity Fα = h0, α, 0ρ0, α, 0i being the target supersinglet. 2. Perform the twirling operation (27) to remove coherences between spin sectors. 3. Perform permutation symmetrization (32) such that the state has the same symmetry as a supersinglet. 4. Take three copies of the state and perform local spin measurements (21) and a Schur transform (23) at each local site n. 5. All parties classically broadcast their measurement outcomes and postselect on the outcome j = 1/2, α = 1. The fixed qubits d = 1, 2 are discarded such that the effective operation is (25). 6. Using only the postselected state, recursively iterate Steps 4 and 5 until a high fidelity supersinglet is obtained.

Supersinglet State Definition

Supersinglets are defined as completely antisymmetric states consisting of several qudits. In this paper, the authors only consider supersinglets that are formed from N qubits, where N is assumed to be an even number The supersinglet state is a type of singlet state with total spin s = 0, meaning it has eigenvalues s = m = 0 in Eq. (4). The multiplicity of such spin zero states is determined by the way that the angular momentum of the N qubits couple. The supersinglet is defined as the singlet state that couples in such a way that there are two groups (called group I and II), each of N/2 qubits, such that each group forms the maximal spin state of s = N/4. This structure involves coupling the group I spins (forming a s = N/4 spin) antisymmetrically with the group II spins (also forming a s = N/4 spin). The explicit wavefunction of the supersinglet is written in Eq. (12).

Allowed Operations and Constraints

The primary constraint for the main distillation sequence is that all operations must only use LOCC. Allowed operations include local measurements, Schur transform, twirling, and permutation symmetrization. Local measurements in the local basis (8) are made which distinguish the spin and outer multiplicity label j, α. The measurement operator Mjα is defined as U(N=3)Sch Πjα = X j m=-j v(j, α, m)ihj, α, m. Twirling operations are defined as ρ → T (ρ) = Z dUU ⊗N ρU†⊗N where U = e−i σ· wθ/2 is a single qubit rotation. Permutation symmetrization involves applying the permutation symmetrizer ρ → W(ρ) = 1/Q X σ∈Q PσρP† σ, where Q is the set of permutations with the symmetry that one would like to enforce. The permutation symmetrizer is not strictly local as it involves the physical interchange of qubits, or applying a sequence of SWAP gates.

Distillation Analysis and Results

The protocol's key property is that the operator (41) leaves the supersinglet state invariant. This invariance is shown by demonstrating that MSN i ⊗3 = √psucSN i. The success probability psuc for obtaining the desired j = 1/2, α = 1 outcome is given by psuc = c / 2. The analysis shows that for an initial state that is exactly a mixture of permutations of singlets (e.g., Eq. (38)), this allows for a way to obtain the same result with greatly reduced computational resources. For noisy initial states, the fidelity reaches a maximum and then degrades because a competing state in the s = 1 sector starts to develop, which twirling operations cannot remove. The threshold for convergence towards the supersinglet is found to be δ = 0, meaning there is an exact balance of the convergence and no change in fidelity is seen.

Experimental Realization

The most likely experimental realization involves a combination of a local quantum memory at each party’s site consisting of 3 qubits, combined with long-distance entanglement distribution realized by photons. The primary operation in the protocol is the measurement following a Schur transformation, which has a complexity scaling as O(d 2). While the protocol is LOCC compliant, some resource-intensive steps are required, specifically Step 3 involves performing a permutation symmetrization. The success probability is expected to drop with N due to the increasing multiplicity of the singlet sector.

Conclusion

The paper successfully introduces a distillation protocol for purifying supersinglet states using LOCC, demonstrating that high fidelities are reachable after only several rounds of purification. The protocol is compatible with existing methods for Bell state distribution and purification. The development of this distillation protocol opens the door for applications such as quantum cryptography, quantum clock synchronization, and quantum metrology.

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The gist: This paper introduces an entanglement distillation protocol for supersinglet states composed of N qubits, which uses local operations and classical communication (LOCC) to generate higher fidelity supersinglet states

How it works

The protocol is a LOCC protocol designed to prepare an entangled state with improved purity starting from multiple noisy copies of the same state. The process involves several key steps, including preparation, twirling, permutation symmetrization, and local measurements 1. Prepare an initial state ρ with total spin zero, with the dominant fidelity Fα = h0, α, 0ρ0, α, 0i being the target supersinglet. 2. Perform the twirling operation (27) to remove coherences between spin sectors. 3. Perform permutation symmetrization (32) such that the state has the same symmetry as a supersinglet. 4. Take three copies of the state and perform local spin measurements (21) and a Schur transform (23) at each local site n. 5. All parties classically broadcast their measurement outcomes and postselect on the outcome j = 1/2, α = 1. The fixed qubits d = 1, 2 are discarded such that the effective operation is (25).

Supersinglet State Definition

Supersinglets are defined as completely antisymmetric states consisting of several qudits. In this paper, the authors only consider supersinglets that are formed from N qubits, where N is assumed to be an even number The supersinglet state is a type of singlet state with total spin s = 0, meaning it has eigenvalues s = m = 0 in Eq. (4). The multiplicity of such spin zero states is determined by the way that the angular momentum of the N qubits couple. The supersinglet is defined as the singlet state that couples in such a way that there are two groups (called group I and II), each of N/2 qubits, such that each group forms the maximal spin state of s = N/4. This structure involves coupling the group I spins (forming a s = N/4 spin) antisymmetrically with the group II spins (also forming a s = N/4 spin) <ref:III.

1. Improved entanglement distillation protocol for supersinglet states allows AI systems to maintain high-fidelity entangled states over long distances, which is crucial for quantum clock synchronization and cryptography applications.

2. The protocol uses only local operations and classical communication (LOCC), making it suitable for practical implementations in distributed quantum networks without requiring impossible entangling operations like CNOT gates between distant parties.

3. AI systems can perform quantum metrology with improved precision because the distillation process avoids a high dimensional Schur transform, allowing for more efficient use of the system's resources.

4. The ability to purify supersinglet states leads to more robust quantum computing components, as these states have zero quantum noise and are useful in decoherence free subspaces for computation.

We introduce an entanglement distillation (purification) protocol for supersinglet states composed of N qubits. The supersinglet state we target is a total spin zero state with zero spin variance, and has a fully entangled structure involving all qubits. In our distillation protocol, three copies of an initial spin zero state are measured in the local total spin basis such that a higher fidelity supersinglet state is generated upon postselection. The initial state can be prepared using conventional Bell state distillation methods distributed in a way to target the supersinglet symmetries. The protocol uses only local operations and classical communications, and is suitable for long-distance applications such as quantum clock synchronization and cryptography, and avoids a high dimensional Schur transform such that it can be used for tasks such as quantum metrology.