Symmetric GHZ Superpositions

• Symmetric superposition of GHZ states is a family of quantum states exhibiting maximal multipartite entanglement and invariance under operations like global bit-flips and qubit permutations.
• The approach employs parametrization and block-encoding methods, such as concatenated GHZ states, to mitigate decoherence and enhance quantum metrological performance.
• The framework underpins rigorous SLOCC classification, geometric representations, and experimental realizations in systems including circuit QED and optical networks.

The symmetric superposition of GHZ (Greenberger-Horne-Zeilinger) states refers to quantum states that exhibit maximal multipartite entanglement and possess invariance under specific symmetry operations, resulting in a highly structured family of states relevant for both theoretical analysis and practical implementation. The canonical GHZ state for $N$ qubits is $$\left| {\rm GHZ}_N \right\rangle = (1/\sqrt{2})\left( |0\rangle^{\otimes N} + |1\rangle^{\otimes N} \right)$$, serving as a paradigmatic example of genuine multipartite entanglement. Extensions and generalizations are studied in contexts ranging from decoherence-resilient encoding schemes to resource-theoretic characterizations and geometric representations.

1. Definition and Symmetry Properties

Symmetric superposition in the GHZ context involves quantum states that are invariant under operations such as global bit-flips, permutations, and coordinated phase rotations about the $z$ axis. The classical example is the GHZ state: $$|{\rm GHZ}_N\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes N} + |1\rangle^{\otimes N})$$ Symmetry-adapted families (GHZ-symmetric states) generalize this structure to mixed states and more complex superpositions. For three qubits, GHZ-symmetric states are parameterized by two real variables $(x, y)$ which encode the overlap with the symmetric basis. For four-qubit and higher systems, the parametrization involves three or more real variables and geometric objects such as tetrahedra (Eltschka et al., 2013, Park, 2016).

Permutation symmetry is central: GHZ-symmetric states are invariant under any permutation of qubits, $\sigma_x^{\otimes N}$ flips, and rotations of the form $U(\vec{\phi}) = \bigotimes_k e^{i\phi_k \sigma_z}$ with a global phase constraint. This restricts the possible state space, facilitating classification via SLOCC equivalence classes, subspace geometry, and resource-theoretic properties.

2. Concatenated and Encoded GHZ States

A key advance in macroscopic quantum superpositions is the use of block-local structure for enhanced stability against decoherence (Fröwis et al., 2012). The "concatenated GHZ" (C-GHZ) state is defined as

$$|\phi_c\rangle = \frac{1}{\sqrt{2}}\left(|0_L\rangle^{\otimes N} + |1_L\rangle^{\otimes N}\right)$$

where each logical qubit $|0_L\rangle$, $|1_L\rangle$ is itself a symmetric GHZ state over $m$ physical qubits: $$|0_L\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes m} + |1\rangle^{\otimes m}), \quad |1_L\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes m} - |1\rangle^{\otimes m})$$ This block encoding yields increased robustness: under local white noise, the off-diagonal ("interference") elements decay as $$\Vert \mathcal{D}(|0_L\rangle\langle 1_L|) \Vert_1 \leq 1 - [1-p(t)]^m$$, with the overall coherence decaying as the $N$th power. Choosing block size $m = O(\log N)$ substantially slows exponential decay, stabilizing macroscopic coherence and thus enabling enhanced metrological sensitivity.

Compared to cluster-state encoding or active error-correction codes, the passive symmetry-based GHZ encoding nearly saturates optimal stability bounds for interference terms, negativity, and distillable entanglement. Tensor network (MPS/MPO) representations enable scalable computation of decoherence effects, norm preservation, and quantum resource measures for high $N$ (Fröwis et al., 2012).

3. Entanglement Structure and SLOCC Classification

The entanglement hierarchy of symmetric superpositions is well characterized by SLOCC classes, most compactly represented in two-dimensional $(x, y)$ or higher-dimensional parameter spaces (Eltschka et al., 2013, Park, 2013, Park, 2016). For three qubits, states fall into classes: separable, biseparable, W-type, and GHZ-type, separated by convex boundaries in the symmetry-adapted space. GHZ symmetry projects the state space onto triangles (for three qubits) or tetrahedra (for four), vastly reducing classification complexity.

Restricted GHZ symmetry (e.g. invariance under pair flips rather than global flips) further compresses possible classes: the four-qubit RGHZ-symmetric states admit only two SLOCC classes ($L_{abc_2}$ and $G_{abcd}$), with the general set described by two parameters. This reduction enables sharp optimal witnesses for entanglement detection and clarifies sharing properties of symmetric superpositions.

4. Geometric and Resource-Theoretic Representations

Symmetric superpositions (including GHZ states) possess unique geometric characterizations. The Majorana representation expresses symmetric states as combinations of $N$ distinct spinors, with GHZ states or superpositions (W + $\overline{\mathrm{W}}$) corresponding to polynomials with $N$th roots determining spinor distribution (Sudha et al., 2023).

Canonical steering ellipsoids visualize two-qubit correlations within symmetric $N$-qubit systems. GHZ states yield degenerate (zero-volume) ellipsoids—no pairwise entanglement, maximal monogamy (maximal $N$-tangle). Superpositions like WWbar can have nonzero volume (indicating partial pairwise entanglement), but with volume decreasing as $N$ increases, indicating increased monogamous restriction (Sudha et al., 2023).

Löwdin symmetric orthogonalization bridges coherent and superposition resource theories: applying LSO on each subsystem transforms nonorthogonal-basis GHZ-like superpositions into orthonormal, maximally coherent states, preserving symmetry and resource hierarchy (Torun, 2022). In formulas: $$|l_0\rangle = \alpha |c_0\rangle + \beta |c_1\rangle,\quad |l_1\rangle = \beta |c_0\rangle + \alpha |c_1\rangle$$ and for the $n$-partite state: $$|\mathrm{GHZ}_{\mathrm{coh}}\rangle = \frac{1}{\sqrt{2}} (|l_0\rangle^{\otimes n} + |l_1\rangle^{\otimes n})$$ with LSO mapping to the nonorthogonal representation for superposition theory.

5. Generation and Experimental Realization

Symmetric superpositions of GHZ type are accessible via several experimental platforms and protocols. In circuit QED, ultrastrong coupling and frequency tuning of a central qubit mediate deterministic generation of GHZ states between photonic modes: $$|\Psi_{\text{final}}\rangle = \frac{1}{\sqrt{2}} (|0,0,0\rangle + e^{i\varphi}|1,1,1\rangle)$$ The protocol involves preparing the qubit in a superposition and tuning its frequency to resonance with the sum of the mode frequencies, enabling third-order processes that transfer excitation into symmetric photonic GHZ superpositions (Macrì et al., 2018).

Optical multiport splitters implement symmetric $N$-photon, $N$-mode GHZ superpositions; interference and the "zero transmission law" ensure post-selection filters out non-GHZ terms. For odd $N$: $$|\Psi^{\text{out}}\rangle = \frac{1}{\sqrt{2}} (|\mu\rangle_1\ldots|\mu\rangle_N + (-1)^{N+1}|\eta\rangle_1\ldots|\eta\rangle_N)$$ Even $N$ require modified input states due to higher-order interference suppression (Bhatti et al., 2023).

In circuit QED and superconducting qubit systems, time-dependent couplings modulated with synthetic magnetic flux enable directional rotation of excitation. The phase structure allows robust generation and control of symmetric GHZ superpositions even at high $N$, with protection against decoherence and low operation error (Feng et al., 2017).

6. Decoherence, Stability, and Macroscopicity

GHZ states' sensitivity to decoherence motivates symmetric encodings and passive stabilization techniques. The C-GHZ encoding, with logical blocks constructed from GHZ sub-states, slows exponential decoherence of off-diagonal elements and entanglement measures such as negativity. The $q$-index quantifies macroscopicity: $\Vert C(\rho) \Vert_1 \propto O(N^2)$ with suitable block sizes extending effective macroscopic entanglement under noise (Fröwis et al., 2012).

Symmetric superpositions of GHZ and W states display resilience in quantum Fisher information under phase damping, preserving metrological usefulness at high levels of noise (Erol, 2017). The role of symmetry and superposition in decoherence resistance is a central feature of real-world quantum metrology and computation protocols.

7. Nonlocality, Monogamy, and Entanglement Witnesses

Multipartite nonlocality in symmetric GHZ superpositions does not automatically follow from genuine tripartite entanglement. GHZ-symmetric states can be fully local (admitting local hidden variable models) in some biseparable regimes and can be "bilocal" even within GHZ and W SLOCC classes, demonstrating subtle hierarchy between entanglement and nonlocality (Zhu et al., 2021).

Optimal entanglement witnesses leveraging GHZ symmetry—constructed as linear combinations of GHZ projectors—simplify detection and classification in experiment, as the symmetry projects full state statistics onto reduced dimensions and boundaries that separate uninteresting (e.g., biseparable) from interesting (e.g., GHZ-type) entanglement (Eltschka et al., 2012). The geometric approach yields straight or tangential boundaries in the symmetry-adapted plane, supporting device-independent discrimination.

Self-testing protocols allow device-independent certification of symmetric superpositions (e.g., GHZ-W mixtures) either analytically (via subsystem projection and CHSH/XOR games) or numerically (via semidefinite programming and swap methods), even using fixed Pauli measurement settings. This expands robust entanglement verification to broad symmetric superposition classes (Li et al., 2019).

---

The symmetric superposition of GHZ states encapsulates a mathematically rich family of maximally entangled multipartite quantum states with profound implications for decoherence protection, entanglement classification, resource theory, and scalable quantum technology. The interplay of symmetry, encoding, and geometric structure underpins both fundamental understanding and experimental control across platforms such as circuit QED, photonic networks, and ion trap systems.