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Journal of Physics A Mathematical and Theoretical

Wigner function for N-ququarts systems

Juan Jesús Jesús Díaz Guevara, Isabel Sainz, A. B. Klimov

Published Apr 11, 2025• 0 citations
Publisher

Abstract

Abstract We have developed a general algebraic method for constructing a nonredundant Wigner function for N-ququart systems using the structure of the underlying algebraic ring and the associated non-euclidian geometry of the 4 N × 4 N pseudo phase-space. Using the stabilizer formalism we show that the Wigner function satisfies the tomographic condition and possesses a generalized covariance under some specific Clifford symplectic transformations allowing to represent a wide class of ququart stabilizer states in form of delta-functions. We establish a relation between the Wigner function of N-ququart states represented in 4 N × 4 N grid with its 2N-qubit counterpart in a F 2 2 N × F 2 2 N discrete phase space and discuss the corresponding drawbacks.

DOI
10.1088/1751-8121/adcbf6

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Abstract We have developed a general algebraic method for constructing a nonredundant Wigner function for N-ququart systems using the structure of the underlying algebraic ring and the associated non-euclidian geometry of the 4 N × 4 N pseudo phase-space. Using the stabilizer formalism we show that the Wigner function satisfies the tomographic condition and possesses a generalized covariance under some specific Clifford symplectic transformations allowing to represent a wide class of ququart stabilizer states in form of delta-functions. We establish a relation between the Wigner function of N-ququart states represented in 4 N × 4 N grid with its 2N-qubit counterpart in a F 2 2 N × F 2 2 N discrete phase space and discuss the corresponding drawbacks.
The apparent absence of a stabilizer-specific phase-space description for qubit systems is not fundamental, but arises from fixing a state-independent discrete Wigner map. Allowing the phase-space representation to adapt to the state through a simple optimization procedure, we prove that every qubit stabilizer state admits an exact delta-function discrete Wigner representation for any fixed phase-space partition. This optimization can be implemented directly at the level of measurement statistics by selecting maximal probabilities across mutually unbiased bases forming the partition. This construction naturally leads to a non-stabilizer witness defined by the optimized value of the Wigner function at the origin. Stabilizer states saturate the universal lower bound of this witness for pure states, whereas a value strictly larger than the bound certifies non-stabilizerness. Although the witness is not a finite-size resource monotone, it becomes sharply concentrated and effectively invariant under random Clifford transformations in the large-qubit limit, suggesting an asymptotically linear growth law for broad families of non-stabilizer states. Our results restore a meaningful geometric phase-space characterization of qubit stabilizer structure and provide an operational diagnostic of non-stabilizerness.
We observe that the discrete Wigner functions (DWFs) of n-partite systems with odd local dimensions are tomographically universal, as reflected in the delta function form of the DWF for any stabilizer. However, in the n-qubit case, this property does not hold due to the non-factorization of the mapping kernel, the explicit form of which depends on a particular partition of the discrete phase space. Nonetheless, it turns out that the DWF for some specific stabilizers, not included in the set used for the construction of the Wigner map, takes on the form of a delta function. This implies that the possibility of classical simulations of Pauli measurements in a given stabilizer state for qubit systems is closely tied to the experimental setup.
We describe the symmetry group of the stabilizer polytope for any number n of systems and any prime local dimension d. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. In the case of qudits, the structure is somewhat richer: For n = 1, it is a wreath product of permutations of bases and permutations of the elements within each basis. For n > 1, the symmetries are given by affine symplectic similitudes. These are the affine maps that preserve the symplectic form of the underlying discrete phase space up to a non-zero multiplier. We phrase these results with respect to a number of a priori different notions of “symmetry,” including Kadison symmetries (bijections that are compatible with convex combinations), Wigner symmetries (bijections that preserve inner products), and symmetries realized by an action on Hilbert space. Going beyond stabilizer states, we extend an observation of Heinrich and Gross [Quantum 3, 132 (2019)] and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments. In particular: the symmetries of a set that behaves like a 3-design preserve Jordan products and are therefore realized by conjugation with unitaries or anti-unitaries. (The structure constants of the Jordan algebra are encoded in an order-three tensor, which we connect to the third moments of a design). This generalizes Kadison’s formulation of the classic Wigner theorem on quantum mechanical symmetries.
We start from Wootter's construction of discrete phase spaces and Wigner functions for qubits and more generally for finite-dimensional Hilbert spaces. We look at this framework from a non-commutative space perspective and we focus on the Moyal product and the differential calculus on these discrete phase spaces. In particular, the qubit phase space provides the simplest example of a four-point non-commutative phase space. We give an explicit expression of the Moyal bracket as a differential operator. We then compare the quantum dynamics encoded by the Moyal bracket to the classical dynamics: we show that the classical Poisson bracket does not satisfy the Jacobi identity thus leaving the Moyal bracket as the only consistent symplectic structure. We finally generalize our analysis to Hilbert spaces of prime dimensions d and their associated d × d phase spaces.
We show that qubit stabilizer states can be represented by non-negative quasiprobability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive state-independent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two---producing an exterior, or Grassmann, algebra---which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each stabilizer state's three-generator Wigner function, and these distributions evolve deterministically to one another under Clifford gates. This corresponds to a hidden variable theory that is noncontextual and local for qubit Clifford gates while Clifford (Pauli) measurements have a context-dependent representation. Equivalently, we show that qubit Clifford gates can be expressed as propagators within the three-generator Wigner-Weyl-Moyal formalism whose semiclassical expansion is truncated at order ${\ensuremath{\hbar}}^{0}$ with a finite number of terms. The TTT gate, which extends the Clifford gate set to one capable of universal quantum computation, requires a semiclassical expansion of the propagator to order ${\ensuremath{\hbar}}^{1}$. We compare this approach to previous quasiprobability descriptions of qubits that relied on the two-generator Wigner-Weyl-Moyal formalism and find that the two-generator Weyl symbols of stabilizer states result in a description of evolution under Clifford gates that is state-dependent, in contrast to the three-generator formalism. We have thus extended Wigner non-negative quasiprobability distributions from the odd ddd-dimensional case to d=2d=2d=2 qubits, which describe the noncontextuality of Clifford gates and contextuality of Pauli measurements on qubit stabilizer states.
The Gottesman–Knill theorem established that stabilizer states and Clifford operations can be efficiently simulated classically. For qudits with odd dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-d qudits that has the same time and space complexity as the Aaronson–Gottesman algorithm for qubits. We show that the efficiency of both algorithms is due to harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm for odd-d and the Aaronson–Gottesman algorithm for qubits are likely due only to the fact that the Weyl–Heisenberg group is not in S U ( d ) for d = 2 and that qubits exhibit state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits.