A quantum quasiprobability looks like an ordinary probability distribution until it becomes negative. That negativity is not a mathematical mistake: it records that no single classical phase-space distribution can reproduce all of the represented quantum statistics under the chosen phase-space convention.
Guide 38 owns Gaussian states and covariance matrices, Guide 39 owns quantum optics, Guide 56 owns contextuality, and Guide 27 owns magic-state resources. This guide owns the representation layer connecting them: continuous-variable Wigner functions, Weyl characteristic functions, displaced parity, negative phase-space regions, discrete Wigner functions and the special role of positivity in stabilizer-like subtheories.
Density operator → characteristic function → quasiprobability on phase space → marginals and moments → negativity/nonclassicality tests → discrete phase-space resource interpretations.
1. Why a quasiprobability is useful
Classical mechanics describes a state by a probability density over position and momentum. Quantum mechanics instead uses a density operator ρ. A quasiprobability representation maps ρ to an ordinary-looking function of phase-space coordinates while preserving enough information to reconstruct ρ.
The cost is that the function need not be nonnegative. Quantum interference is encoded partly through oscillatory positive and negative regions.
2. Continuous-variable Wigner function
For one continuous degree of freedom with position coordinate x and momentum p, one common convention is
W(x,p)=(1/πℏ)∫dy exp(2ipy/ℏ)⟨x−y|ρ|x+y⟩.
Other conventions shift factors of two between y, p and the prefactor. A formula should therefore always be interpreted with its chosen normalisation.
3. Normalisation
The Wigner function integrates to one:
∫dx dp W(x,p)=1.
This remains true even when W takes negative values. Negative regions are compensated by positive regions so the total integral remains one.
4. Position and momentum marginals
Integrating over momentum gives the position probability density:
∫dp W(x,p)=⟨x|ρ|x⟩.
Integrating over position gives the momentum probability density:
∫dx W(x,p)=⟨p|ρ|p⟩.
Thus W can be negative locally while all directly measured marginals remain ordinary nonnegative probabilities.
5. Characteristic function
For bosonic mode operator a, define a displacement operator D(α)=exp(αa†−α* a).
A symmetrically ordered characteristic function is
χ(α)=Tr[ρD(α)].
The Wigner function is a Fourier transform of χ. This duality makes displacement operators the natural analogue of classical Fourier characters on phase space.
6. Wigner function from displaced parity
For one bosonic mode, the Wigner function can be measured through displaced photon-number parity. In a common optical convention,
W(α)=(2/π)Tr[ρ D(α) Π D(α)†],
where Π=(−1)^{a†a}.
This relation turns a phase-space function into an experimentally accessible expectation value.
7. Coherent states are Gaussian and nonnegative
A coherent state |α₀⟩ has a Gaussian Wigner function centred at α₀. It is everywhere nonnegative.
Squeezed Gaussian states also have nonnegative Wigner functions even though they are strongly nonclassical in other senses. Therefore Wigner negativity is a sufficient witness of certain nonclassicality, not a necessary condition for every quantum resource or experiment.
8. Fock-state negativity
The single-photon Fock state |1⟩ has Wigner function
W_1(α)=(2/π)(4|α|²−1)exp(−2|α|²)
under the displaced-parity convention above.
At the origin,
W_1(0)=−2/π.
The negativity follows immediately from odd photon-number parity.
9. Cat-state interference fringes
A coherent superposition such as |α⟩+|−α⟩ produces two broad Gaussian lobes plus an oscillatory interference term between them.
The interference term contains alternating positive and negative fringes. An incoherent mixture of |α⟩ and |−α⟩ keeps the lobes but loses those fringes.
Thus Wigner tomography can visually separate coherent superposition from classical mixture.
10. Wigner negativity volume
A simple global measure of negative volume is
𝒩_W=(∫dξ |W(ξ)|−1)/2.
It vanishes for nonnegative W and is positive when negative regions are present.
The precise phase-space measure dξ and normalisation depend on convention. Comparative studies must use the same convention.
11. Negativity is representation dependent
The Wigner function is one member of a family of phase-space quasidistributions.
- The Glauber–Sudarshan P representation is tied to coherent-state mixtures and can become more singular than an ordinary function.
- The Husimi Q function is smoothed enough to remain nonnegative.
- The Wigner function sits between them and corresponds to symmetric ordering.
A statement such as “the quasiprobability is negative” is incomplete until the representation is named.
12. Symmetric ordering
Wigner functions reproduce expectation values of symmetrically ordered operator products through phase-space integrals.
For example, quadrature moments correspond to moments of x and p with operator ordering handled by the Weyl/symmetric prescription.
This is why Wigner phase space resembles classical mechanics while still encoding operator noncommutativity through the star product.
13. Moyal product
Operator multiplication maps to a noncommutative phase-space product. Schematically,
A_W ⋆ B_W = A_W exp[(iℏ/2)(←∂x→∂p−←∂p→∂x)] B_W.
The leading term is ordinary multiplication; the first correction generates the Poisson bracket.
Quantum mechanics approaches classical Hamiltonian flow when higher ℏ corrections become negligible for the relevant observables and distributions.
14. Wigner equation of motion
The von Neumann equation maps to a Moyal-bracket evolution for W. For Hamiltonians at most quadratic in x and p, the Moyal corrections terminate and W evolves under the same linear symplectic flow as a classical phase-space density.
This explains why Gaussian quantum mechanics is especially tractable and why nonlinear Hamiltonians are required to generate many forms of Wigner negativity from initially Gaussian states.
15. Gaussian channels can wash out negativity
Loss, thermal noise and coarse-graining convolve or smooth phase-space distributions. Negative interference features are often the first to disappear.
For cat states, loss carries which-path information into the environment and suppresses the interference fringes.
Guide 40 owns Gaussian channels; here the Wigner representation makes their action geometrically transparent.
16. Hudson’s theorem for pure continuous-variable states
Hudson’s theorem states that a pure continuous-variable state has an everywhere nonnegative Wigner function if and only if it is Gaussian.
This is a powerful structural result, but its hypotheses matter: it concerns pure states. Mixed non-Gaussian states can have positive Wigner functions.
17. Discrete phase space
Finite-dimensional systems can also be represented on discrete phase spaces.
For odd prime or odd prime-power dimensions, especially clean constructions use finite fields and phase-point operators A_u satisfying orthogonality and covariance properties.
A discrete Wigner function can be written schematically as
W_ρ(u)=(1/d)Tr(A_u ρ).
The exact A_u convention matters, especially for qubits where no single construction has all the same properties as the odd-dimensional case.
18. Discrete Hudson theorem
For odd prime-power dimensions under the standard Gross discrete Wigner representation, pure states with nonnegative Wigner functions are precisely stabilizer states.
This finite-dimensional analogue of Hudson’s theorem makes positivity a sharp structural boundary for stabilizer quantum mechanics in that setting.
19. Clifford evolution preserves positivity in the odd-dimensional stabilizer representation
Clifford unitaries act as permutations/symplectic transformations of discrete phase-space points. Stabilizer measurements and state updates also admit nonnegative stochastic descriptions in the appropriate odd-dimensional formalism.
This supports efficient classical simulation of the stabilizer subtheory and connects Wigner negativity with resources beyond Clifford computation.
20. Magic-state negativity
Magic states injected into stabilizer computation can carry discrete Wigner negativity. In odd prime dimensions, that negativity can be quantified as a computational resource and related to simulation cost.
Guide 27 owns magic-state distillation. This guide supplies the phase-space resource picture.
21. Mana
For a discrete Wigner function W(u), one resource monotone is mana
𝓜(ρ)=log Σ_u |W_ρ(u)|.
It vanishes when W is nonnegative and grows with total quasiprobability negativity.
The logarithm base is a convention. Resource comparisons require the same discrete Wigner representation and dimension assumptions.
22. Contextuality and negativity
Guide 56 owns contextuality. In several stabilizer-plus-magic computational frameworks, Wigner negativity and contextuality are closely related indicators of departure from a classically simulable noncontextual subtheory.
But the equivalence is framework dependent. For qubits, state-independent contextuality and the choice of discrete Wigner representation require extra care. One should not state that every negative quasiprobability in every representation is exactly the same resource as contextuality.
Raussendorf and collaborators analysed these connections explicitly for qubit/rebit computation.
23. Negativity and classical simulation cost
A signed quasiprobability can be sampled by drawing from its absolute value and attaching a sign/weight. The variance of such Monte Carlo estimators grows with the total absolute quasiprobability weight.
This creates a direct relation between negativity and sign-problem-like classical simulation overhead in selected phase-space methods.
The statement is algorithm specific: another classical representation may have a different cost.
24. Quasiprobability error mitigation
Probabilistic error cancellation in Guide 55 also uses signed coefficients: an ideal operation is expanded as a linear combination of implementable noisy operations, some with negative weights.
The sampling overhead scales with the square of the coefficient one-norm in typical estimators.
This is mathematically analogous to negativity-induced Monte Carlo overhead even though the quasiprobabilities live over operations rather than phase-space points.
25. Wigner tomography
In cavity and trapped-mode experiments, displace the state by −α and measure parity. Repeating over a grid reconstructs W(α).
Finite sampling introduces uncertainty at every phase-space point. A visually negative pixel should be compared with its statistical error and reconstruction procedure before being interpreted as significant negativity.
26. Resolution and smoothing
Finite detector resolution convolves the true Wigner function with a response kernel. Sufficient smoothing can remove narrow negative fringes.
Failure to observe negativity therefore need not prove the underlying state has a positive ideal Wigner function.
Conversely, deconvolution can amplify noise and create spurious oscillations unless regularised carefully.
27. Wigner negativity is not entanglement
A single-mode Fock state can have a negative Wigner function without bipartite entanglement. A two-mode Gaussian entangled state can have an everywhere nonnegative Wigner function.
Negativity and entanglement diagnose different resources.
28. Wigner negativity is not Bell nonlocality
A negative phase-space representation does not automatically imply violation of a Bell inequality, and a Bell test depends on measurements and spacelike/locality assumptions rather than a chosen quasiprobability representation alone.
29. Common misconceptions
“Negative W means negative measurement probability.” No. Measured marginals and POVM outcome probabilities remain nonnegative.
“Positive W means classical.” Not universally. Squeezed states and Gaussian entangled states can have positive Wigner functions while exhibiting quantum resources.
“Every discrete Wigner function has the same negativity.” Discrete representations depend on dimension and phase-point conventions.
“Negativity is exactly contextuality.” Their relationship is powerful but framework dependent.
30. Worked synthesis problem
Consider the single-photon state |1⟩.
- Parity of |1⟩ is −1.
- At zero displacement,
W(0)=(2/π)⟨Π⟩=−2/π. - The value is negative even though every detector outcome probability is nonnegative.
- After strong loss, the state becomes a mixture of |1⟩ and |0⟩; the origin parity moves toward positive values.
- When vacuum weight exceeds the odd-parity weight sufficiently, the origin negativity disappears.
This illustrates how Wigner negativity is an experimentally measurable property of parity statistics, not an illegal probability.
31. Practice set
- What makes W a quasiprobability rather than ordinary probability?
- What are its position and momentum marginals?
- How is continuous-variable W measured through parity?
- What is W(0) for |1⟩ in the stated convention?
- Define negative volume.
- State Hudson’s theorem for pure CV states.
- What is special about the discrete odd-prime-power stabilizer Wigner representation?
- Define mana.
- Why does negativity often increase Monte Carlo cost?
- Why is Wigner negativity not equivalent to entanglement?
Answers
- It is normalised and reproduces quantum statistics but may take negative values.
- Integrating over p gives position probability; integrating over x gives momentum probability.
- Displace the state and measure photon-number parity.
−2/π.(∫|W|−1)/2with the appropriate phase-space measure.- A pure continuous-variable state has nonnegative W iff it is Gaussian.
- Pure nonnegative states are stabilizer states and Clifford operations act naturally on phase space.
- The logarithm of the total absolute discrete Wigner weight.
- Signed-weight sampling has variance overhead controlled by the coefficient one-norm/negativity.
- Single-mode negative states need not be entangled, while Gaussian entangled states can have positive Wigner functions.
Sources and further study
[1] Eugene Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Physical Review 40, 749 (1932). Original Wigner phase-space representation.
[2] R. L. Hudson, When is the Wigner quasi-probability density non-negative?, Reports on Mathematical Physics 6, 249–252 (1974). Pure-state Gaussian positivity theorem.
[3] David Gross, Hudson’s theorem for finite-dimensional quantum systems, Journal of Mathematical Physics 47, 122107 (2006). Discrete odd-dimensional stabilizer positivity theorem.
[4] Victor Veitch, Christopher Ferrie, David Gross and Joseph Emerson, Negative quasi-probability as a resource for quantum computation, New Journal of Physics 14, 113011 (2012).
[5] Nicolas Delfosse, Philippe Allard Guerin, Jacob Bian and Robert Raussendorf, Wigner function negativity and contextuality in quantum computation on rebits.
[6] Robert Raussendorf and colleagues, Contextuality and Wigner function negativity in qubit quantum computation.
Continue through Quantum Mathematics — Batch 20
Guide 77: Quantum Zeno Effect, Anti-Zeno Effect, Survival Probabilities and Zeno Subspaces develops measurement-constrained dynamics. Guide 78: Quantum Speed Limits, Mandelstam–Tamm, Margolus–Levitin, Bures Geometry and Control Times develops state-space speed bounds. Guide 80: Quantum Trajectories, Monte Carlo Wave Functions, Quantum Jumps and Diffusive Unravellings develops stochastic conditioned dynamics.