Negative Quantum States: Diagnostics & Resources

• Negative quantum states are defined as quantum constructs where classical nonnegative diagnostics (e.g., Wigner quasiprobability, weak values, local energy density) become negative, highlighting nonclassical behavior.
• They serve as valuable resources in quantum optics and computation by enabling non-Gaussian operations and demonstrating quantum advantage through experimental validation.
• Experimental techniques such as heralded photon production, SNAP-displacement synthesis, and compiled superconducting circuits confirm these states despite challenges in preservation under realistic noise.

Negative quantum states are classes of quantum states or state-like constructions for which a defining diagnostic takes negative values. The diagnostic depends on context: a Wigner function can become negative in continuous-variable and discrete phase-space formulations; a weak value or a post-selected conditional entropy can be negative; a renormalized stress tensor can yield negative local energy density; and non-Hermitian or transport-theoretic response objects can acquire negative occupations, negative density of states, or negative local partial density of states. Across these literatures, the phrase therefore does not denote a single invariant concept, but a family of technically distinct negativities with different operational meanings, constraints, and resource interpretations (Yoshikawa et al., 2013, Kudra et al., 2021, Lee, 2020, 0911.3597, Salek et al., 2013).

1. Scope of the term

The common structure is that a quantity which would be nonnegative in a classical, probabilistic, or equilibrium description becomes negative because the underlying quantum object is not an ordinary probability density. In phase-space quantum optics, the negative object is the Wigner quasiprobability. In post-selected quantum theory, it is a weak value or a conditional entropy derived from a non-Hermitian conditional state. In quantum field theory, it is the renormalized local energy density. In non-Hermitian many-body physics and mesoscopic scattering, the negative quantity is instead a channel-resolved occupation-like eigenvalue, a spectral function, or a local response density (Yoshikawa et al., 2013, Yokota et al., 2016, 0911.3597, Meena et al., 10 Mar 2025).

Context	Negative object	Diagnostic
CV and discrete phase space	Quasiprobability	$W(x,p)<0$ or $W_\alpha<0$
Post-selection	Weak value / conditional entropy	$\Pi_w=-1$, $S_c<0$
QFT and holography	Local energy density / modular energy	$\langle T_{00}\rangle<0$, $K_A<0$
Non-Hermitian and open systems	Occupation-like eigenvalue / DOS	$p'<0$, $A(\omega)<0$
Mesoscopic transport	Local partial DOS	$P_{\mathrm{lpd}}<0$

A recurrent misconception is to treat these negativities as interchangeable. They are not. A negative Wigner function is not a negative-norm state; a negative weak value is not a negative probability; a negative local partial density of states is not a negative total density of states; and negative local energy density does not imply negative total energy. The relevant object, normalization, and operational test must therefore be specified case by case (Yokota et al., 2016, 0911.3597, Meena et al., 10 Mar 2025).

2. Phase-space negativity and non-Gaussian resource states

For a single continuous-variable mode with density operator $\hat{\rho}$, the Wigner function can be written as

$W_\alpha<0$0

In the equivalent convention used in the supplemental material of one of the optical experiments,

$W_\alpha<0$1

For Fock states,

$W_\alpha<0$2

so a single photon has the central value

$W_\alpha<0$3

For the lossy mixture $W_\alpha<0$4,

$W_\alpha<0$5

which becomes negative if $W_\alpha<0$6 (Yoshikawa et al., 2013).

This negativity is used as a witness of nonclassicality and, in the continuous-variable setting, as a resource for non-Gaussian operations, universal continuous-variable quantum computation, and magic-state protocols. The cited works explicitly connect Wigner negativity to the fact that positivity of the Wigner function permits efficient classical simulation, while negativity enables quantum advantage. In that sense, negative phase-space states are not merely unusual visualizations of tomography; they are resource states in a precise computational sense (Kudra et al., 2021).

The same logic extends to finite-dimensional discrete phase space. For Hilbert-space dimension $W_\alpha<0$7, the discrete Wigner function is defined by

$W_\alpha<0$8

with phase-point operators

$W_\alpha<0$9

In the two-qubit case, negative quantum states are defined as normalized eigenvectors of $\Pi_w=-1$0 associated with negative eigenvalues. Among the $\Pi_w=-1$1 quantum nets, the phase-point operators fall into three spectral types, including $\Pi_w=-1$2; the most negative eigenvector is denoted $\Pi_w=-1$3 (Lalita et al., 2024).

The experimental literature realizes such states in both optical and microwave architectures. Optical work has emphasized single-photon and photon-subtracted states, while circuit-QED work has demonstrated Fock, binomial, odd cat, GKP, and cubic phase states, all verified by Wigner tomography. A telecom implementation at $\Pi_w=-1$4 reconstructed

$\Pi_w=-1$5

and interpreted the result as a homodyne-certified negative Wigner function at a wavelength directly relevant for fiber-based quantum communication (Baune et al., 2017).

3. Preparation, storage, release, and protection

The main preparation routes in the cited literature are heralding, photon subtraction, SNAP–displacement synthesis, and circuit compilation for discrete-Wigner negative states. They differ in physical platform, but all aim to preserve the negativity-defining diagnostic under realistic loss and control constraints.

Platform	Method	Reported result
All-optical two-cavity source	Heralded creation in memory cavity, on-demand release by shutter cavity	$\Pi_w=-1$6 at 0, 100, 200 ns; $\Pi_w=-1$7 at 300 ns
3D microwave cavity	$\Pi_w=-1$8–$\Pi_w=-1$9 SNAPs plus $S_c<0$0–$S_c<0$1 displacements	Fidelities: $S_c<0$2 for $S_c<0$3, $S_c<0$4 binomial, $S_c<0$5 odd cat, $S_c<0$6 GKP, $S_c<0$7 cubic
IBM superconducting hardware	Compiled $S_c<0$8 circuits and tomography	Depth 13 for $S_c<0$9; depth 4 for $\langle T_{00}\rangle<0$0; mitigated fidelities $\langle T_{00}\rangle<0$1–$\langle T_{00}\rangle<0$2

The all-optical architecture combines a memory cavity functioning as a below-threshold NOPO with a shutter cavity containing an RTP electro-optic modulator. Idler detection heralds a signal photon that remains stored until the shutter resonance is shifted by a high-voltage pulse. Balanced homodyne tomography, principal-component mode extraction, and maximum-likelihood reconstruction yielded single-photon fractions of $\langle T_{00}\rangle<0$3, $\langle T_{00}\rangle<0$4, $\langle T_{00}\rangle<0$5, and $\langle T_{00}\rangle<0$6 for storage times of 0, 100, 200, and 300 ns beyond an intrinsic 150 ns delay, with corresponding central Wigner values of $\langle T_{00}\rangle<0$7, $\langle T_{00}\rangle<0$8, $\langle T_{00}\rangle<0$9, and $K_A<0$0. The inferred storage lifetime was on the microsecond scale, with fits giving $K_A<0$1 in a time-shifted analysis and $K_A<0$2 in an unshifted fit (Yoshikawa et al., 2013).

In the microwave setting, universal control is implemented by interleaving displacements

$K_A<0$3

with selective number-dependent arbitrary phase gates

$K_A<0$4

The preparation is optimized in two stages: gradient descent over displacement amplitudes and SNAP phase vectors using a cost

$K_A<0$5

followed by pulse-envelope optimization of the SNAP drive. The optimized SNAP implementation reduces the gate duration from approximately $K_A<0$6 to $K_A<0$7 and is reported to be approximately $K_A<0$8 less sensitive to $K_A<0$9 miscalibration and approximately $p'<0$0 less sensitive to qubit-frequency drift than standard comb SNAPs (Kudra et al., 2021).

Discrete-Wigner negative states have also been compiled directly to superconducting hardware. On ibm_brisbane, the synthesized circuits used only $p'<0$1, achieved simulator synthesis loss $p'<0$2, and were validated by maximum-likelihood state tomography with 8192 shots. Reported hardware fidelities were approximately $p'<0$3–$p'<0$4 without readout mitigation and approximately $p'<0$5–$p'<0$6 with mitigation, despite circuit depths of 13 for most negative states (Lalita et al., 2024).

Protection protocols treat these states as fragile resources under non-Markovian noise. With local weak measurement

$p'<0$7

and quantum measurement reversal

$p'<0$8

the final state is

$p'<0$9

For the parameter choices optimized at $A(\omega)<0$0, the pair $A(\omega)<0$1 for $A(\omega)<0$2 gave the strongest reported improvement under non-Markovian amplitude damping and random telegraph noise, including higher concurrence, steering, maximal teleportation fidelity, and near-zero fidelity deviation in the RTN case (Lalita et al., 2023).

4. Post-selection, weak values, and conditional entropies

In pre- and post-selected quantum mechanics, negativity appears in quantities that are neither eigenvalues nor ordinary probabilities. For an operator $A(\omega)<0$3 with pre-selected state $A(\omega)<0$4 and post-selected state $A(\omega)<0$5, the weak value is

$A(\omega)<0$6

For a projector $A(\omega)<0$7, $A(\omega)<0$8 can lie outside the interval $A(\omega)<0$9. In the three-path example with

$P_{\mathrm{lpd}}<0$0

the projector weak values are

$P_{\mathrm{lpd}}<0$1

The cited experiment interprets $P_{\mathrm{lpd}}<0$2 as the operational counterpart of probability one, because the polarization-pointer shift obeys

$P_{\mathrm{lpd}}<0$3

in the weak-coupling regime, so $P_{\mathrm{lpd}}<0$4 produces a shift symmetric in magnitude and opposite in sign to $P_{\mathrm{lpd}}<0$5. The paper explicitly distinguishes this from a negative-norm state (Yokota et al., 2016).

Post-selection also leads to a conditional-state formalism. For pure pre-selection $P_{\mathrm{lpd}}<0$6 and post-selection $P_{\mathrm{lpd}}<0$7 with $P_{\mathrm{lpd}}<0$8, the generalized density operator is

$P_{\mathrm{lpd}}<0$9

Its associated conditional entropy is defined by

$\hat{\rho}$0

which simplifies to

$\hat{\rho}$1

The averaged conditional entropy is

$\hat{\rho}$2

Because $\hat{\rho}$3 for nontrivial post-selection, both $\hat{\rho}$4 and $\hat{\rho}$5 can be negative. In the three-box example above, $\hat{\rho}$6, so $\hat{\rho}$7 (Salek et al., 2013).

These two constructions clarify a frequent confusion. Negative weak values and negative post-selected conditional entropy do not assert that a state has negative probability or negative norm. They quantify how conditioning on a final state reshapes the algebra of observables and the information content assigned to the post-selected ensemble (Yokota et al., 2016, Salek et al., 2013).

5. Negative energy densities, quantum inequalities, and holographic descriptions

In quantum field theory, negativity refers to the renormalized expectation value of the stress tensor. After vacuum subtraction, the local energy density

$\hat{\rho}$8

can be negative. Standard examples include the Casimir effect and non-classical states such as squeezed vacua. For perfect parallel plates separated by $\hat{\rho}$9, the cited review gives

$W_\alpha<0$00

Quantum energy inequalities constrain such sub-vacuum effects by bounding sampled averages. In $W_\alpha<0$01-dimensional Minkowski space,

$W_\alpha<0$02

and in $W_\alpha<0$03 dimensions the optimal constant for a massless scalar is

$W_\alpha<0$04

For Gaussian sampling, $W_\alpha<0$05 (0911.3597).

Along inertial worldlines, these bounds imply the averaged weak energy condition. Along accelerated worldlines, the restrictions can be much weaker. For a particle in sinusoidal motion through a single-mode squeezed vacuum, the integrated energy density

$W_\alpha<0$06

can become arbitrarily negative at a constant average rate. In the nonrelativistic regime, the cited analysis gives

$W_\alpha<0$07

for motion perpendicular to the wavevector and

$W_\alpha<0$08

for motion parallel to it, together with a Raychaudhuri-equation analysis showing possible net defocussing of a congruence of such worldlines (Ford et al., 2013).

Holography supplies two complementary descriptions of negative-energy states. In Lifshitz holography, gravitational-wave perturbations in the bulk reproduce the stress tensor of squeezed vacuum states on the boundary, with the principal negative contribution at first order and a uniform positive “interest” term at second order. For $W_\alpha<0$09, the null-averaged stress tensor is consistent with ANEC and with a weaker near-null QEI-type bound in strongly coupled field theory than in the corresponding free relativistic theory (Lee et al., 2016). In AdS/CFT for ball-shaped regions, the modular Hamiltonian

$W_\alpha<0$10

has a minimum eigenvalue

$W_\alpha<0$11

and the modular vacua saturating this negative bound are identified holographically with zero-temperature hyperbolic black holes. In two-dimensional CFTs, the same work reports the closed form

$W_\alpha<0$12

(Rosso, 2018).

The literature also contains explicit claims of violations of spatial or temporal quantum inequalities for a massless scalar in $W_\alpha<0$13 dimensions, based on double-delta potentials, abrupt removal protocols, and mode-regularized stress tensors. Those papers present a negative constant energy density in a finite interval and argue that Lorentzian smearing violates the quoted bounds. They also note unresolved issues themselves, including the applicability of standard QEI assumptions, the state class after potential removal, and the dependence on renormalization procedure (Solomon, 2010, Solomon, 2010). The status of these claims is therefore tied to those assumptions, rather than being a settled replacement of the standard QEI framework.

6. Non-Hermitian, spectral, relativistic, and other specialized usages

In non-Hermitian many-body systems, negativity can arise in the spectrum of a truncated correlation matrix rather than in a quasiprobability or stress tensor. Exceptional boundary states emerge at exceptional points where the eigenspace is defective. For the restricted projector $W_\alpha<0$14, the relevant eigenvalues satisfy

$W_\alpha<0$15

When $W_\alpha<0$16 is large, one finds $W_\alpha<0$17 and $W_\alpha<0$18, interpreted as anomalously large and negative occupation probabilities. The corresponding entanglement contribution

$W_\alpha<0$19

is real and negative. These states are explicitly distinguished from both topological edge states and non-Hermitian skin states (Lee, 2020).

Open driven-dissipative systems exhibit a different spectral negativity. For a nonlinear van der Pol resonator, the retarded Green function

$W_\alpha<0$20

defines a photonic spectral function

$W_\alpha<0$21

In equilibrium, $W_\alpha<0$22 for $W_\alpha<0$23. In the driven-dissipative case, the cited work finds a negative density of states, $W_\alpha<0$24 at positive frequency, through two mechanisms: population inversion plus lifetime-broadened resonances, and dissipative eigenmode mixing that generates complex residues in the generalized Lehmann representation. The same analysis defines a frequency-dependent effective temperature by

$W_\alpha<0$25

so $W_\alpha<0$26 at $W_\alpha<0$27 implies $W_\alpha<0$28 (Scarlatella et al., 2018).

Mesoscopic transport supplies yet another response-based negativity. The local partial density of states is defined by

$W_\alpha<0$29

Near Fano resonances, the cited work shows that the Argand-loop topology of the transmission amplitude forces this quantity to change sign, so a weak local perturbation can increase rather than suppress coherent transmission: $W_\alpha<0$30 Negative local partial density of states is therefore not a negative total DOS, but a phase-sensitive channel-resolved response quantity (Meena et al., 10 Mar 2025).

Relativistic quantum theory contains negative-energy states in several senses. In a multivector reformulation of the Dirac equation, the sign of the canonical energy density for the massless field is controlled by the relative phase between even- and odd-grade components, while for the massive field it is controlled by spatial parity; the second-order equations inherit the same sign structure. In the two-photon Kapitza–Dirac effect, time-dependent perturbation theory shows that virtual negative-energy intermediate states can dominate the diffraction amplitude, even though no real pair creation occurs because the net energy transfer is zero (Papaioannou, 2020, Wang et al., 12 Jan 2026).

Several domain-specific usages are terminologically narrower. In self-assembled InAs/GaAs quantum dots, “negative charge states” denote states with one excess electron, especially the negatively charged trion $W_\alpha<0$31; the negativity there is electrical charge, not quasiprobability or energy negativity (Benny et al., 2012). In the infinite square well, negative-energy solutions are square-integrable on the finite interval and must be excluded by appropriate boundary conditions; Dirichlet, Neumann, mixed Dirichlet–Neumann, and Robin conditions with nonnegative parameters all remove the unphysical negative-energy sector (Lin, 2017).

Taken together, these literatures show that negativity in quantum theory is structurally plural. Sometimes it is a resource, sometimes a diagnostic, sometimes a response function, and sometimes a terminological marker for charge or spectral sign. What unifies these cases is not a single definition, but the repeated appearance of quantum objects that cease to behave like classical positive distributions once interference, conditioning, renormalization, or non-Hermiticity becomes essential.