Strong Perfect Realism: Quantum & Compression

• Strong Perfect Realism is a framework defining exact realism with vanishing epistemic overlap in quantum systems and exact distribution matching in lossy compression.
• It refines the ψ-ontology debate by replacing full Preparation Independence with weaker independence notions, employing permutation symmetry and de Finetti approximations.
• In information theory, strong perfect realism enforces precise output matching of source distributions, impacting rate–distortion trade-offs with side information and common randomness.

Searching arXiv for the cited and closely related papers on ψ-ontology and strong realism in compression. {"query":"(Mansfield, 2014) Reality of the quantum state: Towards a stronger psi-ontology theorem", "max_results": 5} {"query":"Pusey Barrett Rudolph 2012 theorem arXiv psi-ontology preparation independence", "max_results": 10} {"query":"(Hamdi et al., 20 Jul 2025) Rate-Distortion-Perception Trade-off with Strong Realism Constraints: Role of Side Information and Common Randomness", "max_results": 5} Strong Perfect Realism is a label applied to exact realism requirements in two technically distinct settings. In quantum foundations, the strengthened $\psi$-ontology analysis around Mansfield’s “Reality of the quantum state: Towards a stronger $\psi$-ontology theorem” shows that, under the Subsystem Condition together with permutation symmetry and large symmetric ensembles, the epistemic overlap of distinct pure states is forced to vanish in the infinite-sample limit, thereby recovering full $\psi$-ontology without the full Preparation Independence assumption of Pusey–Barrett–Rudolph (Mansfield, 2014). In information theory, “strong perfect realism constraints” require reconstructed symbols to match the source law exactly, either marginally through $P_{Y^n}\equiv p_X^{\otimes n}$ or jointly through $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$, and they alter the rate–distortion–perception trade-off in the presence of side information and common randomness (Hamdi et al., 20 Jul 2025).

1. Ontological meaning of realism for the quantum state

In the ontological framework used for strengthened $\psi$-ontology, every quantum system has an ontic state $\lambda\in\Lambda$, where $\Lambda$ is a measurable space with $\sigma$-algebra $\mathcal L$. A pure quantum state $\psi$0 induces a probability measure $\psi$1 on $\psi$2, so preparing $\psi$3 picks $\psi$4 according to $\psi$5. A measurement with outcome set $\psi$6 is represented by response functions $\psi$7 satisfying

$\psi$8

The ontological model must reproduce quantum statistics: $\psi$9

Within this framework, the reality question is formulated through the overlap of ontic distributions associated with distinct pure states. The epistemic overlap of $\psi$0 and $\psi$1 is quantified by

$\psi$2

Equivalently, $\psi$3 is the minimum probability of hitting a $\psi$4 compatible with both $\psi$5 and $\psi$6 (Mansfield, 2014).

A $\psi$7-ontic theory is one in which distinct pure states have disjoint ontic supports, so $\psi$8. A $\psi$9-epistemic theory permits nonzero overlap, allowing the quantum state to be statistical in character. In this setting, realism is not merely an interpretive slogan; it is encoded as a structural property of the family $P_{Y^n}\equiv p_X^{\otimes n}$0.

2. From Preparation Independence to weaker independence notions

The Pusey–Barrett–Rudolph argument considers two systems $P_{Y^n}\equiv p_X^{\otimes n}$1 prepared in product quantum states $P_{Y^n}\equiv p_X^{\otimes n}$2 and assumes Preparation Independence: $P_{Y^n}\equiv p_X^{\otimes n}$3 for measurable $P_{Y^n}\equiv p_X^{\otimes n}$4, $P_{Y^n}\equiv p_X^{\otimes n}$5 and preparations $P_{Y^n}\equiv p_X^{\otimes n}$6. If two non-orthogonal states $P_{Y^n}\equiv p_X^{\otimes n}$7 have overlap $P_{Y^n}\equiv p_X^{\otimes n}$8, then under (PI), two independent preparation devices each choosing $P_{Y^n}\equiv p_X^{\otimes n}$9 or $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$0 produce a joint ontic state that lies in the overlap region with probability at least $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$1. Quantum theory, however, admits a conclusive-exclusion measurement on the two-system Hilbert space with outcomes

$P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$2

such that each outcome has zero probability on one corresponding product state. The PBR conclusion is that any ontological model reproducing these conclusive-exclusion statistics must have $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$3 (Mansfield, 2014).

Mansfield’s analysis treats Preparation Independence as too strong, because it rules out even classical correlations. Two weaker assumptions are introduced. The first is Independence up to Classical Correlations: one adds an auxiliary space $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$4, interpreted as a “common past,” and requires conditional factorization,

$P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$5

This is formally analogous to Bell locality with a shared hidden variable.

The second is the Subsystem Condition, a minimal causal independence requirement: $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$6 This requires only that marginals do not signal. It is presented as the minimal requirement needed to speak of “the ontic state of $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$7” in isolation, and it forbids superluminal influences during preparation (Mansfield, 2014).

3. Explicit $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$8-epistemic evasion of the PBR contradiction

Under either (CC) or (SC), the PBR contradiction no longer goes through automatically. Mansfield gives an explicit two-system ontological model for the PBR experiment that violates (PI) but satisfies both (CC) and (SC), reproduces the quantum conclusive-exclusion statistics, and retains $P_{(Y^n,Z^n)}\equiv p_{X,Z}^{\otimes n}$9 (Mansfield, 2014).

The ontic state space is

$\psi$0

with overlap region $\psi$1. The single-system distributions are chosen so that $\psi$2 is supported on $\psi$3 with $\psi$4, while $\psi$5 is supported on $\psi$6 with $\psi$7. Thus the two preparations share the ontic state $\psi$8.

For the joint distribution, one chooses any $\psi$9 with $\lambda\in\Lambda$0; the construction assigns zero probability to $\lambda\in\Lambda$1 for all four preparation pairs $\lambda\in\Lambda$2, $\lambda\in\Lambda$3, $\lambda\in\Lambda$4, and $\lambda\in\Lambda$5, while distributing weight over $\lambda\in\Lambda$6, $\lambda\in\Lambda$7, and $\lambda\in\Lambda$8 so that the single-party marginals coincide with $\lambda\in\Lambda$9 and $\Lambda$0. This verifies (SC). The same probabilities can be mediated by a three-valued $\Lambda$1, so (CC) also holds.

The response functions are defined on joint ontic states $\Lambda$2. One example is

$\Lambda$3

with analogous definitions for $\Lambda$4, $\Lambda$5, and $\Lambda$6. The resulting outcome probabilities satisfy

$\Lambda$7

and they exactly reproduce the quantum table, including

$\Lambda$8

The significance of the construction is negative but precise: conclusive-exclusion statistics alone do not force $\Lambda$9-ontology once full factorization is replaced by weaker independence assumptions.

4. Large-ensemble recovery of exact $\sigma$0-ontology

To recover a $\sigma$1-ontology result under merely the Subsystem Condition, the analysis adds permutation symmetry and passes to large ensembles. For an $\sigma$2-fold ensemble of $\sigma$3-preparation devices, the joint distribution

$\sigma$4

is assumed invariant under any permutation of the $\sigma$5 slots. This permutation symmetry is explicitly stated to be strictly weaker than (PI) or (CC) (Mansfield, 2014).

The protocol is to randomly sample $\sigma$6 subsystems from the symmetric ensemble and perform an $\sigma$7-partite conclusive-exclusion measurement, with $\sigma$8 large enough so that quantum theory admits such a measurement for the overlap angle of $\sigma$9. A finite de Finetti approximation due to Christandl–Toner is then applied: any symmetric $\mathcal L$0 satisfying (SC) can be approximated in trace distance on the $\mathcal L$1-marginal by a classically correlated mixture of product distributions,

$\mathcal L$2

with $\mathcal L$3.

Applying the classically correlated result to each product component yields an overlap bound

$\mathcal L$4

As $\mathcal L$5 for fixed $\mathcal L$6, the right-hand side tends to $\mathcal L$7, forcing $\mathcal L$8 exactly.

In the synthesis surrounding the theorem, this limiting statement is characterized as “strong perfect realism”: even allowing classical correlations or only minimal causal independence, one recovers full $\mathcal L$9-ontology in the ideal large-sample limit. The price is explicit: an auxiliary symmetry assumption and the use of a finite de Finetti bound. The physical motivation is also explicit: these assumptions are presented as milder and more physically motivated than full Preparation Independence (Mansfield, 2014).

5. Strong realism constraints in rate–distortion–perception theory

A second, formally distinct usage appears in lossy compression with side information and common randomness. The source is $\psi$00, the side information is $\psi$01, and the reconstruction is $\psi$02. A blocklength-$\psi$03 rate-$\psi$04 description is $\psi$05, and common randomness is $\psi$06. A D-code has encoder $\psi$07 and decoder $\psi$08; an E-D-code additionally allows the encoder to observe $\psi$09 (Hamdi et al., 20 Jul 2025).

The realism constraints are exact distribution-matching requirements:

Constraint	Formal definition	Interpretation
Perfect marginal realism	$\psi$10	The joint law of $\psi$11 matches exactly that of $\psi$12
Near-perfect marginal realism	$\psi$13	Marginal law converges in total variation
Perfect joint realism	$\psi$14	$\psi$15 has the same joint law as $\psi$16
Near-perfect joint realism	Equality replaced by vanishing TV	Joint law converges in total variation

Under mild uniform-integrability conditions on the distortion measure, any sequence of codes achieving near-perfect realism can be corrected at negligible cost in rate and distortion so as to meet perfect realism exactly. Thus, in this setting, “near-perfect” and “perfect” realism are operationally equivalent up to negligible correction under the stated conditions (Hamdi et al., 20 Jul 2025).

The terminology “marginal realism,” “joint realism,” and “near-perfect realism” is intrinsic to the compression formulation. Here realism does not concern ontic support or $\psi$17-epistemicity; it concerns exact agreement between output and source distributions.

6. Single-letter characterizations, Gaussian cases, and technical implications

For marginal realism with E-D-codes, the achievable region is characterized through

$\psi$18

and $\psi$19 is achievable with perfect or near-perfect marginal realism iff there exists $\psi$20 such that

$\psi$21

For marginal realism with D-codes,

$\psi$22

and every $\psi$23 in the closure of the set satisfying

$\psi$24

is D-achievable with perfect or near-perfect marginal realism. In general, the converse is open except in special cases such as the “common-part” model or large common randomness (Hamdi et al., 20 Jul 2025).

For joint realism, the E-D and D regions take a cleaner form. With

$\psi$25

the closure of achievable $\psi$26 is

$\psi$27

For D-codes under joint realism, with the additional constraints $\psi$28 and $\psi$29, the same inequalities characterize the closure of the achievable region, and in this decoder-only side-information case a full converse holds. The absence of an $\psi$30 term under joint realism is interpreted explicitly: $\psi$31 cannot serve as common randomness under joint realism.

Several corollaries sharpen the operational picture. If $\psi$32 is independent of $\psi$33, one recovers the Saldi–Linder–Yüksel region

$\psi$34

under $\psi$35 and $\psi$36. If $\psi$37 and the alphabets are finite,

$\psi$38

with $\psi$39 and $\psi$40. Side information has a “dual role” in marginal realism when $\psi$41 is available at both terminals: if $\psi$42, its effect is simply to supply $\psi$43 bits of common randomness. In the conditional-independence common-part model, where $\psi$44 and $\psi$45 is finite, the D-code region of Theorem 3 is tight (Hamdi et al., 20 Jul 2025).

The Gaussian examples make the rate penalties explicit. With no side information, $\psi$46 and $\psi$47, the minimum compression rate for any common-randomness rate $\psi$48 and distortion $\psi$49 is

$\psi$50

where $\psi$51 solves

$\psi$52

In particular,

$\psi$53

At small $\psi$54, the penalty of realism vanishes when $\psi$55. In the Wyner–Ziv Gaussian case, where $\psi$56 is bivariate normal with unit variances and correlation $\psi$57, for $\psi$58 and sufficiently large $\psi$59,

$\psi$60

Thus, when $\psi$61 is large enough, side information only at the decoder costs no extra rate versus full side information at the encoder; with no common randomness, the penalty reappears exactly as in the no-side-information case $\psi$62 (Hamdi et al., 20 Jul 2025).

All achievability proofs use random codebooks and the soft-covering lemma to force the output distribution to match the target. In marginal-realism E-D coding, one uses one codebook per $\psi$63, draws $\psi$64 i.i.d. $\psi$65, selects $\psi$66 uniformly, and generates $\psi$67. In marginal-realism D coding, a virtual message $\psi$68 of rate $\psi$69 is introduced at the encoder, the decoder guesses $\psi$70, and the same soft-covering mechanism is recovered. Joint realism uses the same architecture but requires each sub-codebook to satisfy the joint-distribution match $\psi$71.

A plausible implication of the two arXiv usages is that “strong perfect realism” functions as an umbrella phrase for exact realism constraints under weakened structural assumptions. In the quantum setting, exactness means vanishing epistemic overlap in the large-ensemble limit; in the compression setting, exactness means exact distribution matching of reconstruction and source. The shared motif is not a common mathematical object but a common demand for perfect, rather than thresholded or asymptotic-only, realism.