Quantum Uncommon Information

• Quantum uncommon information is defined as quantum correlations or message content that remain inaccessible under specific measurement restrictions, highlighting a divergence between what is present and what can be jointly revealed.
• It encompasses distinct operational quantities—such as the locking gap, locally inaccessible information measured by discord, and the entanglement cost in state exchange—each linked to different tasks.
• This concept underpins practical applications in quantum cryptography, error correction, and information processing by revealing how quantum structure obfuscates classical accessibility.

Quantum uncommon information denotes quantum correlations or message content that remain unavailable under a specified restricted mode of access. In the literature, the term is used in several distinct but related senses: the gap between quantum mutual information and classically accessible mutual information, the locally inaccessible part of mutual information quantified by quantum discord, the entanglement cost of quantum state exchange, the unique component in quantum partial information decomposition, and the information excluded by measurement complementarity or by non-shareability constraints. Across these settings, the recurring theme is that quantum structure makes “what is present” and “what can be jointly revealed” sharply different notions (Dupuis et al., 2010, Fanchini et al., 2011, Lee et al., 2024, Enk, 2023, Ericson et al., 5 Jun 2026).

1. Terminological scope and formal objects

The literature does not treat quantum uncommon information as a single universally fixed scalar. Instead, it introduces several operationally distinct quantities, each tied to a different task or restriction. Some definitions are bipartite and correlation-theoretic, some are communication-theoretic, and some are multipartite decompositions of information into unique, redundant, and synergistic parts.

Framework	Representative quantity	Operational content
Locking / accessibility gap	$U(A{:}B):=I(A{:}B)-I_{\mathrm{acc}}(A{:}B)$	Correlations hidden from all classical measurements
Locally inaccessible information	$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$	Mutual information not accessible by optimized local POVMs
Quantum state exchange	$\Upsilon(A{:}B)$	Minimal net ebits to exchange $A$ and $B$ under LOCC
Quantum PID	$U_A(T;A\setminus B)$, $U_1(T;S_1\setminus S_2)$	Information about a target present in one source but not the other
Complementary information	$r \prec q \prec t$	Post-test information restricted by prior incompatible measurement information

A useful starting point is the contrast with the classical quantity

$$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$

which is the minimum total number of bits two parties must exchange to reconstruct each other’s messages. The quantum state-exchange literature takes this operational idea seriously but shows that no direct closed-form quantum analogue is known; in particular, the naive expression $S(AB)-I(A{:}B)=S(A|B)+S(B|A)$ fails as an operational characterization because quantum conditional entropies can be negative (Lee et al., 2024, Ji et al., 8 Jul 2025). A separate line of work instead identifies “uncommonness” with the part of mutual information that is locally or classically inaccessible, while another identifies it with unique information in a quantum PID or with the unavoidable information loss imposed by incompatible measurements (Fanchini et al., 2011, Xiao et al., 2019).

2. Locking classical information and the accessibility gap

A central meaning of quantum uncommon information is the discrepancy between total quantum correlations and the correlations that any pair of classical measurements can reveal. For a bipartite state $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$0,

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$1

is the quantum mutual information, while

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$2

is the accessible classical mutual information, optimized over measurement superoperators. The quantity

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$3

therefore measures correlations present in the state but not retrievable by local classical readout. The locking framework shows that this gap can be made arbitrarily large (Dupuis et al., 2010).

The strengthened locking definition is formulated in trace distance rather than directly in accessible information. With a classical message register $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$4, entanglement registers $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$5, and ciphertext register $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$6, an encoding is called an $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$7-locking scheme if for every measurement superoperator $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$8,

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$9

Operationally, no measurement on $\Upsilon(A{:}B)$0 produces statistics differing by more than $\Upsilon(A{:}B)$1 from independence from $\Upsilon(A{:}B)$2. This is strictly stronger than a small-$\Upsilon(A{:}B)$3 condition: if the trace-distance condition holds for all measurements, then

$\Upsilon(A{:}B)$4

with $\Upsilon(A{:}B)$5. Thus arbitrarily small accessible information follows from sufficiently small $\Upsilon(A{:}B)$6 (Dupuis et al., 2010).

The main quantitative phenomenon is that removing only a logarithmic-size subsystem $\Upsilon(A{:}B)$7 from one half of a perfectly correlated message can make all remaining measurement outcomes essentially independent of the message, even though the full quantum system still contains a recoverable high-fidelity copy. In the general entangled setting, with $\Upsilon(A{:}B)$8, $\Upsilon(A{:}B)$9, $A$0, and $A$1, Haar-random encodings give an $A$2-locking scheme with probability at least $A$3 provided

$A$4

and

$A$5

In the uniform-message, maximally entangled case, the required key is only of order $A$6 plus constants. Corresponding no-entanglement and projective-measurement variants exhibit the same “small key” structure (Dupuis et al., 2010).

The same framework proves a narrow transition between the locked regime and the decodable regime. For the transmission scenario, there exists a decoding measurement on $A$7 whose average error obeys

$A$8

so once $A$9 is sufficiently larger than $B$0, the message becomes recoverable. Comparing the locking and decoding bounds yields only a logarithmic-size window, in system size and $B$1, between “all measurements look independent” and “high-fidelity decoding exists.” This makes the locking effect generic under Haar-random unitary encodings rather than a special artifact of tailored basis constructions (Dupuis et al., 2010).

The framework has both physical and cryptographic significance. Because random subsystem removal generically produces locking, measurement-based diagnostics can severely underestimate correlations in high-dimensional dynamics, a point explicitly connected to statistical mechanics and black hole physics. Cryptographically, the paper constructs a QKD variant that is “secure” under accessible information but catastrophically insecure under side-information leakage: revealing only $B$2 key bits unlocks the rest of the key. The example shows that $B$3-based secrecy is not composable (Dupuis et al., 2010).

3. Locally inaccessible information, discord, and flow

A second major usage identifies quantum uncommon information with the locally inaccessible part of mutual information. For a bipartite state $B$4,

$B$5

is decomposed into a locally accessible part and a locally inaccessible part. If $B$6 is measured to learn about $B$7,

$B$8

and the corresponding discord is

$B$9

The same construction with $U_A(T;A\setminus B)$0 measured defines $U_A(T;A\setminus B)$1. In this framework, the locally inaccessible information is exactly the discord, and the asymmetry $U_A(T;A\setminus B)$2 records which local measurement is more informative (Fanchini et al., 2011).

Two derived quantities sharpen that asymmetry: $U_A(T;A\setminus B)$3 The first is the average LII, while the second is the balance of LII. For pure bipartite states, discord is symmetric and equals entanglement of formation: $U_A(T;A\setminus B)$4 For mixed $U_A(T;A\setminus B)$5, with purifier $U_A(T;A\setminus B)$6, the relation becomes

$U_A(T;A\setminus B)$7

so entanglement is the average LII internal to $U_A(T;A\setminus B)$8 after subtracting the LII imbalance that the purifier has with $U_A(T;A\setminus B)$9 and $U_1(T;S_1\setminus S_2)$0 separately (Fanchini et al., 2011).

The same formalism gives an operational reading of negative conditional entropy. For pure $U_1(T;S_1\setminus S_2)$1,

$U_1(T;S_1\setminus S_2)$2

and for mixed $U_1(T;S_1\setminus S_2)$3 purified by $U_1(T;S_1\setminus S_2)$4,

$U_1(T;S_1\setminus S_2)$5

Negative conditional entropy therefore arises when the locally inaccessible information between $U_1(T;S_1\setminus S_2)$6 and $U_1(T;S_1\setminus S_2)$7 exceeds that between $U_1(T;S_1\setminus S_2)$8 and the environment. This is directly tied to the state-merging interpretation of conditional entropy (Fanchini et al., 2011).

The paper further defines clockwise and counterclockwise LII flows,

$U_1(T;S_1\setminus S_2)$9

and proves that for pure $r \prec q \prec t$0,

$r \prec q \prec t$1

together with the cyclic balance identity

$r \prec q \prec t$2

This makes “uncommonness” a conserved balance-and-flow structure rather than only a static difference of entropies (Fanchini et al., 2011).

The LII framework is also used to explain entanglement sudden death. For two qubits coupled to independent amplitude-damping reservoirs, entanglement disappears at the first parameter value $r \prec q \prec t$3 satisfying

$r \prec q \prec t$4

where $r \prec q \prec t$5. The environment’s LII imbalance then exactly compensates the average LII within $r \prec q \prec t$6. This does not redefine uncommon information, but it shows how discord-type uncommon information governs the redistribution of correlations under open-system dynamics (Fanchini et al., 2011).

4. Quantum state exchange and the entanglement cost interpretation

A third, operationally very different meaning takes uncommon information to be the minimum quantum resource required for two parties to exchange their shares of a correlated quantum state. If Alice and Bob hold $r \prec q \prec t$7 and $r \prec q \prec t$8 of a tripartite pure state $r \prec q \prec t$9, the task is to transform $$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$0 into $$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$1 by LOCC with free classical communication and preshared pure entanglement, preserving all correlations with the reference. In the asymptotic iid setting, if $$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$2 and $$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$3 are the Schmidt ranks of the initial and final entanglement resources, then

$$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$4

is the quantum uncommon information, or equivalently the asymptotic entanglement cost of quantum state exchange (Ji et al., 8 Jul 2025).

Known general bounds are

$$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$5

The upper bound is the merge-and-send strategy, while the lower bound follows from entanglement monotonicity across the exchange. The improved-bounds construction refines both sides. On the achievability side, a common subspace $$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$6 can be factored out because it is invariant under exchange, and after stretching the state to isolate the uncommon component one obtains

$$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$7

On the converse side, a referee-assisted exchange with reversible decomposition into EPR- and GHZ-type blocks yields

$$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$8

with the hierarchy

$$U(X,Y)=H(X|Y)+H(Y|X)=H(X,Y)-I(X;Y)=2H(X,Y)-H(X)-H(Y),$$9

These results are asymptotic iid and do not supply a closed-form general expression for $S(AB)-I(A{:}B)=S(A|B)+S(B|A)$0 (Lee et al., 2024).

The operational distinction from the classical case is sharp. A naive quantum analogue of the classical identity,

$S(AB)-I(A{:}B)=S(A|B)+S(B|A)$1

fails because conditional entropies can be negative, regularization over $S(AB)-I(A{:}B)=S(A|B)+S(B|A)$2 matters, and LOCC constraints prohibit the naive iteration that would otherwise exploit negative costs. The special cases emphasize this. For pure bipartite states $S(AB)-I(A{:}B)=S(A|B)+S(B|A)$3, one has $S(AB)-I(A{:}B)=S(A|B)+S(B|A)$4 and $S(AB)-I(A{:}B)=S(A|B)+S(B|A)$5, so the bounds collapse to

$S(AB)-I(A{:}B)=S(A|B)+S(B|A)$6

hence $S(AB)-I(A{:}B)=S(A|B)+S(B|A)$7. For product states, the bounds remain

$S(AB)-I(A{:}B)=S(A|B)+S(B|A)$8

and the common-subspace improvement can be absent (Ji et al., 8 Jul 2025, Lee et al., 2024).

Because exact evaluation is generally unavailable, variational estimation has been proposed via the quantum Donsker–Varadhan representation

$S(AB)-I(A{:}B)=S(A|B)+S(B|A)$9

together with

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$00

For entropy estimation, the implemented surrogate is

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$01

with a rank-$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$02 parameterization

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$03

This produces estimates of $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$04, $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$05, $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$06, and decomposition-based lower bounds without full tomography. In reported numerical experiments on random states of 4, 6, and 8 qubits, the loose upper bound stabilized after approximately $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$07 steps for 4 qubits and approximately $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$08 steps for 6 and 8 qubits, while the loose lower bound stabilized after approximately $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$09 steps (Ji et al., 8 Jul 2025).

5. Unique information, redundancy, and synergy

Another line of work identifies quantum uncommon information with unique information in a quantum partial information decomposition. In a capacity-based construction, one considers a tripartite state $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$10 with a qubit target $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$11 and defines superdense-coding capacities

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$12

together with analogous $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$13 and $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$14. From corrected one-bit unique capacities one obtains nonnegative components

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$15

For a qubit target, these induce a mutual-information PID

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$16

Within this framework, unique information has a direct role in quantum error correction: if a subset $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$17 of encoding qubits is erasure-correctable, then its unique information about the logical degree of freedom vanishes, $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$18. Synergy is positive when a logical operator is supported on $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$19 but not on $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$20 or $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$21 separately, and large redundant information underlies the quantum Darwinism mechanism in which many environment fragments carry the same information about the system (Ericson et al., 5 Jun 2026).

The stabilizer-code version makes this especially explicit. If $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$22 denotes the number of independent logical operator cosets supported on subset $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$23, then for the encoded $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$24–logical-qubit state,

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$25

so $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$26 for $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$27. Correctable subsets have $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$28 and therefore carry no unique information. In the 5-qubit perfect code, for $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$29, both $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$30 and $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$31 are correctable but $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$32 supports both logicals, giving $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$33. In the 3-qubit repetition code, a strict subset has only the baseline capacity, while the union of two single-qubit subsets produces a synergistic excess $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$34 (Ericson et al., 5 Jun 2026).

A distinct, measurement-free QPID defines unique information directly from operator-valued conditional states. With

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$35

and similarly $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$36, one forms

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$37

then

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$38

The unique-information atoms are

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$39

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$40

Redundancy and synergy are then fixed by

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$41

In this formulation, unique information is nonnegative by construction, while $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$42 and $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$43 need not be (Enk, 2023).

This operator-based QPID is designed to expose a feature invisible to tri-information

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$44

The paper’s “triadic” and “dyadic” quantized examples both satisfy

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$45

yet have radically different decompositions. In the triadic case, $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$46, so $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$47, $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$48, and $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$49. In the dyadic case, $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$50, so $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$51, $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$52, and $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$53. The same logic is applied to scrambling, where unique information provides a finer description than tri-information because it records asymmetric localization of target information between subsystems (Enk, 2023).

6. Complementarity, copying, non-shareability, and open problems

Measurement incompatibility supplies another precise notion of uncommon information. In the complementary information principle, a pre-test of measurement $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$54 yields distribution $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$55, thereby restricting the compatible states to

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$56

and the feasible post-test distributions for an incompatible measurement $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$57 form

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$58

For every fixed $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$59, there exist unique optimal majorization bounds $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$60 and $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$61 such that

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$62

These bounds are computed by semidefinite programs over cumulative sums of post-test probabilities and are optimal because the majorization lattice is complete. In this setting, uncommon information is the part of post-test information that cannot be jointly sharpened once pre-test information has been fixed; the framework yields universal outer approximations to uncertainty regions and state-independent limits for all joint uncertainty measures monotone under doubly stochastic maps (Xiao et al., 2019).

A different manifestation appears in multi-copy information extraction. There exist two quantum carriers $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$63 and $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$64, both parameterized by the same classical random variable, such that every single-copy measure in a broad class ranks $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$65 at least as informative as $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$66, yet two copies reverse the ordering for averaged fidelity. The single-copy simulation is mediated by a unital positive statistical morphism $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$67 with

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$68

For the explicit construction,

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$69

while for two copies

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$70

so $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$71 for

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$72

The mechanism is that $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$73 is positive but not completely positive, so $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$74 need not preserve positivity; entangled two-copy POVMs can therefore extract correlations invisible at the single-copy level. This is called strongly non-quantitative information and makes precise the idea that quantum carriers can hide classical information that only joint processing reveals (Miyazaki, 2020).

Non-shareability gives yet another operational incarnation. In uncloneable encryption of a classical bit, a QECM scheme encrypts $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$75 into

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$76

on an $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$77-qubit register, then conjugates by a uniformly random unitary $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$78 from the Clifford 2-design. Even if two non-communicating adversaries both receive the decryption key after a cloning attack, their optimal joint success probability is bounded by

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$79

The proof combines one-shot decoupling, smooth min-entropy, quantum de Finetti reduction, an AEP step, and the strong-subadditivity inequality

$$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$80

In this setting, quantum uncommon information is information about the bit that cannot be made common to two separated parties, even with the key, because monogamy of entanglement excludes simultaneous high correlation with both adversaries (Bhattacharyya et al., 9 Mar 2026).

These formulations are not interchangeable, and several open problems remain. For quantum state exchange, no analytical closed form for $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$81 is known, no general method is given to detect maximal common subspaces, and general reversible decompositions tightening the converse are unavailable. For variational estimation, the entropy-specific construction avoids direct Gibbs-state preparation but does not directly generalize to arbitrary $$\delta_{AB}^{\leftarrow}=I(A{:}B)-J_{AB}^{\leftarrow}$$82 in the full DV objective. For strongly non-quantitative carriers, the persistence of the reversal beyond the exhibited two-copy examples and for fully optimized accessible information remains open. These limitations indicate that “quantum uncommon information” is best understood as a family of task-dependent obstructions to classicalization, localization, or shareability rather than as a single settled invariant (Lee et al., 2024, Ji et al., 8 Jul 2025, Miyazaki, 2020).