Entanglement without a secret key

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Family 272 · Mathematical physics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Entanglement is a resource, but a resource for which task? Two laboratories may share a state that cannot be prepared by mixing independent local states. That fact alone does not immediately tell them how to produce identical random bits hidden from an eavesdropper.

Secret-key distillation asks whether many copies of a noisy shared state can be converted into such private bits using local quantum operations and public discussion. Public messages may coordinate the laboratories, but the eavesdropper hears them too. The central question is whether every entangled state contains some recoverable secrecy.

Family 272 reports a negative answer under its specified local-instrument protocol model. It constructs an entangled state on two ten-dimensional systems and claims that no number of copies can even approximate one ideal secret bit beyond a fixed error threshold. The same construction also yields counterexamples to conjectures about composing quantum maps that preserve positivity after partial transposition.

These conclusions distinguish three tasks: possessing entanglement, distilling pure entangled pairs, and distilling classical secret key. Earlier work separated the second from the third. This manuscript claims a separation between the first and the third.

2. Background and essential definitions

States, entanglement and Eve

A quantum state is a positive semidefinite matrix $\rho$ of trace one. For laboratories $A$ and $B$, it acts on a tensor product $\mathcal H_A\otimes\mathcal H_B$. It is separable if it can be written

$$\rho=\sum_j p_j\rho_A^{(j)}\otimes\rho_B^{(j)}, \qquad p_j\geq0,\quad\sum_jp_j=1.$$

Otherwise it is entangled. A product vector is a vector $a\otimes b$. Every nonzero separable positive matrix has at least one nonzero product vector in its range, a useful geometric obstruction to separability.

A purification of $\rho_{AB}$ is a pure state on a larger system $ABE$ whose partial trace over $E$ equals $\rho_{AB}$. Giving Eve the purifying system models all information compatible with the input state, rather than assuming that its apparent randomness is private.

An ideal shared secret bit has joint state

$$\gamma_{K_AK_BE'}=\frac12\sum_{i=0}^{1}|i,i\rangle\langle i,i|\otimes\sigma_{E'}.$$

The registers $K_A,K_B$ agree, each bit is uniform, and Eve’s state $\sigma_{E'}$ does not depend on the bit. The enlarged register $E'$ includes her initial purification and the entire public transcript.

The trace norm is $\|X\|_1=\operatorname{Tr}\sqrt{X^*X}$. The conventional trace distance between states is half the trace norm of their difference. This factor of two matters for the manuscript’s numerical bound.

Positive maps and channels

A linear map $F:M_a(\mathbb C)\to M_b(\mathbb C)$ is completely positive, or CP, if $\operatorname{id}_k\otimes F$ preserves positive matrices for every ancillary dimension $k$. A trace-preserving CP map is a quantum channel.

Coordinate transposition is denoted $T$. A CP map is called PPT here when $T\circ F$ is also CP. Equivalently, its Choi matrix

$$J(F)=\sum_{i,j=1}^{a}|i\rangle\langle j|\otimes F(|i\rangle\langle j|)$$

is positive and remains positive after transposition of one tensor factor. A map is entanglement breaking if applying it to part of any bipartite positive input always produces a separable output. This is equivalent to separability of $J(F)$, and makes sense even without trace preservation.

3. Development history and previous partial results

Classical key agreement established how public discussion and privacy amplification can turn suitable correlated data into secrecy. Entanglement purification then provided a quantum route: nearly pure shared entangled pairs are nearly independent of a purifying adversary.

The discovery of bound entanglement showed that some entangled states cannot yield pure entangled pairs by local operations and classical communication. In particular, positive-partial-transpose entangled states have zero distillable entanglement. But Horodecki, Horodecki, Horodecki and Oppenheim showed that some such states still yield secret key. Their private-state framework explains how auxiliary quantum systems can shield a classical secret even when singlet distillation is impossible.

The remaining converse was harder: could an entangled state have no distillable key at all? The manuscript traces the explicit question to the 2005 “Secret key from all entangled states” entry in Krüger and Werner’s problem collection. Earlier distance bounds between PPT states and private bits depended on the size of an auxiliary shield. A protocol can enlarge that shield as more copies become available, so those bounds did not give the copy-independent obstruction claimed here.

A related classical development is Pauwels, Gisin and Renner’s Bipartite Bound Information Exists, cited by the manuscript in its September 2026 revision. It separates classical secrecy formation cost from asymptotic secret-key rate. The quantum input and purifying adversary in the present family are different. The same cited revision also obtains positive quantum key for a particular two-qutrit family, so those states are not the zero-key example used here.

On the map side, the PPT-composition conjecture asked whether composing two PPT maps must destroy entanglement. Christandl, Müller-Hermes and Wolf proved affirmative cases, including dimension three and Gaussian channels, and developed equivalent formulations. Results on eventual entanglement breaking under repeated iteration addressed another timescale. Failure at the second iterate is compatible with eventual loss after a map-dependent number of iterations.

4. What the new manuscript claims

An entangled state with a uniform secrecy obstruction

The manuscript specifies a density operator $\rho$ on $\mathbb C^{10}\otimes\mathbb C^{10}$ whose range contains no nonzero product vector. It claims that every admitted completed bit-output protocol on every $\rho^{\otimes n}$, $n\geq1$, obeys

$$\left\|\tau_{K_AK_BE'}- \frac12\sum_{i=0}^{1}|i,i\rangle\langle i,i|\otimes\sigma_{E'}\right\|_1 \geq\frac15$$

for every state $\sigma_{E'}$. In conventional trace distance, the lower bound is $1/10$. Thus the error cannot approach zero even for one bit; in particular the distillable key rate $K_D(\rho)$ is zero.

The protocol permits joint processing of all copies within each laboratory, local auxiliary systems and private randomness, and unlimited two-way authenticated public communication. The supplied input is the only shared private resource. Any initial shared randomness is public, independent of input and reference, and recorded for Eve. Conditional on it, the two private auxiliary resources are independent of each other and of the input-reference state.

Definition 4.1 allows ordinary finite protocols with finite records, with the number of rounds growing with input length, as well as specified countable processes. Its measurable histories, local readouts and any aggregated subroutines must define compatible quantum operations on arbitrary inputs with untouched finite references. A completed protocol produces the required outputs almost surely with normalized joint output. Eve retains messages, speaker choices and stopping information. These conditions are part of the theorem’s scope, not optional technical decoration.

The manuscript separately extends the bound to conditioning on any positive-probability event in the complete public transcript. It also excludes approaching an ideal accepted bit with vanishing abort probability, and treats trace-norm limits of outputs when Eve’s full purification and transcript remain recoverable.

Two map counterexamples with different dimensions

The ten-dimensional construction gives PPT maps $\Phi_1,\Phi_2:M_{10}(\mathbb C)\to M_{10}(\mathbb C)$ such that

$$Z=J(\Phi_2\circ\Phi_1)\ne0, \qquad \rho=Z/\operatorname{Tr}Z.$$

The range of $Z$ contains no nonzero product vector, so the composition is not entanglement breaking. The two maps may differ and are not required to preserve trace or the identity.

A separate construction produces a trace-preserving PPT channel $\Theta:M_{21}(\mathbb C)\to M_{21}(\mathbb C)$ whose square $\Theta\circ\Theta$ is not entanglement breaking. This second example uses the same channel twice. Neither dimension is claimed to be minimal.

5. Why the advance matters

Importance: 5/5, provisional. A finite-dimensional entangled state with zero distillable key would answer a central resource question in quantum information. The uniform one-bit obstruction is stronger than merely showing a vanishing asymptotic rate: it prevents an increasingly accurate isolated secret bit from arbitrarily many copies in the stated protocol model.

The accompanying map examples challenge the proposed two-step boundary between PPT noise and complete destruction of entanglement. Together, the results connect an operational impossibility statement to the geometry of positive matrices and quantum maps.

The implication is not that entanglement is useless. It is that usefulness depends on the task and available operations. A state can fail at private-key extraction while retaining mathematical and potentially operational properties absent from separable states. The score measures the significance of the reported claims if correct, without independently verifying them.

6. The main idea

A class closed under taking copies

The operational argument begins with states obtained by applying a PPT map locally on each side of a pure bipartite input. This represented class remains closed under tensor powers. A Choi composition identity places the normalized matrix $Z$ in that class.

The manuscript analyzes a complete public transcript and the four possible final bit pairs. It obtains common positive product factors for their local effects. Auxiliary product inputs and untouched references are used to identify those factors without assuming independence of the actual entangled input. This is why reference compatibility and retention of the full public record enter the protocol definition.

Let $e$ be the probability that the laboratories’ bits disagree. Let $f$ measure overlap between Eve’s unnormalized states for the two agreeing outcomes, summed over the transcript. The manuscript derives

$$f\leq\sqrt{e(1-e)}+e.$$

A nearly ideal bit requires both small disagreement and substantial overlap of Eve’s conditional states: she should not distinguish the two values. If the unhalved trace-norm error is $\eta$, the reported bounds give $e\leq\eta/2$ and $f\geq(1-\eta)/2$. These conditions are incompatible for $\eta<1/5$. Taking the first bit of a longer key cannot increase trace distance, transferring the obstruction to key rates.

Make a state in that class entangled

The construction starts from a PPT map on four-dimensional matrices and twenty special projective directions. Its algebraic arrangement ensures that no nonzero homogeneous quadratic in four variables vanishes at ten of those directions.

The ten-dimensional symmetric square stores quadratic monomials. If the resulting Choi range contained a product vector, its factors would define two nonzero quadratics whose zero sets together cover all twenty directions. Each could cover at most nine, giving the contradiction $9+9<20$.

The manuscript supplies integer data and exact arithmetic certificates for this arrangement. Their role is to specify and support the construction; no certificate execution is needed to understand the geometric mechanism. A separate larger construction uses projective measurements and restricted intersections to obtain entanglement under two PPT maps through a different argument.

Finally, to obtain a channel, two ten-dimensional coordinate blocks alternate the original maps and an additional one-dimensional flag receives missing trace. The resulting dimension is $10+10+1=21$. A local corner of the square’s Choi matrix retains a positive multiple of $Z$; separability of the whole matrix would force that corner to be separable too.

7. Scope, limitations, and open questions

The secrecy statement is tied to the stated resource rules and admitted local-instrument processes. It includes broad finite two-way protocols and specified countable extensions, but should not be paraphrased as covering an unspecified larger model. Presupplying shared secret key or hiding part of the public transcript changes the problem.

The maps in dimension ten are not channels merely because their normalized Choi matrix is a state. Trace preservation is supplied by the distinct dimension-21 construction. The result also does not say that every PPT entangled state has zero key; earlier positive-key examples remain part of the picture.

Minimal dimensions, further characterizations of zero-key entanglement, and behavior under other resource combinations remain questions beyond these claims. Failure of a channel’s second iterate to break entanglement does not exclude entanglement breaking at later iterates.

8. Sources and further reading

The linked private-key and PPT-composition papers explain the principal historical questions. The source manuscript below provides the state, exact map conventions, arithmetic data and Definition 4.1 of the protocol model. Its mathematical and operational claims are reported as stated.

Source manuscripts

• M1. OpenAI, Entanglement with zero distillable secret key in local dimension ten (September 27, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.