Genuinely nonlocal sets with smallest cardinality

This paper establishes the existence of genuinely nonlocal sets with the smallest possible cardinality, demonstrating that three pure states suffice in arbitrary N-partite systems and even two mixed states are sufficient, thereby providing new examples of strongly nonlocal sets and highlighting the role of genuine entanglement in hindering local access to multipartite quantum information.

The Gist

The Big Picture: The "Secret Handshake" Game

Imagine you have a group of friends (let's say $N$ friends) who are scattered across different rooms in a giant mansion. They are all connected by walkie-talkies (classical communication), but they cannot leave their rooms or see each other directly.

Someone hands them a secret "state" (a specific quantum object) and asks: "Which one of these secret objects do you have?"

In the quantum world, there are two types of objects:

1. Product States: These are like simple, independent items. If Friend A has a red ball and Friend B has a blue ball, they can easily figure out what they have by just looking at their own ball and talking on the walkie-talkie.
2. Entangled States: These are like a magical pair of dice. If Friend A rolls a 6, Friend B instantly knows they rolled a 6, even if they are light-years apart. The information isn't in one room; it's woven into the connection between the rooms.

The Problem:
For a long time, scientists knew that if you gave the friends a big list of these secret objects, they might get confused and fail to identify which one they had, even with their walkie-talkies. This is called Quantum Nonlocality (specifically, "nonlocality without entanglement" or "entanglement without nonlocality").

But there was a catch: To make them confused, you usually needed a huge list of objects (like 100 or 1,000 items). The question was: What is the absolute smallest list of objects that can still confuse them?

The Two Main Discoveries

This paper answers that question with two surprising findings, one for "pure" objects and one for "mixed" (blurry) objects.

1. The Pure Case: You Only Need Three

The Old Belief: Scientists thought that to confuse the friends in a multi-room mansion, you needed a massive list of items. They thought the list had to grow as the mansion got bigger.

The New Discovery: The authors found that you only need three specific objects to confuse the friends, no matter how many rooms (subsystems) there are.

• The Analogy: Imagine you have three special keys.
  • Key 1 and Key 2 are "twins" (entangled GHZ states). They are so perfectly linked that if you try to look at them separately, they look identical.
  • Key 3 is a "wildcard" that looks a little bit like Key 1 and Key 2, but not quite.
• The Trick: If the friends try to identify which key they have by only looking at their own room and talking to neighbors, they hit a dead end. The information is so deeply hidden in the connections between all the rooms that they can't solve the puzzle.
• Why it matters: This is a huge breakthrough because it proves that you don't need a giant library of confusing items to create a "lock" that only the whole group can open. Just three items are enough. This is the smallest possible number because any two items can always be distinguished by the friends.

2. The Mixed Case: You Only Need Two

The Twist: What if the friends get many copies of the secret object? Usually, if you have more copies, it's easier to figure out what you have (like taking more photos to identify a blurry face).

• The Old Belief: If you have enough copies, you can always solve the puzzle.

• The New Discovery: The authors found a type of "blurry" (mixed) object where even if the friends have infinite copies, they still cannot tell the difference between two specific options unless they all join hands in one room.

• The Analogy: Imagine two types of fog.

  • Fog A is a mix of many different colored mists.
  • Fog B is a completely different kind of mist.
  • Even if the friends take 1,000 photos of the fog in their own rooms, they can't tell if they are looking at Fog A or Fog B. The "fogginess" is distributed in such a way that local clues are useless.

• Why it matters: This is shocking. Usually, "more data = better answer." Here, "more data" still leads to a dead end. It shows that some quantum information is so deeply hidden that it requires the entire system to be accessed at once, regardless of how many times you try.

The "Secret Sauce": Entanglement is the Key

The paper reveals a fascinating truth: To create these tiny, confusing lists (of 3 or 2 items), you must use Genuine Entanglement.

• The Metaphor: Think of the quantum state as a complex knot.
  • If the knot is just a bunch of loose strings (no entanglement), the friends can untangle it easily by looking at their own strings.
  • To make the knot impossible to untangle locally, you need to tie the strings together in a way that involves everyone in the room simultaneously.
• The Conclusion: The authors show that Genuine Multipartite Entanglement (entanglement involving all parties at once) is the "super-glue" that makes it hard to access information locally. It's not just a fancy feature; it's the fundamental reason why these small sets are so hard to distinguish.

Summary of Why This Matters

1. Efficiency: We used to think we needed thousands of items to create a "quantum lock." Now we know three (for pure states) or even two (for mixed states) are enough.
2. New Tools: Previous methods for finding these locks relied on a specific, rigid technique (called TOPLM). This paper shows that technique is limited and that there are much smaller, more efficient locks that require new ways of thinking.
3. Security: This helps us understand how to protect quantum information. If we know the smallest "lock" possible, we can design better quantum encryption that is impossible to crack without the whole group being present.

In a nutshell: The paper proves that in the quantum world, you don't need a big crowd to create a mystery. Sometimes, just three perfectly entangled friends (or even two blurry ones) are enough to keep a secret that no one can solve unless they are all in the same room.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question in quantum information theory regarding genuine nonlocality: What is the smallest set of orthogonal quantum states that cannot be perfectly distinguished by Local Operations and Classical Communication (LOCC) across any bipartition of an $N$-partite system?

• Context: While "quantum nonlocality without entanglement" (indistinguishability of product states) and local indistinguishability of entangled states are well-studied, genuine nonlocality requires that a set of states remains locally indistinguishable even if all but one subsystem join together.
• The Gap: Previous constructions of genuinely nonlocal sets (often based on the "Trivial Orthogonality-Preserving Local Measurement" or TOPLM technique) required a large number of states (cardinality scaling with system dimension or number of parties). It was unclear if sets with a constant, minimal cardinality (e.g., 2 or 3) could exist for arbitrary $N$-partite systems, particularly for pure states.
• Specific Questions:
  1. Do genuinely nonlocal sets of three orthogonal pure states exist in arbitrary $N$-partite systems?
  2. Do genuinely nonlocal sets of two orthogonal mixed states exist, even in the limit of many copies?

2. Methodology

The authors employ a combination of entanglement theory, PPT (Positive Partial Transpose) criteria, and unextendible product basis (UPB) constructions.

A. Pure States: PPT-Indistinguishability and Embedding

• Bipartite Foundation: The authors start with a known set of three locally indistinguishable states in $C^2 \otimes C^2$: two Bell states ($|\beta_1\rangle, |\beta_2\rangle$) and a third state ($|\gamma_3\rangle$ or $|01\rangle$).
• PPT Proof: They prove that this specific triplet is PPT-indistinguishable (cannot be distinguished even by PPT measurements, a superset of LOCC). This is crucial because PPT-indistinguishability implies LOCC-indistinguishability.
• Extension to N-partite: They construct $N$-qubit states by embedding the bipartite indistinguishable core into every possible bipartition of the $N$-partite system.
  • They utilize an $N$-qubit GHZ pair ($|\Gamma_{1,2}\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes N} \pm |1\rangle^{\otimes N})$).
  • They introduce a third state ($|\Gamma_3\rangle$) constructed from basis vectors excluding $|0\rangle^{\otimes N}$ and $|1\rangle^{\otimes N}$.
  • Key Logic: For any bipartition $L|R$, the GHZ pair is supported on a $2 \otimes 2$ subspace. If the third state has a non-zero overlap with this specific $2 \otimes 2$ subspace in every bipartition, the set remains PPT-indistinguishable across that cut.

B. Mixed States: Non-Orthogonal Unextendible Product Bases (nUPB)

• Many-Copy Limit: The authors address the scenario where multiple copies of the state are available. While pure states become distinguishable with many copies, mixed states may not.
• Construction: They generalize the concept of Unextendible Product Bases (UPBs) to Non-Orthogonal Unextendible Product Bases (nUPBs).
  • They construct a set of fully product states $\{|\Psi_j\rangle\}$ that span a subspace $H_S$ such that the orthogonal complement $H_S^\perp$ contains no fully product states (and no biproduct states in any bipartition).
  • This construction relies on Vandermonde matrices to ensure the "local spanning" condition is met in every bipartition, guaranteeing unextendibility.
• State Definition:
  • State $\sigma$: The normalized projector onto the subspace spanned by the nUPB (a separable mixed state).
  • State $\rho$: Any state supported on the orthogonal complement $H_S^\perp$ (necessarily genuinely entangled).
• Proof of Indistinguishability: Using the property that $\sigma^{\otimes n}$ remains full-rank on a subspace with no product states in any bipartition, they invoke a proposition regarding conclusive local distinguishability to show that $\sigma$ and $\rho$ cannot be distinguished even with $n$ copies.

3. Key Contributions and Results

Result 1: Minimal Pure State Sets (Cardinality = 3)

• Theorem 1: In any $N$-qubit system $(C^2)^{\otimes N}$, there exists a set of three orthogonal pure states that are genuinely nonlocal.
• Composition: Two $N$-qubit GHZ states and one specific third state (which can be chosen to be genuinely entangled).
• Significance: This is the smallest possible cardinality for pure states (since any two orthogonal pure states are always locally distinguishable).
• Strong Nonlocality: The authors prove these sets are also strongly nonlocal (locally irreducible across all bipartitions). This provides the first examples of strongly nonlocal sets with genuine entanglement that do not rely on the TOPLM technique and have a cardinality ($3$) dramatically smaller than previous bounds ($d^{N-1}+1$).

Result 2: Minimal Mixed State Sets (Cardinality = 2)

• Theorem 2: In any $N$-partite system with arbitrary local dimensions, there exists a set of two orthogonal mixed states that are genuinely nonlocal.
• Robustness: This nonlocality persists even in the many-copy limit (i.e., even if the observers share $n$ copies of the unknown state, they cannot distinguish them unless all subsystems join).
• Composition: One separable mixed state (projector on an nUPB subspace) and one genuinely entangled state (supported on the orthogonal complement).

Result 3: Role of Genuine Entanglement

• The constructions demonstrate that genuine multipartite entanglement is a necessary resource for achieving these minimal cardinalities.
• For the pure case, the third state must be genuinely entangled to satisfy the overlap conditions in all bipartitions.
• For the mixed case, the second state $\rho$ must be genuinely entangled because the orthogonal complement of the nUPB contains no product states.

4. Significance and Implications

1. Redefining Minimal Nonlocality: The paper settles the question of the minimum size of genuinely nonlocal sets. It proves that for pure states, 3 is the minimum, and for mixed states, 2 is the minimum, regardless of the number of parties $N$.
2. Limitations of TOPLM: The results highlight a significant limitation of the TOPLM technique, which previously suggested that genuinely nonlocal sets must be large (scaling with dimension). The authors show that sets of size 3 exist, suggesting that new detection techniques beyond TOPLM are required to identify such nonlocality.
3. Entanglement as a Barrier: The work provides strong evidence that genuine multipartite entanglement is not just a resource for quantum communication but a fundamental barrier to locally accessing global quantum information, even when the number of available copies is large.
4. Strong Nonlocality with Entanglement: It resolves open questions regarding the existence of strongly nonlocal sets with genuine entanglement, providing simple, general constructions for all $N$-partite systems.

Conclusion

Xiong et al. successfully constructed the smallest possible genuinely nonlocal sets for both pure and mixed states in arbitrary multipartite systems. By leveraging PPT-indistinguishability for pure states and non-orthogonal unextendible product bases for mixed states, they demonstrated that genuine nonlocality can be manifested with as few as three pure states or two mixed states, fundamentally challenging previous assumptions about the size and complexity required to exhibit this form of quantum nonlocality.