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Flower States Reveal a Hidden Asymmetry in the Quantum World · The Quark Side - Quantum Physics Podcast

• 0:01
  Speaker 1: Welcome to the quark side Quantum Physics podcast, an exploration
• 0:05
  Speaker 1: of the fundamental structure of reality, where quantum laws govern matter, energy,
• 0:11
  Speaker 1: and information. Here, uncertainty is a feature, not a flaw,
• 0:16
  Speaker 1: and understanding begins at the smallest scales.
• 0:27
  Speaker 2: Imagine pulling your car up to the gas station. Right,
• 0:30
  Speaker 2: you open the tank, and you pump in ten gallons
• 0:34
  Speaker 2: of premium high octane.
• 0:36
  Speaker 3: Fuel, right, just the standard morning commute exactly.
• 0:39
  Speaker 2: You pay the attendant, you start the engine, But when
• 0:42
  Speaker 2: you look down at the dashboard, the gauge says you
• 0:44
  Speaker 2: only have one single gallon.
• 0:46
  Speaker 3: I mean, you'd immediately assume there was a leak in
• 0:48
  Speaker 3: the tank, or you know, maybe that the pump itself
• 0:51
  Speaker 3: was rigged to scam you.
• 0:53
  Speaker 2: Yes, your first instinct is that there just has to
• 0:56
  Speaker 2: be a mechanical failure somewhere. But what if the tank
• 1:00
  Speaker 2: is perfectly sealed, Like what if the engine is completely flawless.
• 1:03
  Speaker 2: What if the universe itself just irreversibly text you nine
• 1:06
  Speaker 2: gallons of fuel simply for the privilege of putting it
• 1:09
  Speaker 2: into the car.
• 1:10
  Speaker 3: That would be well, a very frustrating and a very
• 1:13
  Speaker 3: expensive commute. You'd literally be fighting the laws of physics
• 1:16
  Speaker 3: just to get to work.
• 1:17
  Speaker 2: And yet that is exactly the reality you face when
• 1:20
  Speaker 2: you step into the quantum realm. Because usually when we
• 1:22
  Speaker 2: talk about quantum entanglement, we use these mystical, almost poetic
• 1:26
  Speaker 2: phrases sure like spooky action at a distance, right, spooky action.
• 1:31
  Speaker 2: We treat it like it's just some strange philosophical connection
• 1:34
  Speaker 2: between particles. But to understand the absolute bleeding edge of
• 1:39
  Speaker 2: quantum thermodynamics, you have to completely stop thinking of entanglement
• 1:43
  Speaker 2: as magic. You need to start thinking of it as fuel.
• 1:46
  Speaker 3: And that's such a crucial mindset shift because it really
• 1:49
  Speaker 3: is an operational, physical, consumable resource.
• 1:52
  Speaker 2: Yeah, it's the premium gasoline that will, you know, eventually
• 1:55
  Speaker 2: power quantum computers, unhackable quantum cryptography, and next generation communication networks.
• 2:02
  Speaker 1: Right.
• 2:02
  Speaker 3: And just like classical thermodynamics has strict, unbreakable laws about
• 2:06
  Speaker 3: energy loss, like how a traditional engine always loses some
• 2:09
  Speaker 3: energy to heat, dissipation, and friction, quantum entanglement has this
• 2:13
  Speaker 3: massive irreversibility problem.
• 2:14
  Speaker 2: Meaning what you put in isn't what you get out.
• 2:17
  Speaker 3: Exactly, what you put into a complex quantum system is
• 2:19
  Speaker 3: almost never what you get back out. Entropy always collects
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  Speaker 3: its dues.
• 2:23
  Speaker 2: So we are exploring some groundbreaking theoretical physics today that
• 2:27
  Speaker 2: pushes right to the extreme edges of how this quantum
• 2:30
  Speaker 2: fuel is manufactured, manipulated, and ultimately lost forever.
• 2:34
  Speaker 3: Yeah, and we're going to unpack this by looking at
• 2:36
  Speaker 3: a deeply fascinating mathematical blueprint known as Flower states.
• 2:41
  Speaker 2: Right, the Flower States. By the time we're done here,
• 2:43
  Speaker 2: you're going to see how this single mathematical construct completely
• 2:46
  Speaker 2: shatters our understanding of quantum limits, breaks some of our
• 2:50
  Speaker 2: favorite metrics, and reveals a bizarre reality where the universe
• 2:53
  Speaker 2: essentially audits your math.
• 2:55
  Speaker 3: But to really understand why these Flower states are so revolutionary,
• 3:00
  Speaker 3: we first have to establish the rules of the game,
• 3:03
  Speaker 3: specifically the physical constraints of manipulating this quantum fuel in
• 3:08
  Speaker 3: the real world.
• 3:09
  Speaker 2: Okay, let's untack this. So in the real world, like
• 3:12
  Speaker 2: in an actual physical laboratory with lasers and cryogenic scientists
• 3:16
  Speaker 2: are bound by a strict set of rules called LCC
• 3:20
  Speaker 2: that stands for Local Operations and Classical Communication.
• 3:23
  Speaker 3: Yeah, LCC is the gold standard for physical reality. It
• 3:27
  Speaker 3: describes the absolute limits of what two people, let's call
• 3:29
  Speaker 3: them Alice and Bob. Since that's tradition in quantum thought experiments. Well,
• 3:33
  Speaker 3: they can actually achieve when they share an entangle quantum
• 3:35
  Speaker 3: state but are separated by a large distance.
• 3:38
  Speaker 2: I always picture LCC like this, right, Alice and Bob
• 3:42
  Speaker 2: are locked in two completely separate, entirely sound proof rooms. Say,
• 3:47
  Speaker 2: Alice is in New York and Bob is in Tokyo,
• 3:50
  Speaker 2: okay I like this, and they're trying to build this
• 3:52
  Speaker 2: incredibly complex intertwined lego set together, but the rules are
• 3:57
  Speaker 2: violently strict. Alice can only physically touch the lego blocks
• 4:01
  Speaker 2: in her own room. Bob can only touch the blocks
• 4:04
  Speaker 2: in his room.
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  Speaker 3: Right. Those are the local operations part of the acronym.
• 4:07
  Speaker 2: Exactly, and to coordinate their building efforts so that the
• 4:11
  Speaker 2: final structures perfectly correlate, they are allowed to talk to
• 4:14
  Speaker 2: each other over a standard everyday telephone line. They can
• 4:18
  Speaker 2: text each other instructions, but they cannot send actual lego
• 4:21
  Speaker 2: blocks through.
• 4:22
  Speaker 3: The phone, and that is the classical communication.
• 4:24
  Speaker 2: So the workflow just goes like this. Alice looks at
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  Speaker 2: her blocks, runs a specific operation on them, picks up
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  Speaker 2: the phone and says, hey, Bob, I just clicked the
• 4:32
  Speaker 2: red piece into the blue piece based on our blueprint,
• 4:34
  Speaker 2: it's your turn.
• 4:35
  Speaker 3: And then Bob listens, processes that information, does something to
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  Speaker 3: his blocks based on what she said, and then calls
• 4:42
  Speaker 3: her back with his results.
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  Speaker 2: Yeah, they just do this back and forth, over and
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  Speaker 2: over again.
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  Speaker 3: It sounds totally reasonable when you put it like that,
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  Speaker 3: but mathematically LOCC is an absolute nightmare. Really why it
• 4:53
  Speaker 3: is notoriously intractable for physicists to calculate because think about
• 4:59
  Speaker 3: what happens when you try to map out every possible outcome.
• 5:03
  Speaker 3: Every single time Alice makes a choice and calls Bob,
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  Speaker 3: it changes the entire state of the system, which changes
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  Speaker 3: Bob's available moves, which then changes his next phone call,
• 5:14
  Speaker 3: which limits Alice's next move.
• 5:16
  Speaker 2: Oh wow, so it's like a game of chess, but
• 5:18
  Speaker 2: every single time you move upon the rules of how
• 5:20
  Speaker 2: the knights move completely change for the rest of the game.
• 5:23
  Speaker 3: Yes, exactly. If you try to calculate the absolute limits
• 5:26
  Speaker 3: of entanglement under LOCC, you are dealing with this infinitely
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  Speaker 3: branching tree of probability.
• 5:31
  Speaker 2: Okay, that sounds awful to calculate.
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  Speaker 3: It's a computational brick wall very quickly, which is exactly
• 5:36
  Speaker 3: why theoretical physicists invented what are essentially cheat codes.
• 5:39
  Speaker 2: Cheat codes.
• 5:40
  Speaker 3: Yeah, because LOCC is so difficult to calculate, physicists created
• 5:44
  Speaker 3: relaxed frameworks, these outer approximations that are much easier to
• 5:48
  Speaker 3: do math with. And one of the most important of
• 5:50
  Speaker 3: these relaxed rule sets is called non entangling operations or
• 5:54
  Speaker 3: an ease.
• 5:55
  Speaker 2: Okay, so what makes any a cheat code? Like, how
• 5:58
  Speaker 2: does it change the rules for Alice and Bob in
• 6:00
  Speaker 2: their lego rooms.
• 6:01
  Speaker 3: Well, under any rules, we just wave a magic wand
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  Speaker 3: and throw away all those complicated, headache inducing restrictions about
• 6:08
  Speaker 3: local actions and infinite telephone calls.
• 6:10
  Speaker 2: Oh nice, So what replaces it?
• 6:12
  Speaker 3: We replace all of it with one simple, overarching mathematical rule.
• 6:16
  Speaker 3: If a quantum state goes into an operation without any
• 6:20
  Speaker 3: entanglement at all, meaning the particles are completely separable and independent,
• 6:24
  Speaker 3: it must come out of the operation without any entanglement.
• 6:26
  Speaker 2: So separable states must stay separable.
• 6:28
  Speaker 3: That's it. It is vastly more permissive than LCCI.
• 6:31
  Speaker 2: Okay, here's where I have to push back though, If
• 6:34
  Speaker 2: any operations are just a theoretical playground, like if they
• 6:37
  Speaker 2: literally allow things that are physically impossible for alsmov to
• 6:41
  Speaker 2: do and their actual locked lego rooms. Why should you,
• 6:44
  Speaker 2: the listener, even care about them? That's a fair question, right,
• 6:47
  Speaker 2: Like why are we spending so much time doing incredibly
• 6:50
  Speaker 2: complex math on fake physics.
• 6:52
  Speaker 3: It comes down to the power of absolute upper bounds.
• 6:55
  Speaker 3: This is a classic top down approach to physics. Think
• 6:58
  Speaker 3: about the speed of light, right, Okay, Even if we
• 7:01
  Speaker 3: can't build a ship that goes that fast, knowing the
• 7:03
  Speaker 3: absolute speed limit of the universe tells us fundamentally how
• 7:07
  Speaker 3: space time works. It's the same here. If we grant
• 7:10
  Speaker 3: Alis and Bob these magical highly relaxed any cheat codes,
• 7:15
  Speaker 3: and they still cannot accomplish a specific quantum task.
• 7:19
  Speaker 2: Then we know for an absolute fact that the task
• 7:22
  Speaker 2: is definitively impossible in the real physical world under.
• 7:25
  Speaker 3: LCC exactly, it's a structural separator. It helps us find
• 7:28
  Speaker 3: the absolute ceiling of what quantum mechanics will permit.
• 7:31
  Speaker 2: Ah I see.
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  Speaker 3: And because this permissive any power is so vast, it's
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  Speaker 3: generally assumed that it makes entanglement manipulation much more efficient
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  Speaker 3: under any rules. Things that are stubbornly lossy and irreversible
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  Speaker 3: in the real world are supposed to become perfectly reversible
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  Speaker 3: and clean.
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  Speaker 2: Which brings us perfectly to the metrics of success, the
• 7:50
  Speaker 2: quantum fuel economy, so to speak. When we talk about
• 7:53
  Speaker 2: this resource of entanglement, there are two main numbers we really.
• 7:56
  Speaker 3: Care about, right, Yes, two main metrics.
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  Speaker 2: The first is called entanglement cost. This is the amount
• 8:01
  Speaker 2: of pure quantum fuel which is measured in units called
• 8:04
  Speaker 2: EBITs that you have to spend in order to manufacture
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  Speaker 2: a specific messy quantum state from scratch, right.
• 8:09
  Speaker 3: And the second metric is distillable entanglement. This is your yield. Okay,
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  Speaker 3: if I hand you a large number of those messy,
• 8:16
  Speaker 3: noisy quantum states you just build, how many pure EBITs
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  Speaker 3: can you extract or distill back out of them to
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  Speaker 3: use for your quantum computer.
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  Speaker 2: So cost is what you pay to build the state.
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  Speaker 2: Distillable is the pure fuel you get back out. And
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  Speaker 2: in a perfect entirely reversible universe, those two numbers would
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  Speaker 2: be exactly.
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  Speaker 3: The same, but they almost never are. The universe takes
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  Speaker 3: its cut, and to prove just how viciously deep this
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  Speaker 3: irreversibility goes, we finally turn to the stars of the
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  Speaker 3: show The Flower States.
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  Speaker 2: Yes, the Flower States. These are a highly specific family
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  Speaker 2: of maximally correlated quantum states. They are parameterized by an
• 8:54
  Speaker 2: even local dimension, which we write mathematically as D equals
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  Speaker 2: two K. But the reason they are so vital to
• 9:00
  Speaker 2: our understanding of the universe today is that they possess
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  Speaker 2: a property that is exceedingly, almost impossibly rare in physics.
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  Speaker 3: It really cannot be overstated how rare this is. For
• 9:10
  Speaker 3: almost all complex quantum states, calculating every single entanglement measure
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  Speaker 3: perfectly is just impossible. We rely on estimates, guesses, bounds.
• 9:19
  Speaker 2: But not for the Flower states. No.
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  Speaker 3: Flower states belong to an exclusive mathematical group where every
• 9:25
  Speaker 3: major entanglement measure can be explicitly exactly computed without any guesswork.
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  Speaker 3: They are the perfect transparent anatomical specimens for testing the
• 9:34
  Speaker 3: laws of quantum thermodynamics, and what.
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  Speaker 2: We see when we cut them open and look at
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  Speaker 2: the math is shocking. Let's look at the yield first,
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  Speaker 2: the distillable entanglement. If you take a Flower state, no
• 9:43
  Speaker 2: matter how massive or overwhelmingly complex it gets, you can
• 9:46
  Speaker 2: scale that local dimension up to a million, a billion,
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  Speaker 2: a trillion.
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  Speaker 3: The distillable entanglement remains strictly stubbornly capped at exactly one ebit.
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  Speaker 2: Just one single unit of quantum fuel.
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  Speaker 3: It completely plateaus. It does not matter how how large
• 10:00
  Speaker 3: the state is. You only ever get one ebit out.
• 10:03
  Speaker 2: But the entanglement cost the fuel you have to spend
• 10:06
  Speaker 2: to build that flower state in the first place. That
• 10:08
  Speaker 2: scales up aggressively as the dimension grows.
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  Speaker 3: It creates a massive, yawning disparity between what you put
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  Speaker 3: in and what you get out.
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  Speaker 2: So let's ditch the car analogy from earlier, because this
• 10:18
  Speaker 2: isn't just about economic value depreciating. This is a physical
• 10:22
  Speaker 2: destruction of a resource.
• 10:23
  Speaker 3: Oh absolutely, think of it like this.
• 10:26
  Speaker 2: Imagine you spend a month building this massive, intricate, million
• 10:30
  Speaker 2: piece lego castle, used one hundred thousand pure pristine Lago
• 10:35
  Speaker 2: bricks to do it. But to get your raw bricks
• 10:37
  Speaker 2: back out to build something else, the only method the
• 10:39
  Speaker 2: universe allows is for you to throw the entire castle
• 10:42
  Speaker 2: into a furnace, melt it down and try to scrape
• 10:45
  Speaker 2: out the raw.
• 10:45
  Speaker 3: Plastic, right, and you are inevitably going to lose an
• 10:49
  Speaker 3: enormous amount of that plastic to toxic fumes, heat, dissipation,
• 10:53
  Speaker 3: and chemical changes.
• 10:54
  Speaker 2: Exactly that is a much better way to visualize it.
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  Speaker 2: The fuel is physically destroyed in the trend. Yes, so
• 11:01
  Speaker 2: if the fuel is gone, if it's lost to the furnace.
• 11:04
  Speaker 2: This is where I would assume this is just a
• 11:06
  Speaker 2: physical limitation of our clumsy lab equipment, like the LOCC
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  Speaker 2: restrictions are just inefficient.
• 11:12
  Speaker 3: And that is the logical assumption. But here is where
• 11:15
  Speaker 3: it gets really profound. The math proves that this massive
• 11:20
  Speaker 3: destruction of entanglement happens even under those relaxed cheat code
• 11:24
  Speaker 3: any rules.
• 11:25
  Speaker 2: Wait, even with the magical physics cheat codes where we
• 11:28
  Speaker 2: throw away all the local restrictions and phone calls, the
• 11:30
  Speaker 2: fuel still just vanishes into thin air.
• 11:33
  Speaker 3: It still vanishes. This unbridgable gap between the massive cost
• 11:36
  Speaker 3: to build the state and the tiny yields you can
• 11:38
  Speaker 3: extract is called the irreversibility gap, and for flower states,
• 11:42
  Speaker 3: the any irreversibility gap is mathematically proven to be at
• 11:44
  Speaker 3: least one half one half. Yes, specifically, it scales proportionally
• 11:48
  Speaker 3: to one half the logarithm of the dimension.
• 11:50
  Speaker 2: To put how wild that is in perspective, the previous
• 11:54
  Speaker 2: scientific estimate for the absolute worst case irreversibility gap under
• 11:58
  Speaker 2: any rules was about point two sixty two. That was
• 12:01
  Speaker 2: based on a totally different quantum state. Right, the Flower
• 12:04
  Speaker 2: States completely shatter that record. They pushed the lower bound
• 12:07
  Speaker 2: of that gap all the way to zero point five.
• 12:09
  Speaker 3: It completely redefines the boundaries. It proves that this massive irreversibility,
• 12:14
  Speaker 3: this incredibly lossy nature of entanglement, isn't just a quirk
• 12:18
  Speaker 3: of LOCC restrictions or bad lab equipment.
• 12:21
  Speaker 2: It's just baked in.
• 12:22
  Speaker 3: It is fundamentally, unavoidably baked into the macroscopic limit of
• 12:26
  Speaker 3: the universe. The destruction of quantum fuel is a feature
• 12:29
  Speaker 3: of reality, not a bug.
• 12:31
  Speaker 2: Okay, So the Flower States have already proven that the
• 12:33
  Speaker 2: universe is going to ruthlessly tax our quantum fuel no
• 12:36
  Speaker 2: matter what rules we use. But they do something even
• 12:39
  Speaker 2: more aggressive to another darling of the theoretical physics world,
• 12:42
  Speaker 2: don't they?
• 12:42
  Speaker 3: Oh? They do. They completely irrevocably break a highly relied
• 12:46
  Speaker 3: upon metric known as squashed.
• 12:47
  Speaker 2: Entanglement, the fall of squashed entanglement. Historically, this has been
• 12:52
  Speaker 2: one of the most elegant and beloved mathematical tools we have.
• 12:55
  Speaker 2: Under strict physical LOCC constraints, it behaves beautif it is additive,
• 13:02
  Speaker 2: which means the entanglement of two states is just the
• 13:04
  Speaker 2: sum of their individual entanglements. It just stacks cleanly.
• 13:08
  Speaker 3: And it's also monogamous, right.
• 13:09
  Speaker 2: Which in quantum terms means if Alice's particle is maximally
• 13:13
  Speaker 2: entangled with Bob's particle, she physically cannot be highly entangled
• 13:18
  Speaker 2: with Charlie's potocle. There's a strict limit to the quantum
• 13:21
  Speaker 2: bandwidth exactly.
• 13:22
  Speaker 3: It's the Golden Child metric. It always plays by the
• 13:25
  Speaker 3: rules under LOCC. The squashed entanglement always sits perfectly nestled
• 13:30
  Speaker 3: between the distillable yield and the entanglement cost. Is a
• 13:33
  Speaker 3: highly reliable benchmark.
• 13:34
  Speaker 2: And because it is such a robust tool, a major
• 13:37
  Speaker 2: open question in the physics community was whether squashed entanglement
• 13:40
  Speaker 2: could survive outside of LOCC like could it maintain its
• 13:45
  Speaker 2: beautiful structural integrity if we relaxed the rules and took
• 13:49
  Speaker 2: it into the ne operations framework. Since any operations are
• 13:53
  Speaker 2: so much more permissive, it would be incredibly useful if
• 13:56
  Speaker 2: squashed entanglement could act as a stable lower bound for
• 14:00
  Speaker 2: any entanglement cost.
• 14:01
  Speaker 3: But the Flower states step in and say absolutely not,
• 14:05
  Speaker 3: because we can calculate every single metric perfectly. For flower states,
• 14:09
  Speaker 3: we don't have to guess. We can just run the
• 14:11
  Speaker 3: numbers side by side and look at the hard data.
• 14:13
  Speaker 2: Let's hear it.
• 14:14
  Speaker 3: For flower states, under LOCC, the entanglement cost is exactly
• 14:18
  Speaker 3: equal to the squashed entanglement, which calculates out to the
• 14:21
  Speaker 3: logarithm of two times the square root of k.
• 14:23
  Speaker 2: Okay, I want everyone to just keep that shape in
• 14:25
  Speaker 2: their head. The LOCC cost is the logarithm of two
• 14:29
  Speaker 2: times the square root of K.
• 14:30
  Speaker 3: Now, we calculate the entanglement cost for the exact same
• 14:33
  Speaker 3: flower state, but under the relaxed any rules, that cost
• 14:36
  Speaker 3: is the logarithm of one plus the quardative K.
• 14:39
  Speaker 2: So if we look at those two outcomes, the LOCC
• 14:42
  Speaker 2: squashed entanglement is physically larger than the any entanglement cost exactly.
• 14:47
  Speaker 3: And that is a fatal mathematical contradiction. I think because
• 14:49
  Speaker 3: if squashed entanglement were truly a universal monotone, meaning it
• 14:54
  Speaker 3: held its structured reliability across different operational frameworks, it would
• 14:58
  Speaker 3: fundamentally always have to be less than or equal to
• 15:01
  Speaker 3: the any entanglement cost.
• 15:02
  Speaker 2: But it's not. It is strictly larger the golden child breaks.
• 15:06
  Speaker 2: Squashed entanglement completely falls apart the moment you relax the
• 15:10
  Speaker 2: laws of physics.
• 15:11
  Speaker 3: It is definitively not a monotone under non entangling operations. Now,
• 15:16
  Speaker 3: this isn't necessarily a tragedy for physics. What it actually
• 15:19
  Speaker 3: proves is how perfectly inextricably tailored squashed entanglement is to
• 15:24
  Speaker 3: the strict physical reality of locc.
• 15:27
  Speaker 2: So it draws a boundary, a.
• 15:28
  Speaker 3: Firm, uncrossable boundary, between the physical world of local actions
• 15:33
  Speaker 3: and the theoretical playgrounds of any approximations.
• 15:36
  Speaker 2: It's like finding a highly specialized wrench that works flawlessly
• 15:39
  Speaker 2: to build engines here on Earth, but the second you
• 15:41
  Speaker 2: take it into the zero gravity of a space station,
• 15:44
  Speaker 2: the metal literally shatters in your hands.
• 15:46
  Speaker 3: That is a phenomenal comparison.
• 15:48
  Speaker 2: It might be useless in space, but the way it
• 15:50
  Speaker 2: shatters tells you something incredibly profound about the nature of
• 15:53
  Speaker 2: gravity itself.
• 15:54
  Speaker 3: The failure of the tool defines the environment.
• 15:57
  Speaker 2: Okay, I want to pivot slightly here, because everything we've
• 16:00
  Speaker 2: talked about so far assumes a standard real world laboratory environment,
• 16:05
  Speaker 2: and in a standard lab perfection is well negotiable. Right,
• 16:09
  Speaker 2: if you're diluting or distilling quantum entanglement and you have
• 16:12
  Speaker 2: an err rate of say points zero zero zero zero
• 16:15
  Speaker 2: one percent, scientists call it a day. That's known as
• 16:18
  Speaker 2: the vanishing error regime.
• 16:20
  Speaker 3: Yes, and the asymptotic limit of many copies of vanishingly
• 16:23
  Speaker 3: small error is considered a highly successful, completely acceptable protocol.
• 16:28
  Speaker 2: But what if we demand absolute, uncompromising perfection. What if
• 16:32
  Speaker 2: we switch to a strict zero error constraint, if we
• 16:35
  Speaker 2: tolerate absolutely zero noise, zero mistakes, My assumption is that
• 16:40
  Speaker 2: the cost to build these states would astronomically skyrocket.
• 16:43
  Speaker 3: Naturally, you would expect the cost to go up if
• 16:45
  Speaker 3: you demand absolute perfection in any manufacturing process, you have
• 16:48
  Speaker 3: to pay a premium. But here's the first shock of
• 16:51
  Speaker 3: the flower states. Under the relaxed any rules, demanding zero
• 16:54
  Speaker 3: error doesn't change the cost of flower states at all.
• 16:57
  Speaker 3: The exact any cost is completely immune to the strict
• 17:01
  Speaker 3: zero error constraint. It remains exactly the same as the
• 17:04
  Speaker 3: vanishing error cost.
• 17:05
  Speaker 2: But we just established that we don't live in the
• 17:08
  Speaker 2: any universe. We don't have the cheat codes. We live
• 17:10
  Speaker 2: in the LCC universe. We are stuck in the locked
• 17:13
  Speaker 2: rooms with lego blocks and the standard telephones. What happens
• 17:17
  Speaker 2: when Alice and Bob demand.
• 17:18
  Speaker 3: Zero error under exact LOCC. The entanglement cost exhibits an
• 17:23
  Speaker 3: incredibly rich and incredibly strange mathematical structure. Yeah, to evaluate
• 17:29
  Speaker 3: the exact LOCC cost, you have to calculate something called
• 17:32
  Speaker 3: the regularized Schmidt number.
• 17:33
  Speaker 2: Okay, I'm going to need an explain like I'm five
• 17:35
  Speaker 2: for the regularized Schmidt number because that sounds like terrifying math.
• 17:38
  Speaker 3: Think of it like baking a highly specific, incredibly complex
• 17:42
  Speaker 3: quantum cake. The regularized Schmidt number is essentially the absolute
• 17:46
  Speaker 3: minimum number of fundamental, indivisible ingredients you need to perfectly
• 17:51
  Speaker 3: bake that state.
• 17:52
  Speaker 2: Okay, ingredients, got it.
• 17:53
  Speaker 3: In most non trivial quantum states, figuring out exactly what
• 17:56
  Speaker 3: those fundamental ingredients are and how they combine across in
• 18:00
  Speaker 3: potential copies without any errors is an absolute computational brick wall.
• 18:04
  Speaker 2: But true to form, the Flower states allow us to
• 18:07
  Speaker 2: see right through.
• 18:08
  Speaker 3: The wall exactly. They admit an exact analytical solution by
• 18:13
  Speaker 3: leveraging uncertainty relations over sickle groups, which is mathematics pioneered
• 18:18
  Speaker 3: by Terence Tao and Roy Michulm. We find that the
• 18:21
  Speaker 3: exact LOCC cost is deeply intrinsically tied to the number
• 18:26
  Speaker 3: theoretic structure of the local dimension K.
• 18:28
  Speaker 2: I want to make sure you didn't miss what was
• 18:30
  Speaker 2: just said. The physical cost of entanglement, like the actual
• 18:33
  Speaker 2: amount of real world quantum fuel you have to burn,
• 18:36
  Speaker 2: is being directly dictated by number theory, specifically prime numbers.
• 18:41
  Speaker 3: That's right, If the dimension k happens to be a
• 18:43
  Speaker 3: prime number, the zero error LOCC cost violently spikes. It
• 18:48
  Speaker 3: roughly doubles, jumping to the logarithm of k plus one.
• 18:51
  Speaker 2: Why in the world would a prime number make a
• 18:53
  Speaker 2: physical process twice as expensive?
• 18:55
  Speaker 3: Because of how efficiency works in quantum mechanics. Normally, to
• 18:58
  Speaker 3: build a complex state, official Alice and Bob want to
• 19:01
  Speaker 3: break the operations down into smaller, highly manageable, parallel subroutines.
• 19:05
  Speaker 3: You have factor the problem down into smaller chunks. But
• 19:08
  Speaker 3: a prime number, by definition, is only divisible by one
• 19:11
  Speaker 3: in itself. You cannot factor it. You cannot break it
• 19:15
  Speaker 3: down into smaller subroutines.
• 19:17
  Speaker 2: Oh, so they are forced to execute the state preparation
• 19:20
  Speaker 2: as one massive, monolithic, highly inefficient operation.
• 19:24
  Speaker 3: Precisely they had to swallow the whole pill at once.
• 19:27
  Speaker 3: You're paying an enormous amount of premium fuel to build
• 19:30
  Speaker 3: a state perfectly simply because its mathematical dimension is prime.
• 19:34
  Speaker 2: That is insane.
• 19:36
  Speaker 3: And conversely, if k is a perfect square, say nine,
• 19:39
  Speaker 3: which can cleanly break down into three and three, the
• 19:42
  Speaker 3: exact locc cost drops right back down and perfectly matches
• 19:46
  Speaker 3: the standard vanishing error cost.
• 19:48
  Speaker 2: That is wild. The universe isn't just taxing your fuel,
• 19:50
  Speaker 2: it's actually auditing your math and penalizing you for using
• 19:54
  Speaker 2: unfactorable prime numbers exactly. So this raises a huge hypothesis
• 19:57
  Speaker 2: for me. If it is this incredibly resource intent of
• 20:00
  Speaker 2: this deeply tied to number theory, just to build a
• 20:02
  Speaker 2: flower state with zero error, I have to assume extracting
• 20:05
  Speaker 2: that single tiny one ebit of pure fuel backout is
• 20:09
  Speaker 2: a nightmare. Do we need an absolute supercomputer running for
• 20:12
  Speaker 2: a thousand years. Do we need millions of copies just
• 20:14
  Speaker 2: to sift out that one ebit?
• 20:15
  Speaker 3: You would certainly think so. Typically, to achieve the theoretical
• 20:19
  Speaker 3: limit of distillation in quantum thermo dynamics, you need an
• 20:23
  Speaker 3: asymptotic number of copies. You need a massive factory working
• 20:27
  Speaker 3: on thousands of quantum states simultaneously to slowly sift out
• 20:32
  Speaker 3: the pure entanglement.
• 20:33
  Speaker 2: Right, and you'd think we definitely need those any cheat
• 20:35
  Speaker 2: codes again to make it work cleanly.
• 20:37
  Speaker 3: But here is the brilliant final twist of the flower states.
• 20:41
  Speaker 3: Extracting that one single pure ebit from generalized flower states
• 20:47
  Speaker 3: doesn't require infinite copies, it doesn't require a vanishing error margin,
• 20:50
  Speaker 3: and it absolutely does not require the relaxed any cheat codes.
• 20:54
  Speaker 2: So how do they get it out?
• 20:55
  Speaker 3: It can be distilled deterministically with absolute perfection using a
• 20:59
  Speaker 3: simple one shot LOCC protocol.
• 21:02
  Speaker 2: Wait one shot, as in, Alice looks at a single
• 21:05
  Speaker 2: flower state in her room, does one physical operation, makes
• 21:08
  Speaker 2: one single phone call to Bob, and boom, one perfect
• 21:11
  Speaker 2: inhibitive fuel is extracted exactly.
• 21:14
  Speaker 3: The protocol simply involves Alice applying a specific set of
• 21:17
  Speaker 3: Krause operators, sending a single classical message to Bob, and
• 21:20
  Speaker 3: Bob applying a corresponding operation based on her message.
• 21:24
  Speaker 2: Okay, stop you use jargon again. What is a Krouse
• 21:27
  Speaker 2: operator in this context? Speak English fair enough?
• 21:30
  Speaker 3: Think of a crowds operator as a specific physical filter.
• 21:33
  Speaker 3: Imagine Alice places a highly specialized polarized lens over her
• 21:37
  Speaker 3: quantum system that forces it to collapse into a very
• 21:40
  Speaker 3: specific state. Okay, she tells Bob which lens she used.
• 21:44
  Speaker 3: He uses the corresponding lens on his end. No asymptotic limits,
• 21:47
  Speaker 3: no infinite branching trees of probabilities.
• 21:50
  Speaker 2: So the massive theoretical cheat code power of any operations
• 21:55
  Speaker 2: is entirely completely redundant for this specific task. A simple
• 21:59
  Speaker 2: physical one way locc protocol just putting a lens on
• 22:02
  Speaker 2: a camera and making a phone call is perfectly capable
• 22:04
  Speaker 2: of extracting this fuel.
• 22:06
  Speaker 3: It is a striking asymmetry in the laws of entanglement theory. Diluting,
• 22:10
  Speaker 3: like building the state from scratch, is a chaotic, resource
• 22:13
  Speaker 3: heavy nightmare, directly tied to complex prime number theory. Distilling
• 22:17
  Speaker 3: extracting the fuel back out is perfectly clean, simple, and instantaneous.
• 22:21
  Speaker 2: Let's take a step back and look at what we've
• 22:23
  Speaker 2: unpacked here. We started by looking for the edges of
• 22:26
  Speaker 2: how quantum fuel is manipulated, and these flower states provided
• 22:30
  Speaker 2: the ultimate mathematical X ray to see the bones of
• 22:33
  Speaker 2: the universe.
• 22:34
  Speaker 3: They really did. They proved that even non entangling cheat
• 22:37
  Speaker 3: codes have strict limits. They established a massive macroscopic irreversibility gap,
• 22:44
  Speaker 3: proving that quantum fuel is fundamentally inherently destroyed during manipulation.
• 22:48
  Speaker 2: They completely broke the Golden Child metric of squashed entanglement,
• 22:53
  Speaker 2: proving it cannot survive outside strict physical reality.
• 22:56
  Speaker 3: And they reveal this bizarre, beautifully strange number theory at
• 23:00
  Speaker 3: a quirk where prime number dimensions cause the costs of
• 23:03
  Speaker 3: zero error physics to violently skyrocket while extraction remains perfectly
• 23:08
  Speaker 3: beautifully simple.
• 23:09
  Speaker 2: It completely reshapes how we have to think about the
• 23:12
  Speaker 2: thermodynamic rules of the quantum realm, which leaves me with
• 23:15
  Speaker 2: a final thought for you to chew on.
• 23:16
  Speaker 3: Let's hear it.
• 23:17
  Speaker 2: We just saw how a single class of mathematical blueprints,
• 23:20
  Speaker 2: the Flower States, could completely upend our understanding of entanglement.
• 23:25
  Speaker 2: They expose this extreme resource trap where quantum friel just
• 23:29
  Speaker 2: disappears forever into the fabric of reality simply because the
• 23:32
  Speaker 2: universe won't let you divide a prime number. So, if
• 23:35
  Speaker 2: this mechanism was hiding in plain sight, just waiting to
• 23:38
  Speaker 2: be unlocked by one specific mathematical structure, what other foundational
• 23:43
  Speaker 2: laws of physics laws we firmly believe are unbreakable, are
• 23:46
  Speaker 2: secretly breaking down in the quantum realm right now. What
• 23:50
  Speaker 2: other impossible resource traps are out there, quietly burning up fuel,
• 23:54
  Speaker 2: just waiting for the exact right mathematical blueprint to finally
• 23:57
  Speaker 2: expose them.
• 23:58
  Speaker 3: The quantum fuel tank might be hiding quite a few
• 24:01
  Speaker 3: more leaks than we ever imagined.
• 24:03
  Speaker 2: Next time you fill up your car at the gas station,
• 24:05
  Speaker 2: just be glad you aren't paying the quantum prime number tax.