Quantum no-deleting theorem explanation

quantum-mechanics quantum-information quantum-states

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Let me phrase the no-deleting theorem operationally (that is, in terms of explicit actions taken by people) in a way that I think will clarify your confusion.

I'll also address your explicit questions at the end of the discussion here.

The no-deleting theorem says that when you have two copies of an unknown quantum state, you can't delete one of the copies while leaving the other intact. I'll explain the meaning of that soon, but let's start with an easier theorem: when you have a single copy of an unknown quantum state, you can't delete it.

To understand the easier theorem first: Imagine Alice prepares a box in some quantum state and hands it to Bob, but doesn't tell him the state. Now Bob wants to stick the box in his Deletion Machine. Once he sticks the box in the machine, he isolates the machine from the rest of the laboratory. Is it possible for this machine to "delete" the box, such that afterwards, the state of the machine and box together is some fixed, pre-determined state that doesn't depend on the original state of the box? The answer is no, because if the final state of the machine-and-box didn't depend on the initial state, that would mean that multiple possible initial states had evolved into the same final state, a contradiction of the unitarity (or reversability) of quantum mechanics. Even if the machine smashed the box, one could hypothetically put the pieces back together. It's important we demanded the machine is isolated other issues, for instance, the machine could burn the box such that it left the same pile of ashes regardless of the initial state of the box. This would be possible if the radiation from the heat of the fire escaped the system, in a sense carrying away the information about the state of the box.

The actual no-deleting theorem is slightly less trivial. Now we start with two copies of the state. Let's say Alice prepares two boxes, box 1 and box 2, in the same quantum state. She doesn't need to clone any states in order to do this: she just knows the state she wants to make, and she makes it twice. Now Alice hands Bob the two boxes. She doesn't tell him what state they're in; he just knows they're both in the same state. Is it possible for Bob to "delete" the state on box 2? That is, he wants to stick both boxes in his new Deletion Machine, and isolate the machine from the rest of the laboratory, then have the machine act such that it leaves box 1 on the same state, but it brings box 2 and the rest of the machine to some final state that doesn't depend on the initial state of the box.

It turns out that Bob can't build such a machine. The actual mathematics is worked out wherever you look it up (say, https://en.m.wikipedia.org/wiki/Quantum_no-deleting_theorem) but I wanted to explain the conceptual content.

To address your explicit questions: Questions (1) and (2) are answered above: Alice can prepare two boxes in the same state just by making them the same way, according to some known experimental procedure. That's not cloning (which would be if she started with a box in some unknown state and then made another identical box).

To address question (3): Bob is not allowed to measure the state of the box or delete it, in our definition of deleting. He needs to use the Deletion Machine, which is isolated from the rest of his laboratory. The machine could measure the state of the box in the sense of the von Neumann picture of measurement (see https://en.m.wikipedia.org/wiki/Measurement_in_quantum_mechanics#von_Neumann_measurement_scheme) but then the state of the machine would change in a way that depended on the state of the box.

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BBmath

Updated on October 28, 2020

Comments

BBmath over 2 years

Quantum no-deleting theorem states "given two copies of some arbitrary quantum state, it is impossible to delete one of the copies."

1) How can we have copies of arbitrary (meaning unknown) quantum states ? Can somebody give a realistic example ?

2) If no-cloning theorem states that "it is impossible to create an identical copy of an arbitrary unknown quantum state" what makes us believe that we can have copies as in question 1 ?

3) What it means to delete a quantum state ? If I do a measurement on an arbitrary unknown quantum state have I not delete that quantum state ?

Thanks !
- Peter Shor over 5 years
  
  If you know a quantum state, you can theoretically make as many copies of it as you want. You can then give two of them to somebody who doesn't know the state.
- BBmath over 5 years
  
  This theorem which underpins QM information conservation now refers to known quantum states (by someone at least) ?
- Peter Shor over 5 years
  
  It refers to quantum states that are unknown to the person trying to delete them. They may or may not be known to somebody else. But if a quantum state is known to the person trying to delete it, they can indeed delete it.
- BBmath over 5 years
  
  Known quantum states can be only made through a measurement. If this information is know and shared through classical channel another observer can delete those states ?
- Peter Shor over 5 years
  
  If an observer knows what a quantum state is, he can delete it; it doesn't matter how he got the information.
- Sean E. Lake over 5 years
  
  Re: obtaining two copies of an unknown state. You're being confused by the nomenclature. Instead of thinking of them as having been "copied", think of them as known to be identical due to how they were produced. For more, see: entanglement.
BBmath over 5 years

thanks for your great answer, especially explaining the deleting process. Can I understand it as the initial state and the Deletion Machine are entangled and hence the Machine can not be left in a final, predetermined (classical) state ?