Quantum Entanglement and Cryptography. Deepthi Gopal, Caltech

Transcription

1 + Quantum Entanglement and Cryptography Deepthi Gopal, Caltech

2 + Cryptography Concisely: to make information unreadable by anyone other than the intended recipient. The sender of a message scrambles/encrypts it to obscure the meaning; the recipient decrypts it. What does this require? That the recipient has enough information to always accurately unscramble the message.

3 + Cryptography Historically, this relied only on both parties knowing the method used to encrypt the message. At this point, the main method of security was simply keeping your algorithm secret. This is not very useful. Shannon s maxim: the enemy knows the system. Any decent cryptographic system should remain secure even if the enemy knows how the message was created.

4 + Keys Most modern cryptography relies in some way on the concept of a key : this is an extra piece of information (generally a large number) that controls the behaviour of the encoding algorithm, and that the recipient needs to know to decrypt the message. If the key is private, the algorithm can be public without trouble! Symmetric key encryption: there is one key for encryption and decryption, and both parties share it.

5 + Keys This of course assumes that the key is unwieldy enough to be unguessable. A 128-digit random number is quite easy to generate. This requires a would-be spy to pick through or so possibilities. How long would that take? A billion modern computers performing a billion operations a second would do it in about a trillion years. This seems fine.

6 + Keys What we ve been describing assumes that the sender and the recipient have a shared secret key. This means they need to communicate to establish the key! How can they be certain that when the key is established, there are no eavesdroppers?

7 + Keys In classical physics? They can t. Mathematically, it turns out that it is always theoretically possible for a third party to obtain classical information without drawing attention to himself. There are some clever ways to get around this, like publickey cryptography: Instead of having one shared key, any given recipient has a public encryption key and a private decryption key, mathematically interrelated. Then any sender can encrypt a message that can be decrypted by the recipient using his totally private decryption key alone.

8 + Keys Public-key encryption is probably the best thing out there. Generally, the strength of public key algorithms are based on the computational difficulty of various mathematical problems for example, large integer factorisation. As computation gets better, this is going to be harder to maintain.

9 + Quantum effects? We ve been stressing classical. How would quantum mechanics help with cryptographic problems? For one thing, it s been shown that if we can build a quantum computer, the problem of large number factorisation becomes computationally very reasonable. This is going to cause trouble! So if quantum computation is implemented, we re going to need quantum algorithms. We need to understand a little more about quantum mechanics first.

10 + Quantum mechanics On a small enough scale, the physical ideas that we take for granted often break down completely. Classically: given the position and velocity of an object, what is its trajectory? The quantum mechanical equivalent: given this object is here right now, what is the probability that it will be there later?

11 + Superposition principle Normally, if we have two boxes, and know that there is a ball hidden in one of them, we expect one to definitely contain a ball, and one to be empty. Under the principles of quantum mechanics, though: until we have opened a box, in some sense the ball exists in both boxes at once a superposition of the two mutually exclusive alternatives.

12 + Superposition principle What do we mean when we say that something is in two places at once? Think about double-slit interference, with a beam of electrons.

13 + Superposition principle Notice that there are blank spaces in the interference pattern! Experimentally, when we cover one of the slits, electrons will fill these spaces. The electrons are able to decide whether both slits are open or not, despite only passing through one. It is as though they are at both slits simultaneously (to check whether both are open)!

14 + Quantum entanglement When we re dealing with more than one particle, superposition leads to the phenomenon of entanglement. Here is the basic idea: objects can be linked in a way that causes them to have a very deep dependence on each other, even if they are separated by millions of kilometres. What? Disturbing one instantaneously disturbs the other, irrespective of separation.

15 + Quantum entanglement Imagine that we have two electrons, with opposite spin, in separate sealed boxes: one given to Alice, one to Bob, they are far apart. Remember superposition! Two cases exist simultaneously: Alice has spin up; Bob has spin down. Alice has spin down; Bob has spin up. Before Alice looks, her electron is neither spin up nor spin down; it is in an indefinite state that can only be described by referring to both electrons.

16 + Quantum entanglement Alice opens her box! If she finds a spin up electron, then Bob has a spin down electron. Regardless of the distance between Alice and Bob, Alice s act of looking into her box instantaneously affects Bob s electron. Remember Schrödinger s cat? Until we look into the box, it s both dead and alive.

17 + Quantum measurement Alice s opening the box is equivalent to making a measurement in quantum mechanics. A measurement is simply an observation of some particular piece of information about a system! (It does not tell you the state of the system directly.) Alice observed that her electron was spin up; from this she deduces that the current state of the system of boxes is spin up in her box, spin down in Bob s. Her observation changed the state of the system measurement disturbs the system!

18 + Entanglement and measurement There s a neat little example called the mean king problem. A physicist is asked by a king to prepare an atom in any state she chooses. The king then measures the spin of the atom along one of three axes. He does not tell her which axis. The physicist is then told to perform any measurement she likes, and is then told along which axis the king measured. She must now tell him the value he obtained.

19 + The mean king problem The answer? Use entanglement! The king has allowed us to prepare the atom however we want, and make whatever measurement we want. So, we prepare the atom in an entangled state with another particle. Remember the electrons? The external particle will reflect the king s measurement. So we have more information.

20 + Measurement and quantum cryptography What we ve discussed summarises as follows: measuring a quantum system disturbs the system. If Alice sends Bob a piece of quantum information, Eve the eavesdropper risks disturbing the information by observing it! Which lets Bob know that it s been intercepted. So eavesdroppers are more detectable. What about better encryption?

21 + Quantum key distribution In fact, there s a nice demonstration of how quantum effects allow Alice and Bob to set up a secure shared key. Remember that this was a problem with symmetric key encryption! Let s imagine that Alice is transmitting photons to Bob. We re going to need to describe a little more physics first.

22 + Photon polarisation Polarisation describes the direction of the oscillating fields that make up a wave of light (electric and magnetic).

23 + Polarisation A polarised photon either can or cannot pass through a polarisation filter; if it does, then it will be aligned with the filter regardless of initial state. The chance of a photon passing through a filter polarised the same way is 1, while for a perpendicular it s 0; at 45 degrees it is exactly ½.

24 + Quantum key distribution Let s return to Alice and Bob. Alice polarises photons in one of 4 directions: 0, 45, 90, and 135 degrees. Bob receives them and measures the polarisation in one of two bases. Note that he cannot measure more than once! He has already disturbed the system. If Alice sends Bob a photon polarised at 90, he can find this out by measuring in the 0-90 basis, but measuring in the basis won t help.

25 + Quantum key distribution 1. Alice sends Bob a collection of photons. 2. For each photon, Bob chooses a measurement. Note that these measurements tell him the polarisation along the lines illustrated above, for each photon. The measurement was therefore correct if Alice s photon is parallel to one of the lines. 3. These are the results of Bob s measurement, which gives him guesses for the polarisation of each photon. He keeps them to himself! But he tells Alice which measurement he used for each photon, in order.

26 + Quantum key distribution 4. Alice then tells Bob which measurements were appropriate for the photon in question. They both keep the cases in which Bob should have measured correctly, and translate into 1s and 0s: the horizontal or vertical cases correspond to 1, and the diagonal cases correspond to 0. This lets Alice and Bob generate a binary sequence that is only known to them, and they can keep going as long as they would like! It is not even necessary for their communication to be secret as long as only they are in possession of the photons.

27 + State discrimination This generalises the photon exchange! Supposing Alice has a system that can be in a finite number of states. (Like our photon(s).) Bob would like to guess the state. Quantum mechanics means he can t directly observe it; so he measures and then guesses. x! Alice! Bob! guess x! pick x! measurement!

28 + State discrimination Variations on this problem go a long way towards explaining real-life information transmission issues. Here s an example involving extra, cheating information (it s a little like the mean king!): xy! Alice! Bob! guess x! pick x, y! measurement! y!

29 + State discrimination What are we doing here? We re trying to choose the best possible measurement for Bob to make, in all possible cases. For the case involving the extra information, it turns out there s a neat little result; the extra information is useless exactly half the time.

30 + Why is all this useful? Again, as computation becomes faster and better, we need cryptographic systems that are more secure. Well-implemented quantum cryptography ought to be impossible to break into undetected. State discrimination? Imagine receiving a transmission containing more than one signal mixed together. It would be nice to distinguish these well. Think of being played two pieces of music at the same time!

31 + State discrimination is still useful! Consider a noisy communication channel, like a noisy phone line. It would be nice if we could extrapolate extra information from an echo. Remember the photons? The better Bob s ability to discriminate between photons is, the faster a usable key can be generated. So, better measurement schemes lower computation time!