EPR and Entangled States

Transcription

1 . ENTANGLED STATES Alice Bob Source of particle pairs Suppose we have two entangled spin-½ particles (A and B) described in basis by the state ψ = 2 ( * ( * Given the (hopefully by now familiar) relations (, = 2 ( (. + (. ), and (, = 2 ( (. (. ), in homework you will show that the entangled state is equivalent to ψ = 2 (. *. (. *. A) Particle A s Z-spin is measured and yields +ħ/2. That is, the spin state of A is found to be (,. Which of the following is a good prediction for particle B s spin state after Particle A s measurement? (a) *. (b) *. (c) *, (d) *, (e) We cannot predict Particle B s spin state from the knowledge of particle A s spin state even if the chosen basis is the same. - Discuss how you made this prediction. B) Particle A s Z-spin is measured and yields spin up. What is the likelihood of finding particle B s spin state in -basis as *.? (In other words, what is the probability that a measurement of X-spin for Particle B yields up?) (a) % (b) 75% (c) 5% (d) 25% (e) % (f) Other/not determined - Discuss how this likelihood was found. C) N= of these entangled pairs are sent out. Each pair constitutes an event. i) If Alice and Bob both measure Sz, how often does Alice measure Spin up? How about Bob? A. Heckler, the Ohio State University 26 (modified by CU Boulder)

2 ii) How often do Alice and Bob both measure the SAME result (up or down) for a given event? iii) If Alice measures Sz, and Bob measures Sx, how often to Alice and Bob both measure the SAME result (up or down) for a given event? iv) Are Alice s measurements causing? That is, can she send him a message instantly using these spins? A. Heckler, the Ohio State University 26 (modified by CU Boulder) 2

3 2. QUANTUM CRYPTOGRAPHY. As above, we have an entangled initial state and 2 observers. When I say measures a below, I mean spin down, and measuring a means spin up. SG means Stern-Gerlach device. Assume Alice and Bob (the two observers) always choose either X or Z SG orientations to measure. A) Bob uses a SG oriented the and measures a. If Alice measures a, which one of the following is true? (a) Alice must have an X oriented SG. (b) Alice must have a Z oriented SG. (c) Alice can have either an X or Z oriented SG. (d) None of the above has to be true. B) Bob uses a SG oriented the and measures a. If Alice measures a, which one of the following is true? (a) Alice must have an X oriented SG. (b) Alice must have a Z oriented SG. (c) Alice can have either an X or Z oriented SG. (d) None of the above has to be true. C) If Alice uses a SG oriented the and measures a, what is the likelihood that Bob measures a? Hint: Recall that Bob randomly (with 5/5 odds) orients his SG either or. (a) % certainty (b) 75% certainty (c) 5% certainty (d) 25% certainty (e) % certainty D) If Alice uses a SG oriented the and measures a, what is the likelihood that Bob measures a? Hint: Recall that Bob randomly orients his SG either or. (a) % certainty (b) 75% certainty (c) 5% certainty (d) 25% certainty (e) % certainty E) If Bob and Alice both use SGs oriented the, circle ALL statements that are true (There may be more than one) (I) If Alice measures a, Bob must measure a. (II) If Alice measures a, Bob must measure a. (III) If Alice measures a, Bob may measure either a or a. F) Bob uses a SG oriented the and Alice uses a SG oriented the,, circle ALL statements that are true (There may be more than one) (I) If Alice measures a, Bob must measure a. (II) If Alice measures a, Bob must measure a. (III) If Alice measures a, Bob may measure either a or a. A. Heckler, the Ohio State University 26 (modified by CU Boulder) 3

4 G) Circle all of the following statements that are in general correct: (I) If Bob measures a, he can always infer the result of Alice s measurement. (II) If Bob measures a, he can always infer the result of Alice s measurement. (III) If Alice and Bob both have their SGs oriented in the same direction and they know this, Bob should be able to infer the result of Alice s measurement. H) Complete the following table by recording in the third column whether or not Bob knows with certainty what Alice measures, from his knowledge of his own measurement AND Alice s SG orientation: Alice s SG Orientation SG Orientation Bob is Certain or Uncertain I) In the above table/scenario, which of the following is true? (a) Bob always knows with % certainty what Alice measures. (b) Bob sometimes knows with % certainty what Alice measures. (c) Bob never knows with % certainty what Alice measures. J) Complete the following Table with the measurements made by Bob in each case (write,, or where you are not certain because it could be either or ). Alice measures a Alice measures a the the the the measurement (,, or -) A. Heckler, the Ohio State University 26 (modified by CU Boulder) 4

5 K) Alice and Bob complete trials to generate a key (a string of s and s, that they and ONLY they know). Below is a chart of results that Bob sees. He knows his SG orientation and Alice s SG orientation (sent immediately after measurements on a public channel), and Bob knows the (secret) result of the measurement from his own orientation. Remember also, that when creating the key inverts his observed value (e.g., a becomes a, since is what Alice observes for valid trials.) When creating the Key, the rule is that you discard all results where the orientations were not the same! Alice SG orientation Bob SG orientation Bob measurement KEY X X X Z Z Z Z X Z X X X Z Z Z Z X X Z X L) Given the data set above, what is the shared key? With a shared secret key, you can now send a secret message! In practice, Alice and Bob will publicly announce the first N (some # they have decided on) elements of the key in order to check for eavesdropping. That is what the final portion of the Tutorial investigates. A. Heckler, the Ohio State University 26 (modified by CU Boulder) 5

6 3. EAVESDROPPING DETECTION. Alice Eve Bob Source of particle pairs Source Now there is a third party, Eve, who intercepts every particle sent to Bob with her own SG which she randomly orients or Z direction (She does NOT know the S-G orientations of Alice or Bob at the time she measures). Eve immediately generates a replacement particle to send to Bob in its place. If Eve measures a the, she produces and sends a particle to Bob which is in the *, state. (If Eve measures a the, she produces and sends to Bob a particle in the *. state, etc. ) Since Alice and Bob discard all measurements for which their SGs are not oriented in the same direction, (both in direction, or both X), we ignore such mismatched cases for the remainder of this tutorial. A) If Alice and Bob both have their SGs oriented X, which one of the following statements is true about whether Eve can be % sure about what Alice had noted as the corresponding bit of the key (after Alice and Bob inform each other of their SG orientations over the public channel). (a) Eve will be certain about this bit of the key if her oriented the. (b) Eve will be certain about this bit of the key if her oriented the. (c) Eve will be certain about this bit of the key regardless of the orientation of her SG. (d) Eve can never be certain about this bit of the key. B) If Alice and Bob have their SGs oriented Z and Eve has her SG oriented the, how does Eve s measurement compare to the one Bob would have performed had Eve not interfered? (a) Eve s measurement is the same as the one Bob would have made. This situation is such that Eve s presence will go undetected. (b) Eve s measurement is the opposite to the one Bob would have made (e.g., If Eve measures a then Bob would measure a and vice versa). (c) Eve s measurement is either the same as or opposite to the one Bob would have made with equal likelihood. (d) None of the above. C) If Alice and Bob have their SGs oriented Z and Eve has her SG oriented X, how does Eve s measurement (i.e., or ) compare to the one Bob would have made had Eve not interfered? (a) Eve s measurement is definitely the same as the one Bob would have made. (b) Eve s measurement is opposite the one Bob would have made (e.g. if Eve measures a, Bob would have measured a or vice versa). (c) Eve s measurement is either the same as or opposite to the one Bob would have made with equal likelihood. (d) None of the above. A. Heckler, the Ohio State University 26 (modified by CU Boulder) 6

7 D) Alice and Bob have their SGs oriented X and Eve has her SG oriented X. If Eve generates a replacement particle and sends it to Bob, as described above, what is the likelihood that Bob will make the same measurement that he would have without Eve s interference? (a) % (b) 75% (c) 5% (d) 25% (e) % E) Alice and Bob have their SGs oriented X and Eve has her SG oriented Z. If Eve generates a replacement particle and sends it to Bob, as described above, what is the likelihood that Bob will make the same measurement that he would have without Eve s interference? (a) % (b) 5% (c) 25% (d) % F) If both Alice and Bob have their SGs oriented X, what is the likelihood that Bob will detect Eve s interference (i.e, Alice and bits will not match when they later compare their records? (a) 75% (b) 5% (c) 25% (d) % G) Summarize your answers above by completing the following table. Enter the likelihood that Bob will detect Eve s interference (Alice and bits will not match if Alice and Bob were to compare some bits, e.g., every th bit of their key) for each of the following cases. Alice and SG Orientation Eve s SG Orientation Likelihood of Detecting Interference H) What is the overall likelihood that Bob will detect interference for a bit in the key (if he compares many bits with Alice) due to eavesdropping by Eve. (To think about: If we check, say, bits with Eve present, what are the odds that Bob FAILS to notice the eavesdropper?) I) Why does this system necessarily depend on quantum mechanics rather than classical mechanics? A. Heckler, the Ohio State University 26 (modified by CU Boulder) 7