Vom Bit zum Qubit

1. 00:01Hello and welcome to our next video on the lecture from bit to qubit. In the last lecture we looked at
2. 00:09how to describe a quantum bit how to make measurements, we understood how this works, then we can
3. 00:15also look at the first application, which is how the quantum computer can help us encrypt messages securely
4. 00:26To do that, let's just look at a very simple encryption scheme.
5. 00:30Namely the Vernam method the Vernam procedure is a procedure on relatively simple, is however theoretically hundred percent
6. 00:40safe.
7. 00:41That is, in this procedure we have, for example, a message for example have A very short messages
8. 00:47want to send the letter Q We can convert the letter Q into bit with the help of the ASCII code, that means here these
9. 00:56first row of bits of this bit string, that represents the letter Q.
10. 01:04Now, if we want to encrypt this letter, encrypt this message, we need to have, so to speak
11. 01:10another key and this key is simply a series of random bits and this series of random bits is
12. 01:18is the same length as our message, and if we want to encrypt the letter Q with this key, then we have to use
13. 01:27we have to add the bits from our message and the bits from our key bit by bit.
14. 01:37We just do that, that is we start from the beginning once we have practically the zero from our message the zero
15. 01:43from our key if you add them, then you get 0 out again the second bit is more complicated because we have
16. 01:52from our message the bit one from our key the bit one if it add would come out actually two
17. 01:59and when adding bit by bit so to speak we only have to subtract two until either zero or one comes out
18. 02:07that means one plus one is equal to two, if you subtract two, then 0 comes out again, the third 3. bits these are relative
19. 02:16simply zero plus zero equals zero, then we have again at the 4th 1 plus one equals two mod two gives the 0
20. 02:25at the fifth bit we have zero plus one, that gives one.
21. 02:29That means the encrypted message now has my one for once, and so on.
22. 02:35Since the key consists of random bits, the encrypted message is also a random sequence of bits
23. 02:43that means, depending on which key we apply, any other letter can come out, that is only if
24. 02:51we know the key, we can also decrypt the message again and how do we decrypt this message by using
25. 02:58we just add again the key to this encrypted message.
26. 03:03Therefore we have written the key again in the fourth row and we add bitwise again
27. 03:10zero plus zero gives 1 0 plus one gives one and so on.
28. 03:15And then we notice In this last row, the ASCII code for our letter Q appears again.
29. 03:23This method is already relatively old and we know that this method is safe provided that the key is
30. 03:31is as long as the message.
31. 03:34That is, the problem now, so to speak, is we have to somehow exchange this key, that is the person, the message
32. 03:41encrypts the message and the person decrypts the message, they both have to have the same key, that is, they have to have the same
33. 03:47for example have met before and have exchanged this key And the problem is The procedure is
34. 03:54secure only as long as the key is exactly as long as the message.
35. 03:58That means that it is relatively costly and it would be great, of course, if we had the possibility of how to also make this key
36. 04:06could transmit my key, for example, over the Internet without it being intercepted.
37. 04:14This task can help us just the quantum computer or the quantum bits
38. 04:23that means how do we send it now
39. 04:25A key using quantum bits?
40. 04:28We'll just assume that Alice wants to encrypt a message, so to speak, and she wants that message to be
41. 04:36to Bob.
42. 04:37This means that Bob also needs this key to decrypt this message.
43. 04:43And what would they be looking at?
44. 04:45Is just sending the key, and Alice does that using photons.
45. 04:51We now have Alice again sending single photons to Bob, that is, each photon corresponds to exactly one bit, and Alice
46. 04:59can now polarize the photon in different directions, for example she can again polarize the photon horizontally
47. 05:08which would correspond to a 0, or vertically, which would correspond to a 1.
48. 05:16Now of course Alice can use polarization in between and she uses a total of four different polarizations
49. 05:24namely except zero
50. 05:25it also uses the state plus The state plus is exactly between the quantum bit or the polarization direction for
51. 05:33zero without polarization direction for one that means the vector describing this polarization direction which we call plus, we can use
52. 05:43write it as one by root 2 0 vector plus one by root two the one vector and we just also introduce a
53. 05:534 polarization.
54. 05:54The polarization we call minus and the vector describing this polarization we can write as simply
55. 06:02eims by root 2 0 minus one by root 2 1 and Alice is now always randomly looking for exactly one of these four polarization directions
56. 06:13and then sends this photon to Bob
57. 06:17Bob can now make a measurement and to do this he must decide which direction to measure in.
58. 06:24That means he can either adjust this polarization beam splitter to distinguish zero from one, or he can
59. 06:32adjust the polarization beam splitter to distinguish plus from minus one.
60. 06:39that means in one case our measuring directions, just given by the polarization are zero and one and in the second case are
61. 06:47measuring directions given by polarization plus and minus
62. 06:53we have now for example, that is we have always so to speak a measuring direction as either horizontal or vertical
63. 07:00that just corresponds to the zero and the one or diagonal d, then it's either plus or minus, and both Alice can sort of
64. 07:11pick her direction in which she prepares the photon, and Bob can pick which direction she measures.
65. 07:21That is, for the first photon in this example, Alice has chosen to measure a photon in polarization direction horizontal
66. 07:29or vertically.
67. 07:30And that was that she had decided to use zero.
68. 07:34Bob also happened to have chosen to measure in the horizontal-vertical direction and by that, so to speak.
69. 07:42the directions, the preparation and the direction of measurement are identical, the result in Bob's case is the same as what Alice
70. 07:49prepared.
71. 07:50That means they both get the zero out.
72. 07:53In the second case, Alice decided to take one of the diagonals and polarize in the plus direction
73. 08:02and this photon she now sends to Bob Bob, however, because Bob does not know in which direction Alice has prepared the photon
74. 08:09has again decided for horizontal vertical.
75. 08:15That is, if the state of the photon in the state plus is Bob measures horizontally or vertically
76. 08:24Then we can calculate with which probability Bob gets so to say which measurement result and in that case he gets
77. 08:33which in 50% of the cases zero as measurement result and 50% of the cases one as measurement result, that is the measurement result is total
78. 08:42random and in our case it has now measured zero.
79. 08:47In the third case again Alice and Bob, have chosen the same direction.
80. 08:53That means Alice prepares diagonally in the minus direction Bob measures in the diagonal direction and accordingly gets
81. 09:01exactly the same result out the same state still Alice prepared and so on.
82. 09:09That is, if Alice and Bob decided to go in the same direction, then you have, so to speak, the same bit for the
83. 09:17encryption.
84. 09:19If they chose different direction, so to speak, the result is random with Bob.
85. 09:26That is, if Alice and Bob want to make sure that they have the same key, how do they agree with each other now?
86. 09:32so to speak, which measuring directions you have used, that is Alice and Bob can now for example officially communicating
87. 09:40everyone is allowed to listen in on which measurement direction you used.
88. 09:45That means now see only I have prepared horizontally vertically or I have measured in diagonally.
89. 09:52But we do not say which condition you have prepared exactly, or what you have measured exactly, and so you can
90. 10:00find out if you have the same measurement direction or a different measurement direction if you have the different
91. 10:07measurement direction, then throw away the result.
92. 10:11That is, of our five photons, photon 2 or photon three would be thrown away, so to speak, which are the red columns.
93. 10:19If we don't know there just what the result is of Alice and Bob, then in the green columns there Bob has in the same direction
94. 10:27in which Alice has also prepared and therefore we know that the key transmission may work.
95. 10:34has.
96. 10:36If now Bob has measured the state zero, so to speak, this corresponds to the key only that the minus state corresponds to the
97. 10:46so to speak also to the one.
98. 10:48That was made out before, that is it corresponds to the key one and the one state just also to the one.
99. 10:54Now it looks relatively complicated.
100. 10:56I mean here are average 50% of the qubits throw away why are Alice and Bob making this so complicated?
101. 11:07That's because they just don't know if they're being tapped or not.
102. 11:12Here in this example we have a 3rd person Eve, that comes from the English of eavesdropper, which means the one who is
103. 11:20person who listens to the message and what Eve can do she can sort of the photon that Alice sends to Bob
104. 11:29and send another photon to Bob in return.
105. 11:34However, Eve also does not know in which direction Alice has prepared, mostly the do not discuss in which direction.
106. 11:40she should measure.
107. 11:43Now we have here just in the line of Alice are in the line of Bob, they do again exactly the same as before
108. 11:51that we just now have this measurement in between.
109. 11:54Different things can happen, namely it can be that Eve, so to speak, also happens to choose the same Mess
110. 12:03direction as well as Alice and Bob, and then everything goes through again as before, that is, Alice prepares in horizontal
111. 12:10vertical for example the zero QuBit, Eve measures the QuBit in the horizontal vertical direction therefore gets zero
112. 12:19and also sends the QuBit in the zero state to Bob. Bob measures again in the same direction and gets the zero out too
113. 12:28If we now look at the third column we have an example where Alice and Bob measure in the same direction
114. 12:37but Eve has chosen the wrong measuring direction, that means Alice and Bob in this case measure in the diagonal direction
115. 12:46direction and Eve measures in the horizontal, vertical direction, that is, if Alice now measures the photon in the polarization minus
116. 12:55and Eve measures in the horizontal vertical direction, then the measurement result is totally random here.
117. 13:03In the example, for example, she has measured the one and accordingly she prepares again a photon with the polarization
118. 13:10one and sends it on to Bob and Bob now measures but again in the diagonal direction in that case can be
119. 13:19Again we know that Bob's measurement result gives the state minus 50% of the time and gives plus 50% of the time
120. 13:32that means with the probability of 50 % in this case Bob gets a different measurement result than what Alice prepares
121. 13:41although both of them have chosen, so to speak, in the same measurement base according to the same operation base and this is possible only if
122. 13:52in between someone has disturbed the photon
123. 13:55if someone has made a measurement in between.
124. 13:58We predicted that in the video as well.
125. 14:01If we make a measurement in between, then the final result can change.
126. 14:07That is, What now Alice and Bob can do to discover if someone has been eavesdropping on them is you don't tell just
127. 14:14officially what direction they chose to measure, so to speak, the red
128. 14:23columns out.
129. 14:26That is, they are not just sorting out the qubits,where they had the different measurement directions,
130. 14:33but they also decide to compare a few of the qubits that they could use for the key
131. 14:41and they compare if they have the same measurement result for these qubits if they always have the same measurement result, then
132. 14:49Alice and Bob can say with relatively high probability that they have not been tapped.
133. 14:56Because, in principle, in 25% of the cases, if someone taps every qubit, a different result should come out
134. 15:06But if they find that a few of the qubits are different, then Alice and Bob know that they have been tapped.
135. 15:16and that the key is therefore insecure and they should try again Send another key.
136. 15:24That is, quantum cryptography can help us exchange keys, and we can test whether that key is
137. 15:32has been intercepted or not.

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