Vom Bit zum Qubit

1. 00:00Hello and welcome to the next video of the course from Bit to QuantumBit in this video we want to look a little bit more into the
2. 00:09QuantumBits and with the measurements of these QuantumBits.
3. 00:16We have already met here an example of a QuantumBit, namely the photon, where depending on whether the photon is
4. 00:24polarized vertically or horizontally, we can store a 0 or a 1 in general, so to speak, a quantum bit.
5. 00:34bit, any quantum system that has two microscopic states that you can distinguish from each other such as
6. 00:45polarization.
7. 00:47But there are also quite a lot of other quantum systems, which you can store just one bit Another example is
8. 00:56for example the state of an atom.
9. 00:59So if we now imagine again this Borsche atom model, then we can say Okay, if it's electron a little bit closer to the
10. 01:06nucleus, then it is, for example, our zero, and if the electron is a little bit further away from this nucleus, then
11. 01:14that is a 1 and
12. 01:20we can get this ground state or this probability distribution where the electron is we can get
13. 01:28also with the help of vectors, so who for example had a lecture at the university with the linear
14. 01:36algebra, perhaps learned that there are also functions, which are basis vectors. There is also the vector space
15. 01:45of functions,
16. 01:47that is, a quantum bit is just any two-dimensional vector space or a state in that two-dimensional
17. 01:58vector space, that is, the state of a quantum bit, which we generally denote by the letter psi, that is, it is
18. 02:08a vector is a two-dimensional vector, which has two base vectors, namely once the zero with the prefactor
19. 02:17c0 and the part of the basis vector one with the prefactor c1 and these prefactors c0 and c1.
20. 02:29We can interpret these as probabilities, that is, we've got ourselves our quantum bit our quantum bit
21. 02:36it becomes psi in this state and then we make measurements and then we find that we have a probability of
22. 02:44c0 magnitude square just output of this measurement is zero, a probability c1 magnitude square is the
23. 02:55measurement output one what is magnitude squared?
24. 02:58these numbers c0 in c1.
25. 03:02These can be arbitrary complex numbers, real numbers, these are the numbers we can plot on our number line
26. 03:110 1 2 3 7.5 pi -3
27. 03:18And if we calculate the magnitude square of these real numbers, it is simply the square of this number.
28. 03:25Who has already got to know complex numbers, that is complex numbers are for example the root of minus one, the
29. 03:34has also learned how to calculate the square of the absolute value.
30. 03:38But for everybody else it's not bad, you can understand this course even if you know only real numbers and then
31. 03:45is, so to speak, the probability that we measure the qubit in the zero state simply given by c0 squared.
32. 03:55We have already learned that a photon cannot be polarized only horizontally and vertically.
33. 04:02So just can be polarized also in any direction and that we express this with an arbitrary vector theta
34. 04:09which is then a superposition between the base vector ket zero the part cos theta half and the base vector
35. 04:17ket one with the fraction sinus theta half.
36. 04:21But who says we can also measure only in horizontal or vertical?
37. 04:26In principle, we can also rotate our polarization filters arbitrarily, i.e. we can change our measuring direction similarly to
38. 04:35the direction of the polarization of the photon
39. 04:38Also by a vector express here the vector m and also the vector m can be again with our basis vectors ket null
40. 04:47and ket one.
41. 04:48That means we say now our measurement vector m is just defined by an angle mu half and has then the fraction
42. 04:56cos mu half at the ket vector zero plus sin theta half times the ket vector one and now we want to know how big is
43. 05:06the probability that our photon with the polarization described by the ket theta passes through this ppolarization filter
44. 05:13which is described by the ket m and for this we have to look at, so to speak, how these two vectors
45. 05:22namely the measurement vector shown here in blue and the polarization vector shown here in orange in connection with each other.
46. 05:30stand.
47. 05:32We already know that if the two vectors are parallel then the photon will pass with 100 % if
48. 05:40the vectors are orthogonal to each other.
49. 05:43This would be the case if we had, for example, horizontally polarized photon and a filter that only filtered vertically polarized
50. 05:50passes through.
51. 05:51Then the probability that the photon gets through is just zero, if now, so to speak, the measurement vector and the polarization vector are
52. 06:00enclose an arbitrary angle, then we can calculate the probability that the photon passes through
53. 06:07by looking at the projection of the polarization vector onto the measurement vector.
54. 06:13The projection, we've indicated it here by the red dashed line, which means we have to look at this area in
55. 06:20these curly brackets, and this length of the curly brackets gives the root of the probability and
56. 06:29this length.
57. 06:30We can calculate that if we know the angle between these two vectors in our case, where we sort of go from
58. 06:37our polarization vector and the measurement vector using angles, we can determine them relatively easily
59. 06:46This is namely theta minus mu half and then the length of these curly brackets is exactly given by the cosine
60. 06:53of this angle.
61. 06:55But what do we do if we now have our polarization vector and measurement vector not so nicely specified with angles
62. 07:04how can we calculate this we can use a little trick namely we can use the addition theorems for the
63. 07:11sine and cosine and then we know that the cosine of theta minus mu/2 is just given by cosine theta
64. 07:21half times cosine mu half plus sin theta half times sine mu half and if we look at this carefully we see that
65. 07:30this cos theta half is he same prefactor which is in front of the ket zero in the polarization vector of the photon and that
66. 07:39cosine mu/2 is exactly the fraction that precedes ket zero in the measurement vector and the same is sin theta
67. 07:47half and the sin mu/2 is just the part of the polarization vector and the measurement vector in the direction of ket one That means
68. 07:58we can calculate this probability by multiplying these prefactors before ket zero, so to speak
69. 08:06plus the prefactors before ket one.
70. 08:09And that just defines exactly the scalar product between these two vectors theta for the polarization and m for the meas.
71. 08:17Direction
72. 08:20Let's take a closer look at it now.
73. 08:23So how can we describe general measurements on a quantum bit, As I said, we have two vectors for once
74. 08:32the vector describing the state of our QuBit We can either do that using this ket notation, which is
75. 08:41means our qubit psi is in the state c0 times the basis vector ket zero plus c1 times the basis vector ket one Or else
76. 08:54if we have chosen a coordinate system, so to speak, then we can just use it normally like a column vector
77. 09:04and write it down.
78. 09:06And in the same way we also need a vector which defines the direction of measurement We have the measurement vector ket m
79. 09:16and for every vector in mathematics there is also a dual vector, which is in general hermitian conjugate vector
80. 09:26is.
81. 09:27That is Writing We do not do the whole thing then, so to speak, in which we begin with the vertical line and afterwards
82. 09:35we have the square bracket.
83. 09:36That was our vector our ket vector, but just as bra vector, The bra vector starts with the square bracket and has
84. 09:45then afterwards the vertical line.
85. 09:48the same as if we have, so to speak, to the coordinate system on the one hand column vectors and on the other hand
86. 09:56row vectors and these prefactors these cs c zero c one.
87. 10:04So we have to take the complex conjugate, so to speak, if we have complex numbers or if we have the simple case
88. 10:12where we have real numbers there is simply c0*, so the complex conjugate is the number itself
89. 10:22and if we want to know the probability that our qubit psi is in state m, then we can use this
90. 10:32probability by calculating the scalar product between these two vectors and taking the whole
91. 10:40squared.
92. 10:42We have to multiply this prefactor, so to speak, which is in front of the zero in each case.
93. 10:48If we have complex numbers, then we just have to take for the c0 the complex conjugate and have to multiply the prefactors
94. 10:57both times in front of the one are the close and then we just go the magnitude square of this sum.
95. 11:06And that just gives the probability that our qubit psi is in state m
96. 11:14and if you look at it more closely, then we find that there are certain things that are different now, if we take the measurement
97. 11:22on a quantum bit, compared to a measurement on a classical bit, namely the measurement can be the state
98. 11:33of a quantum bit.
99. 11:35So here in this example we have polarized light again and we have the first filter and the first filter lets
100. 11:42here just vertically polarized light through, that is the horizontally polarized light, that is absorbed and
101. 11:51now we put a second filter on it in this second filter is now just that lets only horizontally polarized light through
102. 11:59and if we have these two filters there one lets it only vertically polarized light through the other one horizontally
103. 12:08polarized light through, then we know that in the end there's no light coming out at all because light can't be simultaneously
104. 12:14horizontally and vertically polarized at the same time.
105. 12:18But we do this little trick We put now between these two filters a third filter, this one
106. 12:25filter, which we have rotated by 45 degrees.
107. 12:29At the very beginning we have the light again, the light comes through and through the first filter just comes vertically
108. 12:36polarized light.
109. 12:38To me, this vertically polarized light hits this polarizing filter that's rotated 45 degrees, it's a single
110. 12:47photon, then that single photon will have a 50% probability of getting through.
111. 12:53But if now this photon comes through, then the polarization direction of this photon is rotated exactly by 45 degrees.
112. 13:03polarized.
113. 13:05If now have the light rotated 45 degrees, which is now hitting this last filter of the horizontal is not letting through
114. 13:15then we have again 50% probability that the light will get through, that is in our arrangement where we have
115. 13:23have these three polarizing filters there is only 25% probability not twice 50% the light comes out at the very end
116. 13:34That is, if we have these two arrangements, so to speak, and we now put someone else and say
117. 13:43you may decide whether we put in this third one of this middle position filter or not, then we can just discover
118. 13:51whether this filter is there or not, by seeing if this comes through or not, that is what we have to remember
119. 13:59is that a measurement can change the state of a qubit.
120. 14:05Here in the case as said by the visible through this polarization filter stepped by 45 degrees when the light comes through,
121. 14:13then the light is in any case turned to 45 degrees
122. 14:20and what we can conclude from it is just once by this probability interpretation that
123. 14:28if somebody now gives us a single photon and says no idea in which direction it is polarized, make a
124. 14:36measurement figure it out, then we can't do that with a single photon,
125. 14:43that means they can't distinguish with the single measurement if a photon is horizontally polarized for example
126. 14:49or rotated by 45 degrees and the second thing which is relatively important is that we cannot copy qubits, that is
127. 15:02with single measurements we can't determine if in which direction a qubit is polarized, if we copy something
128. 15:10with the classical copier the classical copier measures, so to speak, what is there and then pushes that onto another
129. 15:16sheet.
130. 15:17And we can't do the same thing with a QuantumBit because we can't measure with a single measurement,
131. 15:24in which state a qubit is, but we generally need an infinite number of measurements to determine a state exactly
132. 15:32and these infinitely many measurements, we can't describe them in a single
133. 15:39qubit but
134. 15:41We need practically an infinite number of copies of this qubit, and only then can we measure out, so to speak, exactly what
135. 15:48are
136. 15:49The state of this quantum bit is.

To enable the transcript, please select a language in the video player settings menu.