Multiple Systems | Understanding Quantum Information & Computation: Lesson 02

This lesson extends the description of quantum information presented in the previous lesson, including quantum states, measurements, and operations, to situations in which there are multiple systems. Such situations arise naturally in the context of information processing, both classical and quantum: large information-carrying systems are typically constructed using collections of smaller systems, such as bits or qubits.

Timecodes for lesson sections:

0:00 — Introduction
2:35 — Lesson overview
3:46 — Classical states
10:34 — Probabilistic states
16:44 — Tensor products of vectors
25:02 — Measurements of probabilistic states
32:00 — Operations on probabilistic states
36:08 — Tensor products of matrices
39:25 — Operations on probabilistic states (continued)
41:28 — Quantum states
48:38 — Measurements
56:29 — Unitary operations
1:04:37 — Conclusion

#ibmquantum #learnquantum #qiskit

[John.Watrous]

A few people have asked in the comments how we get the matrix representation at 34:41. I gave this as a reply to one such comment, but I'll re-post it in a separate comment for greater visibility:

Think about the deterministic operation where we set Y equal to X. The action on standard basis vectors looks like this: 00 —> 00, 01 —> 00, 10 —> 11, 11 —> 11. In words, the second bit gets replaced by the first. The matrix representation of this deterministic operation is the first one on the right-hand side of the equation (not including the factor of 1/2). In a similar way, setting X equal to Y gives the second matrix on the right-hand side (again, not including the factor of 1/2). If we perform one of the two, each with probability 1/2, we average the matrices to get the matrix on the left-hand side.

[igorpavicevic]

This is literally as good as many uni lectures if not even better than when I was still a student - amazing job on this one aswell; can't wait to work with it and refer this video series to other people trying to get into the field. Thank you!

[eugeniovargas]

Although I am already familiar with Q information Q - ML and Python, your explanation is so good, that certain "gaps" I had, your lecture, filled them happily. I am now "inspired" ; for lesson 2. There is always something to learn. Knowledge has many surfaces where the sunlight shines and glows differently. Thanks. E

[firstnamelastname]

the simplicity and correctness and relevance of the examples is striking

[samykamkar]

This is great, thank you for these videos!

[phillustrator]

Wow, I had no idea Matt Leblanc was into quantum computing now.

[thedaveelectricriskband]

I’m a physician by profession, with a passion for physics. I was struggling for the last three months to understand when a composite quantum state is said to be entangled. Your discussion on tensor products, and when probabilities in a compound system can or can’t be written as products of the probabilities of single systems, came as a revelation.

I was wondering if composite quantum systems have analogs of basis vectors in single systems, and then you started talking about Bell states. So now I know.

Thank you! You’re a wonderful teacher!

[quantumradio]

Thanks for the video series. Looking forward to gates & circuits coming up next!

[tomasrabas]

At 31:55 is probably a small mistake on the slides in the sence: "1. The probability that a measurement of Y yields an outcome a \in \Sigma is...".
I believe there should be "1. The probability that a measurement of Y yields an outcome b \in \Gamma".
Otherwise, totally amazing lesson :)

[stankotomic]

Awesome lecture, the whole series. Around 53:37, small misspelling in Measurement/example, the coefficient in front of |10> has to be with the sign "-" and the coefficient in front of |11> has to be with the sign "+", or similar to restore the normalization condition to the unity of the mixed state.

[piotrfrackiewicz]

Dear Professor, excellent presentation. May I use your presentation for my classes? I would like to prepare slides based on the material covered in your lectures and possibly supplement the lecture with my own examples.

Best regards,

[mttsteel]

On Bell states derivation.
I hope what I'm writing here is meaninful to you, at last.
I noticed that the two Bell states |phi_plus> and |phi_minus> appear to be very similar to the result of Hadamard Operator H acting on the two base states |0> and |1>, but substituting them with |00> and |11>.
A similar argument can be expressed on |psi_plus> and |psi_minus>.

So, one can think a bigger "Hadamard operator", say B, to be applied to the four base states |00>, |01>, |10> and |11> and use it to get the four Bell states from them.

Starting from the fact that:

H = (X + Z) / sqrt(2)

I exploited some block-matrix B adding these two tensor products and normalizing the same way:

B := ( X § X + Z § 1 ) / sqrt(2)

here "§" stands for "tensor-product-operator" and "1" stands for identity matrix.

The definition of B, amazingly for me, looks similar to the above equation of H ~ X+Z.

Expanding the tensor product using block-matrices as intermediate results, so that B looks like

{ 0 X } { 1 0 } { 1 X }
{ X 0 } { 0 -1 } { X -1 }
+ =
sqrt(2) sqrt(2) sqrt(2)

and expanding completely:

{ 0 0 0 1 } { 1 0 0 0 } { 1 0 0 1 }
{ 0 0 1 0 } { 0 1 0 0 } { 0 1 1 0 }
{ 0 1 0 0 } { 0 0 -1 0 } { 0 1 -1 0 }
{ 1 0 0 0 } { 0 0 0 -1 } { 1 0 0 -1 }
+ =
sqrt(2) sqrt(2) sqrt(2)

The columns of matrix B are the coefficients of the four Bell states, so that:

B |00> = |phi_plus>
B |11> = |phi_minus>
B |01> = |psi_plus>
B |10> = |psi_minus>


_Matteo

[lenishpandey]

I didnt get the reference of " the whole is greater than the sum of the parts" said at 11:51

[liamazar]

At 1:01:54 why does swapping phi- not change the state to -(phi-); wouldn’t the action from psi- apply to that one too?

[vishuviswanathan]

Thanks for the great presentation.
I am not clear at 1:03:26 when you say that for controlled-Not with Y as control, you swap the order of tensor products. I would appreciate help (may be, it has been taken up earlier and I am not able to locate it). Thanks

[aarushchaubey]

suggestions for material/textbooks/exercise books to practice these questions?

[suleymanemirylmaz]

Hello sir, thank you for your efforts, it is a great video.
I came across N00N status on the internet. I learn it makes precise phase measurement when optical interferometer is used in quantum optics. Can such a situation be used in quantum algorithms? If so, where is it usually used?

[CodeWarrior]

Hi there! I have a question at 52:50 . You said earlier that tensor products in quantum state represents independence of individual states. But when we measure a state X from compound system XY, the quantum state of Y changes. Doesn't it mean that the independence has collapsed along with the measurement of individual state X?

[BeeepBoop]

Hi! I'm following along and find this series to be very good, however I'm struggling with the example matrix U at 57:20. To test my knowledge I'm trying to decompose it into a matrix A tensor B acting on X and Y respectively. If I'm correct A should be a 3x3 matrix and B should be a 2x2 matrix.

So far, so good.
However, I cannot for the life of my figure out how (for example) u_41 can be 0.

If my understanding is correct u_41 = a_21 * b_21. This implies that either a_21 or b_21 is zero.

If we assume a_21 is 0, then that should also imply that u_31 is 0 as we can rewrite it as a_21 * b_11, but it isn't.
If we assume b_21 is 0, then that should also imply that u_21 is 0 as we can rewrite is as a_11 * b_21, but it isn't.

The same sort of logic can be applied to all 0-valued entries in U as far as I can tell. So I seem to arrive at the contradiction that A and B must have all non-zero entries that somehow multiply to 0 once they are tensored. So either there's an error or (much more likely) a gap in my understanding. Any clarification would be greatly appreciated!

[FatemehJaffary]

what is a qubit? and How can we find how many qubits do we have?