I Quantum computation and entropy

[antonantal]

So they are unitary? But:So they are not unitary?

I'm confused.

This is why I keep asking for a reference.

My understanding is that error detection is not unitary (involves measurement) but error correction is. Of course one could put both detection and correction under the same umbrella of "error correction".

 

[otennert]

Do you have a good reference on quantum computing that goes into all this in more detail?

Well the whole discussion teased me into re-starting the studying of "Jürgen Audretsch: Entangled Systems", which, to me, seems to be an excellent starting point for physicists with a good background on QM to study the basics of quantum information theory. I think at least this one I can recommend. But of course, there are now lots and lots of new books on that subject, the still classic one being "Nielsen/ Chuang: Quantum Computation and Quantum Information", which to Quantum Computing is like Jackson is to Classical Electrodynamics, as it were.

 

[antonantal]

Don't go by what you "would say". Do you have any actual references? Such as textbooks or peer-reviewed papers on the topic?

Nielsen / Chuang: Quantum Computation and Quantum Information
chapter 11.3.3 Measurements and entropy

"Suppose, for example, that a projective measurement described by projectors $P_i$ is performed on a quantum system, but we never learn the result of the measurement. If the state of the system before the measurement was $\rho$ then the state after is given by $$\rho^{'}=\sum_{i} P_i \rho P_i$$
The following result shows that the entropy is never decreased by this procedure, and remains constant only if the state is not changed by the measurement"

Then it goes on to calculate the entropy of the state after measurement:
$$S(\rho^{'})=-tr(\rho^{'} \log\rho^{'})$$
I will now show that this is equivalent to the entropy that I have calculated in post #7 (except Boltzmann's constant).
The state before measurement is $\Psi=\alpha|0\rangle+\beta|1\rangle$, so: $$\rho = | \Psi \rangle \langle \Psi | =
\begin{bmatrix}
\alpha^2 & \alpha\beta \\
\alpha\beta & \beta^2 \\
\end{bmatrix}
$$
The projectors $P_i$ are: $$
P_0 =
\begin{bmatrix}
1 & 0 \\
0 & 0 \\
\end{bmatrix}
, P_1 =
\begin{bmatrix}
0 & 0 \\
0 & 1 \\
\end{bmatrix}
$$
Then:
$$\rho^{'}=\sum_{i} P_i \rho P_i =
\begin{bmatrix}
\alpha^2 & 0 \\
0 & \beta^2 \\
\end{bmatrix}$$
Since the matrix is diagonal, its logarithm is:
$$\log\rho^{'} =
\begin{bmatrix}
\log(\alpha^2) & 0 \\
0 & \log(\beta^2) \\
\end{bmatrix}$$
So, the entropy is:
$$S(\rho^{'})=-tr(\rho^{'} \log\rho^{'}) = -\left[ \alpha^2 \log(\alpha^2) + \beta^2 \log(\beta^2) \right]$$
which is equivalent to what I have calculated before.

 

[otennert]

Nielsen / Chuang: Quantum Computation and Quantum Information
chapter 11.3.3 Measurements and entropy

Then it goes on to calculate the entropy of the state after measurement:
$$S(\rho^{'})=-tr(\rho^{'} \log\rho^{'})$$
I will now show that this is equivalent to the entropy that I have calculated in post #7 (except Boltzmann's constant).
The state before measurement is $\Psi=\alpha|0\rangle+\beta|1\rangle$, so: $$\rho = | \Psi \rangle \langle \Psi | =
\begin{bmatrix}
\alpha^2 & \alpha\beta \\
\alpha\beta & \beta^2 \\
\end{bmatrix}
$$
The projectors $P_i$ are: $$
P_0 =
\begin{bmatrix}
1 & 0 \\
0 & 0 \\
\end{bmatrix}
, P_1 =
\begin{bmatrix}
0 & 0 \\
0 & 1 \\
\end{bmatrix}
$$
Then:
$$\rho^{'}=\sum_{i} P_i \rho P_i =
\begin{bmatrix}
\alpha^2 & 0 \\
0 & \beta^2 \\
\end{bmatrix}$$
Since the matrix is diagonal, its logarithm is:
$$\log\rho^{'} =
\begin{bmatrix}
\log(\alpha^2) & 0 \\
0 & \log(\beta^2) \\
\end{bmatrix}$$
So, the entropy is:
$$S(\rho^{'})=-tr(\rho^{'} \log\rho^{'}) = -\left[ \alpha^2 \log(\alpha^2) + \beta^2 \log(\beta^2) \right]$$
which is equivalent to what I have calculated before.

To be quite frank: I think the real issue under discussion here is not a calculational one, it is a conceptual one, and also one that has to do with terminology. The calculation above itself is trivial, but that's not the point I think.

Let me come back to your original question, which seemed to me a rather basic one: do you think it is answered to a better part?

 

[antonantal]

To be quite frank: I think the real issue under discussion here is not a calculational one, it is a conceptual one, and also one that has to do with terminology. The calculation above itself is trivial, but that's not the point I think.

Let me come back to your original question, which seemed to me a rather basic one: do you think it is answered to a better part?

It is not clear yet which is the correct justification why entropy increases after measurement.
Is it this:

After the measurement, we can find the qubit in state $|0\rangle$ with probability $\alpha^2$ or in state $|1\rangle$ with probability $\beta^2$. So, we now have a mixed state which is an ensemble of 2 states each with its probability. This means the entropy of the system is now $$S=-k_B \sum p_i \ln(p_i)=-k_B \left[ \alpha^2 \ln(\alpha^2) + \beta^2 \ln(\beta^2) \right] > 0$$ since $\alpha^2, \beta^2 \in (0,1)$, so their logarithms are negative.
This shows that entropy has increased.

or this:

The usual view of why quantum measurement increases entropy is that it destroys information: it changes the system's state via a non-unitary, probabilistic process, so we can no longer tell what the original state was--since whichever result we get, $\ket{0}$ or $\ket{1}$, could have been produced by many different measurements from many different initial states.

The point of my previous post was to show that Nielsen / Chuang seem to have followed the same logic as I did.

 

[otennert]

As I explained, what you say above is correct: the entropy has increased, as the system is in a mixed state after the (non-selective!) measurement.

 

[vanhees71]

Do the error correction operations involve the ancillas? It didn't appear that way from your previous description; it appeared that those operations only operate on the qubits that are intended to store the desired information.

I'm not an expert in quantum computing, but as far as I understand it, the idea is to have some decoherence but to realize each q-bit of your quantum computer by many "physical q-bits" to have this possibility of "error correction". The holy grail are of course to engineer as protected as possible q-bits like superconductor topological structures.

 

[gentzen]

Do the error correction operations involve the ancillas? It didn't appear that way from your previous description; it appeared that those operations only operate on the qubits that are intended to store the desired information.

Yes, the error correction operations always involve ancillas. I personally think that "Chapter 5 Quantum error correction" in N. David Mermin "Quantum Computer Science - An Introduction" is a good place to start learning about this topic. The working of "standard" error correction schemes is nicely illustrated in Figure 5.3:

Fig 5.3 How to detect and correct the three possible single-bit-flip errors shown in Figure 5.2. One requires two ancillary Qbits (the upper two wires), each initially in the state $\ket{0}$, coupled to the codeword Qbits by cNOT gates. After the cNOT gates have acted each ancilla is measured. If both measurements give 0, then none of the erroneous NOT gates on the left have acted and none of the error-correcting NOT gates on the right need to be applied. If the upper measurement gate shows $x = 1$ and the lower one shows $y = 0$, then the uppermost of the three erroneous NOT gates has acted on the left. Its action is undone by applying the uppermost of the three NOT gates on the right. The other two possible 1-Qbit errors are similarly corrected.

Here the boxes around M with an x or y in a small box on top denote measurement operations, whose (binary) result is denoted x and y respectively. The boxes around X to the power of some combination of optionally negated x and y represent an X operation if the exponent evaluted to 1 (or true), and an identity operation if the exponent evaluated to 0 (or false).

The working of "non-standard" MWI like error correction schemes is nicely illustrated in Figure 5.4:

Fig 5.4 Automation of the error-correction process of Figure 5.3. The three controlled gates on the right - one of them a doubly controlled Toffoli gate with multiple targets - have precisely the same error-correcting effect on the three codeword Qbits as does the application of NOT gates contingent on measurement outcomes in Figure 5.3. The final state $\ket{\Psi}$ of the ancillas (which is also the state that determines the action of the three controlled gates on the right) is $\ket{00}$ if none of the erroneous NOT gates on the left has acted. It is $\ket{10}$ if only the upper erroneous NOT gate has acted, $\ket{11}$ if only the middle one has acted, and $\ket{01}$ if only the lower one has acted.

Note that both figures contain two ancilla qubits, which are both initialized to $\ket{0}$. This known state is what allows them to absorb entropy from the main qubits. With measurements, the state of the ancilla qubits is known once again after they absorbed the entropy, and hence the ancillas are "ready" to absorb entropy once again. Without measurement, the ancillas still aborbed the entropy, but they cannot be reused yet.

 

[gentzen]

So they are unitary? But:

So they are not unitary?

I'm confused.

This is why I keep asking for a reference.

The book by Mermin mentioned in my previous answer is an excellent first introduction. Scott Aaronson's old lecture notes are a good follow-up to learn in very few pages which closely related topics Mermin did not even mention:

• Lecture 27: Quantum Error Correction (8 pages, 15 pages in 2.0)
• Lecture 28: Stabilizer Formalism (9 pages, 11 pages in 2.0)
• Lecture 29: Experimental Realizations of QC (9 pages, removed in 2.0)

Scott's newer lecture notes 2.0 are more detailed (for example, its Figure 27.5 is the analog to Figure 5.3 from Mermin in my previous answer), but the lecture on experimental realization of QC has been removed. Nielsen / Chuang is great on the fundamentals and the physics. It is old (2000), but chapter "7 Quantum computers: physical realization" is still very instructive to read. By the way, the same is true for the experimental details from Ashcroft/Mermin, despite being outdated. The temptation is to omit experimental details, because they will soon be outdated. Trying to keep them up-to-date is hopeless, so the temptation to remove them once they became outdated is even stronger. Ashcroft/Mermin's decision to keep their book unchanged is very wise, but it took me a long time to understand this.

 

[antonantal]

Very good resources @gentzen. Thanks for sharing!