the statement “The quantum state of a qubit is obtained by reading across a row … |0011>” seems quite problematic to me.
First, the quantum (pure) state of the considered qubit is a priori not well defined any more at this stage, the global state of the 7-qubits register must be considered as a whole because of entanglement (see next chapter)
Second, even if we could assign this qubit a well defined quantum state, using the notation “0011>” for it is very misleading, as this is used also for the tensor product of qubit states…
Hi Dominique,
All that the statement you referenced is saying is that the quantum state of the first qubit, q[0], is in a superposition of 4 states: |0>, |0>, |1>, |1>. This is simply abbreviated for ease of reading to |0011>.
You are right that that the entire state must be considered as a whole. This is precisely the mega-qubit.
Hope this helps.
Regards,
Nihal
Hi Nihal,
Well, no, what I am trying to say is that q[0] isn’t in a superposition state, which is also called a pure state by physicists. What you can assign it as best at this stage (after it has interacted with other qubits via the gates), is a mixed state, represented mathematically by a density matrix, but I fear this out of scope of this book. You can only consider a pure (“blended”) state for q[0] if the state of the mega-qubit can be expressed as the tensor product of that would-be state of q[0] and that of the rest of the qubits, which is the case at the beginning, but no more when the qubits get entangled.
And I maintain that the notation |0011> for a superposition state is clumsy…
Hope this helps, too.
Regards,
Dominique
q[0] is not a statistical ensemble of pure states (mixed states). q[0], and in fact, all the other qubits have been put in superposition as a result of the gates that acted on the qubits.
Also, bear in mind, the mega-qubit is a conceptual devices to help computer professionals understand how quantum gates act on quantum states in a way that they are familiar with. In later chapters, the tensor product is introduced to mathematically analyze these types of circuits. The mega-qubit is a way to develop a heuristic feel for how quantum circuits act.
I agree we can consider the QuBits aren’t coupled to the environment (ideally – that’s the whole point of constructing a quantum computer, and the reason why it is so difficult…)
But statistical mechanics isn’t the only case where you need mixed states (and hence density matrices to represent them). I suggest you read https://en.wikipedia.org/wiki/Quantum_state#Mixed_states, or that you ask a quantum physicist if you have one at hand, it will be handier than by emails.
The problem stems from the (mathematical) fact that in the tensor product of two vector spaces, the elements can generally not be written as the tensor product of vectors from each of this spaces, but only as linear combinations of such products (likewise a function of two variables can generally not be factored as the product of two one-variable functions…).
In our case, the global state for the set of QuBits (the mega-qubit) can be factored as a tensor product of independent states for the individual QuBits at the beginning of the circuit, and also after the mono-QuBits gates (H, X) are applied. Then we can still speak of a superposition (“blended”) state for the individual QuBits. After the gates are applied that affect multiple QuBits at once, this is no more possible. The whole QuBit set is in a superposition state, but it has no meaning to say this of individual QuBits (unless you can prove it, like when applying involutive gates twice in the following chapter).
I think the best would be to remove the sentence altogether in the next edition.
Regards,
Dominique
Thank you for your comments and suggestions for clarifications. I appreciate you taking the time to read the book carefully.
I appreciate your thoughtful observation regarding the statement about qubit states in the bookd on quantum computing.
You’re absolutely right that in the realm of quantum mechanics, the concept of individual qubit states can become entangled with the overall system, as explained in later chapters.
For a more in-depth understanding of qubits and their intricacies, you might find this article helpful: What are Qubits? Exploring the Building Blocks of Quantum Computing - Quantum AI. It provides valuable insights into qubit representation and the foundations of quantum computing. Your input adds to the discussion and helps improve the learning experience for everyone.
Many thanks for your kind comments and for the reference.