Exploring Qubits: The Basics of Quantum Information and Computing

[bluecap]

https://en.wikipedia.org/wiki/Qubit
(copy and paste)
"In quantum computing, a qubit (/ˈkjuːbɪt/) or quantum bit (sometimes qbit) is a unit of quantum information—the quantum analogue of the classical bit. A qubit is a two-state quantum-mechanical system, such as the polarization of a single photon: here the two states are vertical polarization and horizontal polarization."

It mentioned "A qubit is a two-state quantum-mechanism system." In an electron in an atom that is in superpositions of different position eigenstates. Can we say those position eigenstates are also qubit assuming there is a way to store or extract the information? Is the reason qubit is mostly related to two state quantum mechanical system is because there is a way to extract information in this two state setup? Then if there is a way to extract information in the electron position eigenstates (multiple state setup), it can be called qubits too?

 

[secur]

Qubit, by definition, has only two states. With three states it would be called a qutrit. Any number of states can be used, theoretically. But two, qubit, is by far the most common. Position eigenstates - in fact any "entangled" quantum states - might be used.

 

[bluecap]

Qubit, by definition, has only two states. With three states it would be called a qutrit. Any number of states can be used, theoretically. But two, qubit, is by far the most common. Position eigenstates - in fact any "entangled" quantum states - might be used.

Position eigenstates has more than 2 states.. in fact it can be 1 million states.. so what is it technically called?
Kindly confirm if we only focus on 2 state qubit because there is an algorithm, called shor algorithm, that can produce outputs. If there were none.. then it's not even considered as qubit? Is this correct thinking? Or without any algorithm, does the position eigenstates has corresponding qubit term?

 

[Truecrimson]

A common name is qudit, for a d-state system. As @secur said, ant number of states can be used. You should be able to do universal quantum computation, i.e. everything you can do with a quantum computer, with qudits or continuous variables instead of qubits.

(By the way, position eigenstates are not physical because we can't resolve two positions that are arbitrarily close to each other. So in practice we digitize it as you said, to 1 million states for example.)

In the end though, what d in the qudit is depends on how we encode information. You can use the ground state and the first excited state of an oscillator to encode a qubit, for instance. In fact, this is how superconducting qubits are usually constructed. You quantize an electrical circuit to get a harmonic oscillator and add a Josephson junction to modify the spacing between the energy levels so that there is a unique transition frequency that you can use to manipulate only the first two levels without disturbing the other levels.

 

[bluecap]

A common name is qudit, for a d-state system. As @secur said, ant number of states can be used. You should be able to do universal quantum computation, i.e. everything you can do with a quantum computer, with qudits or continuous variables instead of qubits.

(By the way, position eigenstates are not physical because we can't resolve two positions that are arbitrarily close to each other. So in practice we digitize it as you said, to 1 million states for example.)

In the end though, what d in the qudit is depends on how we encode information. You can use the ground state and the first excited state of an oscillator to encode a qubit, for instance. In fact, this is how superconducting qubits are usually constructed. You quantize an electrical circuit to get a harmonic oscillator and add a Josephson junction to modify the spacing between the energy levels so that there is a unique transition frequency that you can use to manipulate only the first two levels without disturbing the other levels.

Ok. I think my questions is not really related to qubits.. I just want to know how the superposition of positions of electrons in atoms may be able to encode information. A superposition has all eigenstates at the same time so theoretically can't this hold any information.. for example you can put value into the eigenstates as there or not there. Would anyone know of any papers regarding this?

 

[Strilanc]

Yes, you can store quantum information in the position. For example, "send the photon through slit A or through slit B" makes a decent qubit. You can understand most variations of the double-slit experiment via basic facts about qubits.

 

[bluecap]

Yes, you can store quantum information in the position. For example, "send the photon through slit A or through slit B" makes a decent qubit. You can understand most variations of the double-slit experiment via basic facts about qubits.

Ok. But what about in the unmeasured states in superposition. Since time reversal can occur. Can't information be stored in the eigenstates and via time reversal and forwarding store information in the pure state (or superposition)?

 

[Strilanc]

What do you mean by time reversal?

Obviously you can put information into "did the photon go through A or through B". Send a photon through A for yes, send it through B for no. You can also send superpositions of yes and no, which is what makes it a qubit instead of a bit.

 

[bluecap]

What do you mean by time reversal?

Obviously you can put information into "did the photon go through A or through B". Send a photon through A for yes, send it through B for no. You can also send superpositions of yes and no, which is what makes it a qubit instead of a bit.

I'm describing pure state where there is no choices for mixed state or decoherence.. meaning no slits or even screen. I'm describing pure superpositions like an electron in an atom. Here there is many eigenstates of positions. Can we put information in the eigenstates in terms of occupied or not and via time passage, makes it like magnetic recording in a magnetic tape? imagine the information in the magnetic tape as 1 or 0 or the eigenstates in the electron in the atom as occupied or not and the movement of the tape is the movement of time in the superposition?

 

[Truecrimson]

There are two different meanings of the word "state" being used here (and elsewhere). The "state" in d-state systems is the number of orthogonal states. But we know that even a qubit has an infinite number of quantum states which are superpositions of the two orthogonal states. Information can be encoded in either way. However, the amount of information that you can get out from a d-state system is at most 1 classical dit by Holevo's theorem. You can encode more than 1 dit in superposition states, but unless you have a large number of qudits that encode that identical information and then measure all of them, you can't get all of the information out.

One paper that discusses the amount of information that can be encoded using all superposition states is https://arxiv.org/abs/quant-ph/9601025 (equation 11 in particular).

 

[cosmik debris]

Position eigenstates has more than 2 states.. in fact it can be 1 million states.. so what is it technically called?
Kindly confirm if we only focus on 2 state qubit because there is an algorithm, called shor algorithm, that can produce outputs. If there were none.. then it's not even considered as qubit? Is this correct thinking? Or without any algorithm, does the position eigenstates has corresponding qubit term?

A voltage be say anywhere between 0 and 10 volts but we only use two values with some sort of thresh holding to distinguish them, we call those bits. They can be any two values in a huge analogue range. The same is similar for qbits, there may be many states but we choose two.

 

[bluecap]

There are two different meanings of the word "state" being used here (and elsewhere). The "state" in d-state systems is the number of orthogonal states. But we know that even a qubit has an infinite number of quantum states which are superpositions of the two orthogonal states. Information can be encoded in either way. However, the amount of information that you can get out from a d-state system is at most 1 classical dit by Holevo's theorem. You can encode more than 1 dit in superposition states, but unless you have a large number of qudits that encode that identical information and then measure all of them, you can't get all of the information out.

One paper that discusses the amount of information that can be encoded using all superposition states is https://arxiv.org/abs/quant-ph/9601025 (equation 11 in particular).

Can you share any paper that gives the details of how to encode information in the nonorthogonal quantum states? I'd like to test something about decoding it by using Many Minds and trying to access all branches at the same time in the Many Worlds and multiplexing the branches to decode the information.

Also when encoding information in the nonorthogonal quantum states. Is it encoding in in pure state or as mixed state? When the quantum data is sent to me.. is it in the form of pure state or mixed state? Or combined meaning only the superposition is important?