The Hidden Information in Quantum Superposition

A simple Description of Quantum Superposition

A qubit is a two-level system that can exist in two classical states, logic 0 and logic 1, but also in a third state called the superposition state. This third state is used for computation in a quantum computer. We have to be very careful how we describe this superposition state.

It is not just a mixture of two classical states. As Prof. Caslav Brukner describes it in an interview, it is a completely new state: "The concrete state is 'neither' or 'undefined' - it does not exist. Quantum physicists call this 'non-state' superposition." (Translated from German to English)

We don't really know what happens during superposition and it is subject to interpretation and philosophy ??. The statement that a qubit is in several states at once during superposition is not a fact but an interpretation and is controversial among physicists (see Schr?dinger's cat ??). What happens during superposition remains hidden to us due to the destructive act of observation, which creates information that only exists in the classical states.

We can only describe the result of a qubit in superposition with the mathematical construct of the psi function. It is defined as a linear combination of the two states |0> and |1>. The Dirac notation |> is used to distinguish quantum from classical states.

|Ψ> = α |0> + β |1>

The probability of finding the qubit in the state |0> is |α|^2 and |β|^2 for the state |1>. Since you can only find the qubit in one of the two states, this implies that |α|^2 + |β|^2 = 1.

Since α and β are complex numbers, you can use the Euler equation to translate α and β into the angles θ and ?. Describing the qubit superposition with two angles is more intuitive since you can describe logic gates as rotations. ??

Controlling the Superposition

While the logic gates of a classical bit can only switch its state between the states 0 and 1, the logic gates of a quantum bit can control the two continuous angles. The logic gates are physically implemented by applying a fine dose of energy to the qubit, for example in the form of a current or a laser pulse.

But why is the superposition not destroyed when a quantum computer applies energy to the qubit from an external apparatus? Can a qubit not only survive if it is not in contact with the outside world?

The superposition is not destroyed if the information goes only unidirectionally from the external apparatus to the qubits. ?? Since the information on the qubit remains hidden, it does not count as an observation. As soon as there is an interaction from the qubit to the external apparatus - wanted or unwanted - the qubit is forced to reveal its information, which can only exist as the classical state 0 or 1.

Unfortunately, today's quantum computers cannot prevent the qubit from slowly leaking its information to the environment and thus continuously losing its superposition.

The Hidden Information

Nielsen and Chuang describe the power of superposition for quantum computation in the hidden information inside the qubit: "In a sense, in the state of a qubit, Nature conceals a great deal of ‘hidden information’."

While the qubits is in superposition, it represents a vast information space that grows exponentially with the number of qubits interacting (entangled) with each other.

However Nielsen and Chuang state: "in principle one could store an entire text of Shakespeare ... However, this conclusion turns out to be misleading, because of the behavior of a qubit when observed.". We can only read out one bit of information per qubit per measurement, since the qubit is in one of two classical states when observed. ??

In theory, the continuous variables that describe the superposition have an infinite number of decimal places in which to write any data. ?? But to store this amount of information in the continuous variables would require, for example, absurdly accurate logic gates to rotate the qubit to the specific angle, and an extremely large number of measurements to resolve the value of the two angles bit by bit.

During computation, the amount of hidden information that can be exploited is limited by errors in the logic gates and errors due to external noise, both of which can be reduced by good engineering of the quantum computer. ????

?? To learn more about the qubit, read chapter 2.1 of Nielsen and Chuang's book "Quantum Computation and Quantum Information".