Classical Systems

Part of Womanium Quantum + AI 2024 program

States for Physical Systems

A physical device $X$ has some finite, non-empty set of states, for example the set ${0, 1}$ or ${a, b, c}$ .

For a set of 2 states, if we have a single device (aka, bit) then we can have four unique functions on it, these being two constant functions, the identity function and negation.

$x$	$f_{0}$	$f_{1}$	$f_{2}$	$f_{3}$
0	0	1	0	1
1	0	1	1	0

In other words each function has to give a unique state representation for each of the possible states, here 2, since each output can be either one of the states that means it will be one of 2 for each of the 2 inputs, therefore we get $2^{2} = 4$ .

A general way of writing this formula is $functions = 2^{2^{n}}$ , where n is the number of devices (bits in our case).

Probability Representation

If our information of the device's state is incomplete, then we may represent it as a probability vector of all possible states. Let's say that we have a single bit that can be either in the 0 state or the 1 state with equal probability, then we can represent it as the vector $\hat{v}$ .

\hat{v} = (\begin{matrix} 0.5 \\ 0.5 \end{matrix})

An alternative way, is to just state on of the probabilities, let's say the probability of being in state 0, and call that $p$ . So the probability of being in state 0 is $p$ , 0.5 in our example, and state 1 is $1 - p$ , also 0.5 in our example.

Info

A probability vector is valid if and only if all its elements are non-negative and their sum is equal to one.

However when measuring the device, we don't measure $\hat{v}$ or $p$ , but rather a state with some probability.

Using the probability vector representation allows us to easily apply gates on the device in the form of matrices.

Example

Applying the negation gate

[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] \hat{v} = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] (\begin{matrix} p \\ 1 - p \end{matrix})

Multiple Devices with Incomplete Information

We can be represent a system with multiple devices as the tensor product of multiple probability vectors.

\hat{v_{1}} \otimes \hat{v_{1}} \otimes \dots \otimes \hat{v_{n}}

Example: Two devices

\hat{v_{1}} \otimes \hat{v_{1}} = (\begin{matrix} a \\ b \end{matrix}) \otimes (\begin{matrix} c \\ d \end{matrix}) = (\begin{matrix} a c \\ a d \\ b c \\ b d \end{matrix})

In our 2-bit system, the vector elements will represent the probabilities for the states 00, 01, 10 and 11, respectively.

In a classical system, a probability vector $\hat{V}$ representing multiple devices must be decomposable into a tensor product of valid probability vectors, each corresponding to a single device.

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