Studying Quantum Information‐ Multiple Systems - RPIQuantumComputing/QuantumCircuits GitHub Wiki

Both classical and quantum information processing rely on the analysis of multiple systems. In the previous wiki page within this series, we studied classical and quantum information within a singular system. Now, we will study multiple systems: these are mostly easily thought of as collections of singular systems, such as bits or qubits.

As previously stated, the most intuitive way to consider multiple systems is to view multiple systems as a collection of singular systems. In this way, we can study each individual system, the state that it is in, and how the state of each individual system within the multiple-system set affects the states of other systems within the medium.

Much of the time, there is a high degree of correlation when analyzing multiple systems. Within multiple systems, when a singular system's state is measured, it will possibly affect the other systems' states, provided these other systems are in the set of multiple systems being analyzed.

Let us consider an example in which A is a system with classical state set X, and B is another system with a classical state set Y. According to our definition of classical system sets in the prior wiki page, X and Y must be finite and nonempty.

Up until now, we have represented classical states as probabilistic and finite, storing our probabilities in column vectors. However, we will now think of classical states as strings. There are very common occurrences in which each individual state set is associated with individual character types. When looking at this from the perspective of multiple systems, it will be reasonable to consider sums of individual states as strings. Consider 10 bits. Since the classical states of all of these bits are all {0, 1}, the classical state sets of these systems are all the same. When written as strings, classical state sets can look like this: 0000000000 0000000001 0000000010 0000000011 0000000100 ⋮

1111111111 ​ Probabilistic States In the prior section, we addressed the idea of probabilistic states within a singular system. Now, when attempting to understand multiple probabilistic states with multiple systems, it will be beneficial to view probabilistic states collectively(think that they all form a single system).

Like in singular systems, probabilistic states of multiple systems are represented by row and column vectors. Also, the vector can be written as a linear combination of two different vectors, much like the notation seen with vectors from high school. The column vectors have indices placed in correspondence with the underlying classical state set of the system being considered.

When it comes to the representation of a probabilistic state of multiple systems as a product(linear combination) of multiple vectors, one must decide on an ordering of how the product's elements are ordered.

Independence There exists among the different probabilistic states a special type of such a state: where the systems are not dependent on one another. This is an independent system. When none of the system's classical states are dependent on one another(classical states do not depend on the classical states of individual systems within the systems), the set of multiple systems is said to be independent. We cannot measure the classical state of a hitherto unmeasured system within a set of multiple systems through a revelation in which another classical state is discovered. A special type of probabilistic state of two systems is one in which the systems are independent. Intuitively speaking, two systems are independent if learning the classical state of either system has no effect on the probabilities associated with the other. That is, learning what classical state one of the systems is in provides no information at all about the classical state of the other.

The aforementioned idea, independence, can be precisely illustrated through a tensor product calculation. Within this context, a tensor product can be defined as something concrete and simple.

Measuring Proper Subsets Rather than measuring every system within the state of multiple systems, we may choose to measure a specific subset within the set of multiple systems. In this case, each and every system which is measured will have a specific outcome.

There are multiple ways in which we can measure systems such that different results will yield. We can go from the beginning, the end, or we can take a subset of the whole system. Regarding column or row vectors, we can choose states of systems in which the vectors are linearly independent. By the same token, we can choose subsets whose row space is a subspace of a particular dimension. Additionally, we can choose the null space, row space, or column space of a particular matrix as our choice of measurement.

It is also possible for us to take multiple systems, independently performing operations on each individual system. As an example, let us consider two systems, A and B. We can perform a certain independent operation on system A. By the same token, we can also perform another independent operation on System B. Two matrices, matrix F and matrix G will map the independent operations done on A and B, respectively. Thus, the rows and columns in both F and G will correspond to the classical state sets of the systems A and B, respectively. To discern the matrix that represents both of the operations acted on a compound system whose constituents are both A and B, we will have to introduce the idea of tensor products.