Independent Systems - The learning of one system’s classical state will not have any effect on the probabilities associated with the second system
- Lack of independence implies correlation between the systems
- Systems independence can be shown using a tensor product
- Tensor products are the default and natural way of multiplying state vectors in this context. is the product vector/state here
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The tensor product of two vectors is bilinear (both arguments are linear)
- This can be generalized to multilinearity when 3 or more independent vectors are multiplied
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When we choose to view multiple systems as a single system, we can figure out how a measurement works for the systems given that we have measured each system
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When only some systems are measured, the knowledge gained will (in general), affect the remaining systems
Stochastic Matrix - A matrix whose columns are probability vectors. This is also called a Markov matrix as it’s part of a Markov chain
- Multiple quantum systems can be viewed as a collective of single quantum systems
- Quantum systems → many column vectors representing states. Each column vector can have complex numbers and should have a Euclidean Norm of 1
- Euclidean Norm is multiplicative - The product of all Euclidean Norms of the systems will be 1
- When two quantum states are correlated, they are entangled states ← Simplification
- Any quantum state that isn’t a product state is an entangled state
- Examples of non-product/entangled quantum states are
- Bell states
- GHZ states
- W state
When doing a partial measurement (measuring the probability of an outcome in system X), there is no correlation with the outcome in Y at all. If there was any, it would imply FTL communication, which is realistically impossible (for now). To calculate the partial probability then, you sum up the probabilities of all outcomes in Y. They add up to 1 separately, so it’s as good as saying that all of Y is accounted for and it has nothing to do with X
- Every vector that is both a quantum state vector and a probability vector must be a standard basis vector
- Every unitary matrix is invertible
- If a square matrix transforms every standard basis vector into a probability vector, then it must be stochastic
- Composition of unitary operations is represented by matrix multiplication
- The Bell states represent an orthonormal basis for the space corresponding to two qubits
- The tensor product of two unitary matrices represents the independent application of the two operations to two parts of a compound system
- The Euclidean norm of the tensor product of two vectors is equal to the product of the Euclidean norms of the two vectors