Measurements

Last updated Jun 25, 2022

• Type note
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• On quantum computing

In this chapter, we will learn about topics such as Entanglement, Superdense Coding, Quantum Teleportation, Quantum Key Distribution, etc.

• Previously, we learned about Product states, which can be factored into the tensor product of single-qubit states, and Entangled states, which could not.

• A product state has no entanglement, i.e. if we measure a single qubit in a product state, it does not affect the other qubit.

• If we measure a single qubit in an entangled state, it can affect the other qubits. For example, consider the entangled state $|Φ^+⟩ = (|00⟩ + |11⟩) /\sqrt2$. If we measure the left qubit, we get $|0⟩$ or $|1⟩$, each with probability $1/2$, and the state collapses to $|00⟩$ or $|11⟩$, respectively. So, if we measure the left qubit and get $|0⟩$, we know that the right qubit is also in the state $|0⟩$, and same is the case if we get $|1⟩$, we know that the right qubit is also in the state $|1⟩$.
• Maximally entangled states are those Entangled states where measuring one qubit completely determines (with certainty) what the other qubit will be.

• Partially entangled states are those Entangled states where measuring one qubit does not completely determine what the other qubit will be.
• Various ways the quantify the amount of entanglement have been proposed, called Entanglement measures.

• The EPR paradox uses Locality to question the meaning of quantum mechanics.

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• Quantum correlation is the average or expected value of the product of their measurement results.

• We can use a quantum processor to show quantum mechanics is right, and nature is not described by a local hidden variable theory.

• In $1980$, Alain Aspect and others showed that the Bell inequalities are indeed violated by entanglement, and the universe is not locally realistic (i.e., not described by local hidden variable theories).

• The collapse of an entangled state occurs faster than light. Although this doesn’t mean that information can travel faster than light. This is called the no-signaling principle.

• Our explanation of quantum computing and quantum mechanics, that quantum states are superpositions and measurement collapses the state, is known as the Copenhagen interpretation.
• It is not the only interpretation, however, and others explain the EPR paradox differently.

• Classical correlations can be shared among multiple parties.

• Entanglement is monogamous. Two qubits Maximally entangled with each other are not entangled at all with another qubit.

• To send classical information, we only need to send half the number of qubits as we would bits.

• If we want to send a choice out of 4 possible choices, we need to use send two bits since two bits have four possible states.

• Quantumly, we only need to send $1$ qubit but it should be entangled with a second qubit that the receiver already has.