Understanding Quantum Entanglement and Bell States

From One Qubit to Many: The Tensor Product

To describe a system with more than one qubit, we need to expand our mathematical toolkit. If we have two qubits, say A and B, each residing in its own 2-dimensional Hilbert space ($H_A$ and $H_B$), the combined system lives in a larger Hilbert space formed by the tensor product of the individual spaces, denoted $H=H_A\otimes H_B$.

The dimension of this new space is the product of the individual dimensions. For two qubits, the space is $2\times 2=4$ dimensional. The basis vectors for this 4D space are formed by taking all possible combinations of the basis vectors of the individual qubits:

• $|0{⟩}_A\otimes |0{⟩}_B$, which we write as $|00⟩$
• $|0{⟩}_A\otimes |1{⟩}_B$, which we write as $|01⟩$
• $|1{⟩}_A\otimes |0{⟩}_B$, which we write as $|10⟩$
• $|1{⟩}_A\otimes |1{⟩}_B$, which we write as $|11⟩$

A general two-qubit state is a linear combination of these four basis states:

where the sum of the squared magnitudes of the coefficients must equal 1.

The following reading provides a formal introduction to this concept. Your strong background in linear algebra will make this quite accessible.

Separable vs. Entangled States

Now that we have the framework for two-qubit states, we can make a crucial distinction. Some two-qubit states are simply combinations of definite single-qubit states. These are called separable states. They can be written as a tensor product of the individual qubit states: $|\psi {⟩}_{sep}=|{\psi }_A⟩\otimes |{\psi }_B⟩$. In such a state, the two qubits are independent; measuring one has no effect on the other.

However, most states in the two-qubit Hilbert space are not separable. These are the entangled states. An entangled state is one that cannot be factored into a product of individual qubit states. This mathematical property has a profound physical consequence: the measurement outcomes of the two qubits are correlated, regardless of the distance separating them.

The following video segments from an MIT OpenCourseWare lecture by Professor Zwiebach provide an excellent conceptual walkthrough of this distinction.

As the video highlights, if two qubits are in the state $\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$, they are individually in an uncertain superposition. Before measurement, there's a 50% chance of finding the first qubit to be $|0⟩$ and a 50% chance of finding it to be $|1⟩$. However, the moment you measure the first qubit and find it to be, say, $|0⟩$, you instantly know with 100% certainty that the second qubit, no matter how far away, will also be found in the state $|0⟩$. Their fates are linked.

This "spooky action at a distance" led Einstein, Podolsky, and Rosen (EPR) to question the completeness of quantum mechanics in their famous 1935 paper. The following reading explores this "paradox."

The Four Bell States

The simplest and most important examples of entangled states are the four Bell states. They form a complete orthonormal basis for the two-qubit Hilbert space, often called the Bell basis. They are maximally entangled, meaning the correlations between the qubits are as strong as possible.

Here they are for easy reference:

1. $|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$

   • The measurement outcomes are perfectly correlated. If the first qubit is 0, the second is 0. If the first is 1, the second is 1.

2. $|{\Phi }^{-}⟩=\frac{1}{\sqrt{2}}(|00⟩-|11⟩)$

   • The outcomes are also perfectly correlated. The minus sign introduces a phase difference, but the measurement probabilities for the basis states $|00⟩$ and $|11⟩$ are identical to $|{\Phi }^{+}⟩$.

3. $|{\Psi }^{+}⟩=\frac{1}{\sqrt{2}}(|01⟩+|10⟩)$

   • The measurement outcomes are perfectly anti-correlated. If the first qubit is 0, the second is 1. If the first is 1, the second is 0.

4. $|{\Psi }^{-}⟩=\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$

   • The outcomes are also perfectly anti-correlated. This state is also known as the "singlet state" and is the one used in the EPR paradox reading.

These states are the building blocks of many quantum algorithms and communication protocols, including quantum teleportation and some forms of Quantum Key Distribution (QKD), like the E91 protocol, which we will study later in the course.

Generating Bell States

Entanglement is not just a mathematical abstraction; it can be generated in the lab (or on a quantum computer) using a simple sequence of quantum gates. This is a topic for our next module, but seeing how it's done provides crucial insight. The standard circuit to create the Bell states from the computational basis states involves just two gates: a Hadamard gate and a Controlled-NOT (CNOT) gate.

The following video gives a detailed mathematical walkthrough, which should be very clear given your background.

The image below shows the output from running quantum circuits that create three of these Bell states on a simulator. Notice the results: for the first two states, the outcomes are only 00 and 11, each with roughly 50% probability, demonstrating perfect correlation. For the third state, the outcomes are only 01 and 10, demonstrating perfect anti-correlation.

Conclusion

In this lesson, we made the critical leap from single-qubit systems to the interconnected world of multiple qubits. You now have the mathematical and conceptual tools to understand one of quantum mechanics' most defining features.

Here are the key takeaways:

• Multi-qubit systems are described using the tensor product, which creates a larger Hilbert space whose dimension is the product of the individual dimensions.
• A multi-qubit state is separable if it can be written as a product of individual qubit states. Otherwise, it is entangled.
• Entanglement creates non-local correlations between qubits. A measurement on one qubit instantaneously influences the state of its entangled partner(s), a phenomenon highlighted by the EPR paradox.
• The four Bell states ($|{\Phi }^{+}⟩,|{\Phi }^{-}⟩,|{\Psi }^{+}⟩,|{\Psi }^{-}⟩$) are the canonical examples of maximally entangled two-qubit states.
• Entanglement can be reliably generated using simple quantum circuits, such as one combining a Hadamard and a CNOT gate.

We have seen that measuring one part of an entangled system determines the state of the other part. In our next lesson, we will formalize the measurement process itself. We will learn how to apply the Born rule to precisely calculate the probability of obtaining each possible measurement outcome for any given quantum state.