Exploring Quantum Entanglement
“Quantum mechanics : real black magic calculus” — Albert Einstein.
At the end of the 19th century, many physicists were convinced that physics had reached its pinnacle. In the words of Lord Kelvin, ‘There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.’ However, this sentiment was soon proven wrong with the advent of two groundbreaking discoveries in the early 20th century, indicating how little we know about the reality. These two are Einstein’s theory of relativity and Quantum mechanics. One describes the behavior of objects in high speed(Special relativity) and high mass(General relativity) while the other describes the physics in the atomic and subatomic scale. The modern quantum mechanics is based on dual nature(particle and wave) of an object. It challenged classical notions of determinism and measurement precision. Quantum mechanics, unlike classical mechanics, introduced indeterminism and limitations on simultaneous measurements of independent observables i.e. uncertainty. Rather than describing particles solely as particles or waves, quantum mechanics presents a more nuanced view, acknowledging their dual nature. However, even with these advances, some physicists, including Einstein, remained skeptical and sought to uncover potential gaps in the theory. In 1935, Einstein, along with collaborators Podolsky and Rosen, published the EPR paper, in which they explored a phenomenon that would come to be known as quantum entanglement, or as Einstein famously termed it, ‘Spooky action at a distance.’ Let’s delve deeper into this fascinating aspect of quantum mechanics.
At first, let’s know what is quantum entanglement. When two or more than two quantum objects are correlated to each other in such a way that the state of one is dependent on the state of others or in the mathematical sense, a state which can’t be reduced into the product states of two or more than two state vectors. So, it’s a nonlinear phenomena.
There can be several reasons for entanglement like conservation of angular momentum, Pauli’s exclusion principle etc. For example, let’s consider a He atom at the ground state. It has electronic configuration 1s². So, according to Pauli's exclusion principle, these electron’s spins will be opposite to each other. So, if we can know one electron’s spin, we will be able to know other electron’s spin instantly. We don’t know the state of them but know the possible outcomes, the spin state up or |u⟩ and state down or |d⟩. Now the probability of being in the state up-down or down-up is 50 %. So, if the state vector of this entangled system is denoted by |Ψ⟩ then,
Where the 1 and 2 denotes the 1st and 2nd electron and the presence of minus sign denotes that they obey Fermi-Dirac statistics(i.e. antisymmetric state ).
But it does not imply that we must interact two objects directly in order to entangle them. We can also entangle them without ever directly interacting with each other. This is called remote entanglement. For example, consider two particles, A and B, that initially have no direct interaction. If each particle becomes entangled with a third particle, C and D respectively, and then particles C and D are brought together and interact in such a way that their entangled states are swapped, particles A and B will become entangled with each other, despite never directly interacting. Let’s denote the initial states of particles A, B, C, and D as ∣𝜓ₐ⟩, ∣𝜓𝐵⟩, ∣𝜓𝒸⟩ and ∣𝜓|𝒹⟩, respectively. If systems A and B become entangled with systems C and D, respectively, we can represent their joint entangled state as:
Then, if systems C and D interact in a way that swaps their entangled states, the resulting state can be represented as:
Now, systems A and B are entangled with each other, despite never directly interacting. It’s state vector looks kind of different than those entangled states we face often as it has some additional complexities. But this is currently out of scope for this article.
While it may seem that we can use quantum entanglement to communicate faster than the speed of light! it’s not correct. Let’s understand via an example,
Imagine Alice and Bob have two entangled particles, like two magic coins. These coins are special because when Alice flips her coin and sees if it’s heads or tails, she knows what Bob’s coin will show, no matter how far apart they are. This happens instantly!
Now, Alice wants to send a message to Bob using these magic coins. She thinks of a clever code: heads means “yes” and tails means “no”. So, she flips her coin and gets heads. But here’s the tricky part: even though Alice knows what her coin shows, she can’t control it to make it show heads or tails. It’s like flipping a regular coin — it’s random.
So, when Bob looks at his coin, he also sees a random result. He doesn’t know if Alice got heads or tails. Even if they agreed on the code beforehand, the random outcomes of their coin flips mean they can’t use the coins to send a message faster than the speed of light.
In other words, even though the magic coins are connected in a special way, they can’t be used to send messages instantly. That’s because the outcomes of their flips are random and unpredictable, just like regular coin flips. So, while entanglement is fascinating, it doesn’t break the rules of physics by allowing faster-than-light communication despite entangled particles exhibiting instantaneous correlations.
Before discussing further about quantum entanglement, let’s know about some terminologies which are important for later discussion.
1.Pure state: It refers to a state in which the system is described by a single, definite quantum state vector. Mathematically, a pure state can be represented as a unit vector in a complex vector space, typically denoted by ∣𝜓⟩. For example a quantum system with single qubit can be represented as |ψ⟩=α∣0⟩+β∣1⟩ where the complex probability amplitudes 𝛼 and 𝛽 describe the probability of measuring the qubit in the state ∣0⟩ or ∣1⟩, respectively. The normalization condition ∣𝛼|^2+∣𝛽∣^2=1 ensures that the total probability of measuring the qubit in any state is 1.
2.Mixed state: Suppose you have a bag full of n number of balls balls among which n₁ balls are black and n₂ balls are white (n₁+n₂ = n). The probability of picking up a white ball randomly from the bag will be n₂/n and for black balls it will be n₁/n. Now consider the same case for a quantum mechanical system where you shall have quantum particles of different states instead of balls of different colours i.e., it is a statistical ensemble of pure states: {{| Ψ₁⟩, w₁}, {| Ψ₂⟩, w₂} ……, {| Ψₙ⟩, wₙ}} where wᵢ is the classical probability corresponding to | Ψᵢ⟩ state.
3.Density matrix: The density matrix, denoted by 𝜌, is used to describe the state of a quantum system, whether pure or mixed. For a pure state ∣𝜓⟩, the density matrix is given by:
𝜌=∣𝜓⟩⟨𝜓∣
Here, ∣𝜓⟩ is the state vector of the system, and ⟨𝜓∣ is its conjugate transpose (bra) vector.
For a mixed state, which is a statistical ensemble of pure states, the density matrix is a sum of the density matrices corresponding to the individual pure states weighted by their respective probabilities:
Where 𝑝𝑖 is the probability of the system being in the pure state ∣𝜓𝑖⟩. Like other Quantum mechanical operators, it is also Hermitian. Another two important characteristics of density matrix are it’s trace is always 1 and it’s positive semi definite. It provides a complete description of the state of a quantum system, including information about both the coherence between quantum states and the statistical mixtures present in mixed states.
4. Maximal and Partial entanglement: let’s consider an two qubit entangled state ∣Ψ⟩=α∣00⟩+β∣11⟩. Now if α = β. It will mean that the probability of ending up in either states is 50 percent, this state is called maximally entangled state, e.g. Bell states. Otherwise it’s known as partially entangled state.
Now, if we look at the state vectors of an entangled system, we will see that it looks like a state vector of a single qubit like it’s a single entity and the system behaves like a pure system overall i.e. if we take density matrix of an entangled system(maximal), it will behave as it’s a pure state. The off diagonal elements of a density matrix denotes coherence. In the case of maximally entangled state, these coherence factors are 0.5 which denotes maximum coherence i.e. a pure state. Now if we take the density matrices of it’s subsystems by taking the partial trace i.e. individual qubits, then the coherence becomes zero, indicating maximally mixed states. This is a very interesting property of Quantum entanglement where all the quantum object collectively behaves like a pure state and individually behaves like a maximally mixed state. This is why these entangled states can’t be written as a product state of the individual subsystems. If it was possible then the individual subsystems would be at pure state not mixed. This underscores the non-classical nature of entangled states. For example, let’s consider a maximally entangled state
or in matrix from,
now the density matrix of this system will be,
Here we can see it has maximum coherence. Let’s find out the density matrices of its subsystems,
After putting the values, we get,
Here we can clearly see, the coherence is zero i.e. the individual subsystems are in maximally mixed state.
If we want to measure the the amount of entanglement between two subsystems of a larger quantum system, we use a concept called Entanglement entropy or “Von Neuman Entropy”, named after it’s discoverer. It quantifies the degree of entanglement between the subsystems by considering the Von Neumann entropy of the reduced density matrix obtained by tracing out one of the subsystems. Mathematically, for a bipartite quantum system consisting of subsystems A and B, the entanglement entropy S is given by:
where ρA is the reduced density matrix of subsystem A obtained by tracing out subsystem B from the total density matrix of the system. If we apply this formula to the above example, we get S = 1 for both subsystems i.e. highest entropy or maximal entanglement. Entanglement entropy is particularly significant in the context of quantum many-body systems, where understanding the entanglement structure is crucial for investigating quantum phases of matter.
Finally, quantum entanglement challenges classical physics with its non-intuitive properties, defying notions of determinism and locality. From Einstein’s skepticism to its pivotal role in emerging technologies, entanglement continues to intrigue scientists and inspire groundbreaking discoveries. Its implications for the nature of reality and its potential for revolutionizing fields like quantum computing underscore its significance. As we journey deeper into the mysteries of entanglement, we embark on a quest that promises to reshape our understanding of the universe and unlock new frontiers in science and technology.
