The Magic of Entanglement in Quantum Information Theory
Imagine a world where information is transmitted instantly, where data is always secure, and where communication is seamless. It sounds like science fiction, but thanks to quantum entanglement, it’s becoming a reality. Entanglement is a concept that defies intuition, where two particles become so intricately linked that they share a single quantum state, regardless of how far apart they are. This bizarre phenomenon has puzzled scientists since it was first described by Einstein, Podolsky, and Rosen in 1935 ( also called EPR Paradox ), who famously dismissed it as “spooky action at a distance.” But as we dive deeper into the world of quantum mechanics, we’re discovering that entanglement is not only real but also incredibly useful. In this blog, we’ll explore the magic of entanglement and how it’s being harnessed to create secure communication and unlock new technologies. So buckle up, because we’re about to take a journey into the weird and wonderful world of quantum entanglement.
What is eBit?
First we start with an e-bit, short for entanglement bit, is a unit of measurement used to quantify the degree of entanglement between two qubits in a quantum state. See my terrible handwritten notation here:
Sorry for disturbing your eyes! Let’s go. To understand entanglement, it’s helpful to compare it with classical correlation. In classical physics, correlation between two particles is simply a matter of probability. For example, if two coins are flipped and one is found to be heads, the probability of the other being tails is 50%. However, in the quantum world, the correlation between two particles can be much stronger and more complex. Two particles can be entangled in a way that their states are completely dependent on one another, even if they are separated by distant galaxies!
The concept of an e-bit comes into play when we start to use entanglement as a resource for performing certain quantum information tasks. Just as a classical bit is the fundamental unit of classical information, an e-bit is the fundamental unit of entanglement that can be used to perform quantum operations that are impossible with classical information.
Quantum Teleportation Protocol
While the name “teleportation” may evoke images of Star Trek-style transporters, where matter is dematerialized and rematerialized at a different location, quantum teleportation is a bit different. We’re not actually transporting physical objects from one place to another — sorry to disappoint any Trekkies out there! Instead, we’re teleporting quantum information.So, in a way, quantum teleportation is like the transporter, but instead of sending Captain Kirk to a different planet, we’re sending his quantum state. Of course, instead of a starship, we’re using an e-bit and some classical communication. But hey, it’s still pretty cool! And, Let’s get down to protocol,
Protocol
The protocol involves a sender (Alice) who wants to transmit a qubit to a receiver (Bob), but instead of physically sending the qubit, Alice and Bob share an entangled quantum state (one e-bit) and use two bits of classical communication to complete the process.
To set up the protocol, Alice and Bob first share an e-bit, with Alice holding a qubit and Bob holding a qubit, and together they are in the entangled state. Alice then comes into possession of a third qubit that she wishes to transmit to Bob. Alice doesn’t know the state of this qubit, so she can’t just send it directly to Bob. Instead, she uses the shared entangled state and the two bits of classical communication to teleport the qubit to Bob.
The protocol assumes that it’s not possible for Alice to physically send the qubit to Bob, which is a reasonable assumption in many scenarios, such as if they are far apart or if Alice doesn’t know Bob’s exact location. It’s also impossible to transmit quantum information using classical communication alone, as classical information can easily be copied and broadcast, leading to cloning, which is impossible due to the no-cloning theorem state.
- The state of Q is unknown to Alice and Bob in the beginning of the transmission
- Correlation ( including entanglement ) between Q and other systems must be preserved by transmission
- No cloning theorem implies that if Bob receives the transmission, Alice must no longer have the qubit in it’s original state
1. Alice performs a Controlled-NOT operation
where Q is the contol and A is the target
2. Alice performs a Hadamard operation
3. Alice measures A and Q, obtaining binary outcome a and b respectively
4. Alice send a and b to BobBob performs these two steps:1. If a=1,then Bob applies an X operation to qubit B
2. If b=1,then Bob applies a Z operation to qubit B
- Take aways: Teleportation is not an application of quantum information it’s way of performing quantum communication. Quantum teleportation relies on the existence of highly entangled states, which can be difficult to create. Entanglement distillation is a process used to create these states by transforming lower quality entangled states into higher quality ones. This process is essential for successful quantum teleportation.
Here is a quantum circuit implementation of the teleportation protocol
from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister
qubit = QuantumRegister(1, "Q")
ebit0 = QuantumRegister(1, "A")
ebit1 = QuantumRegister(1, "B")
a = ClassicalRegister(1, "a")
b = ClassicalRegister(1, "b")protocol = QuantumCircuit(qubit, ebit0, ebit1, a, b)# Prepare ebit used for teleportation
protocol.h(ebit0)
protocol.cx(ebit0, ebit1)
protocol.barrier()# Alice's operations
protocol.cx(qubit, ebit0)
protocol.h(qubit)
protocol.barrier()# Alice measures and sends classical bits to Bob
protocol.measure(ebit0, a)
protocol.measure(qubit, b)
protocol.barrier()# Bob uses the classical bits to conditionally apply gates
with protocol.if_test((a, 1)):
protocol.x(ebit1)
with protocol.if_test((b, 1)):
protocol.z(ebit1)protocol.draw()
This implementation of teleportation is written in the Qiskit programming language and uses superconducting qubits. If you’re interested in exploring other implementations of teleportation, I would also recommend checking out Pennylane, which uses photonic qubits. It’s fascinating to see how different hardware can be used to achieve the same quantum task.
Superdence Coding Protocol
Superdense coding is a fascinating protocol that allows for the transmission of two classical bits of information using only one qubit of quantum communication and one e-bit of entanglement. This is a remarkable feat that demonstrates a concrete and striking use of entanglement in the context of quantum information theory.
What makes superdense coding so exciting is that it effectively doubles the classical information-carrying capacity of sending qubits through the use of shared entanglement. This is made possible by a famous theorem in quantum information theory known as Holevo’s theorem, which states that without the use of a shared entangled state, it is impossible to communicate more than one bit of classical information by sending a single qubit.
Furthermore, superdense coding has a complementary relationship with teleportation, which makes it even more interesting. While teleportation allows for the transmission of one qubit using two classical bits of communication, superdense coding achieves the opposite by allowing for the transmission of two classical bits using one qubit of quantum communication.
Protocol
* Alice has two classical bits that she wishes to transmit Bob
* Alice is able to send a single qubit to Bob
* Alice and Bob share an e-bit- Take aways: that without the e-bit Alice and Bob’s connection would be ‘impossible’. Also Holevo Theorem implies tha 2 classical bits of communication cannot be reliable transmitted by an single qubit alone.
1. Alice and Bob share an entangled pair of qubits,
which can be represented as the state |Φ+⟩ = 1/√2 (|00⟩ + |11⟩)
2. Alice wants to send two classical bits of information,
which she can encode into one of four possible states:00 ,01 ,10 , or 11.
3. Alice performs one or two gates on her qubit,
depending on the values of the two classical bits she wants to send. Specifically, she performs the following gates:
* If the second bit is 1, she applies a Z gate to her qubit:
* If the first bit is 1, she applies an X gate to her qubit:
4. Alice sends her qubit to Bob.When Bob receives the qubit, he performs the following operations:1. He applies a controlled-NOT (CNOT) gate to the entangled pair,
using Alice's qubit as the control and Bob's qubit as the target
2. He applies a Hadamard gate to his qubit
3. He measures both qubits in the standard basis.
The outcomes of the measurements correspond to the two classical bits that
Alice wanted to send. Specifically:
* If Bob measures 0 and 0, Alice sent the bit string 00.
* If Bob measures 0 and 1, Alice sent the bit string 01.
* If Bob measures 1 and 0, Alice sent the bit string 10.
* If Bob measures 1 and 1, Alice sent the bit string 11.
In short, superdense coding allows Alice to send two classical bits of information using just one qubit of quantum communication, thanks to the use of shared entanglement between Alice and Bob.
Here is a quantum circuit implementation of the superdence coding protocol
from qiskit import QuantumCircuitprotocol = QuantumCircuit(2)# Prepare ebit used for superdense coding
protocol.h(0)
protocol.cx(0, 1)
protocol.barrier()# Alice's operationsif b == "1":
protocol.z(0)
if a == "1":
protocol.x(0)
protocol.barrier()# Bob's actions
protocol.cx(0, 1)
protocol.h(0)
protocol.measure_all()protocol.draw()
This protocol is brilliant in its simplicity! Alice gets to select which Bell state she wants to share with Bob, then she sends him her qubit and Bob performs measurements to determine which Bell state Alice chose. It’s like a game of quantum “guess the Bell state”, where Alice and Bob get to cooperate and win every time thanks to their shared entanglement.
What is the difference between Quantum Teleportation and Superdence Coding protocols?
Quantum teleportation and superdense coding are two incredible quantum protocols that harness the power of entanglement to achieve seemingly impossible tasks.
Quantum teleportation allows us to send the state of one qubit from Alice to Bob using only classical communication and a pre-shared entangled pair of qubits. This protocol is absolutely mind-boggling because it allows for the “teleportation” of quantum information, which is something that was once thought to be impossible. It’s like something straight out of science fiction! But it’s Real…!
On the other hand, superdense coding is a protocol that allows us to transmit two classical bits of information from Alice to Bob using only one qubit of quantum communication and a pre-shared entangled pair of qubits. This might not sound as flashy as quantum teleportation, but it is still an incredibly powerful protocol that shows just how useful entanglement can be.
In essence, these two protocols achieve complementary aims. Quantum teleportation allows us to send quantum information using classical communication, while superdense coding allows us to send classical information using quantum communication. Both protocols showcase the remarkable capabilities of quantum mechanics which leads a whole new experience to share the information securely and instantly.
Cool, We have come so far even learn how to teleport quantum information! But hold on, it’s not all serious science or is it? Let’s have some fun playing non-local games like the CHSH game.
Playing Non-local Games: CHSH
The CHSH game was first described in a 1969 paper by John Clauser, Michael Horne, Abner Shimony, and Richard Holt. These researchers were interested in the strange properties of quantum mechanics and were trying to come up with an experiment that could test whether quantum mechanics was a complete theory or whether there were additional “hidden” variables that we didn’t know about per EPR Paradox. When Einstein said ’ God doesn’t play dices’ he tought that there should be “hidden variable” effects this action which is why he called entanglement as ‘spooky action at distance’.
To do this experiment, they came up with a game called CHSH game that involves two players, Alice and Bob, who are each given a question by a referee. The questions are chosen randomly from a set of possible questions, and Alice and Bob must each give an answer. The referee then checks whether Alice and Bob’s answers are consistent with a certain set of rules. If they are, Alice and Bob win the game; if not, they lose.
What makes the CHSH game interesting is that the rules are such that there is no way for Alice and Bob to win the game if they are only allowed to communicate classically (i.e., by sending messages to each other that can’t be faster than the speed of light). However, if Alice and Bob are allowed to share an entangled quantum state, they can win the game with a probability(%85) that violates the rules of classical physics.
For decades, researchers have been working tirelessly to eliminate the CHSH loopholes and test the Claus-Horne (CH) inequality in order to prove the existence of nonlocal quantum entanglement. And finally, after years of hard work and technological advances in quantum optics, experimentalists were able to achieve this incredible feat. In 2013 and again in 2015, Anton Zeilinger’s group at the University of Vienna and Paul Kwiat’s group at the University of Illinois at Urbana-Champaign respectively, were able to violate the CH inequality with experiments that involved a whopping 56 researchers. This groundbreaking achievement soundly refuted the theory of Local Realism and firmly established the existence of nonlocal quantum entanglement. It’s no wonder why scientists are buzzing with excitement over this amazing achievement!
What makes the Entanglement magical?
Just like a magician, entanglement can do things that seem impossible in human eyes. For example, imagine a magician who can make two coins behave as if they are connected, even if they are far apart. If one coin is flipped and lands heads up, the other coin will also land heads up, no matter where it is. This seems like magic, but it’s actually an example of entanglement in action.
In quantum mechanics, entangled particles behave in a similar way like twins. When two particles are entangled, they become connected in a way that seems to defy our classical understanding of the world. If we measure one of the particles, we can instantly determine the state of the other particle, no matter how far apart they are.
This strange behavior of entangled particles has been experimentally verified and has led to many exciting applications in quantum information science. For example, quantum entanglement is essential to quantum cryptography, where it is used to transmit messages that are completely secure from eavesdroppers. The applications of generating entangled keys to secure information and communication in the future is like discovering a treasure map to a hidden cache of precious gems and thrill of the hunt is what drives scientist forward with their reseach, and the possibility of discovering the treasure of nature is what keeps them motivated.
This is just a beginning, The CHSH game is just one example of how entanglement can be used to test the limits of our classical understanding of the world and pave the way for new discoveries and applications in quantum mechanics. Hope you enjoyed your first nonlocal game!
Conclusion
And Nobel goes too… Entanglement ! The 2022 Nobel Prize in Physics has been awarded toAlain Aspect, John Clauser, and Anton Zeilinger for their groundbreaking work on observing entanglement through Bell tests on entangled photons. Their research has provided a concrete and testable way to observe entanglement, which has deepened our understanding of quantum mechanics and opened up new possibilities for quantum information processing. The CHSH game, a clean and elegant example of a Bell test, has been instrumental in this line of work, and its importance cannot be overstated. This Nobel Prize is a well-deserved recognition of the significance of this research and its impact on our understanding of the universe. Congratulations to the laureates!
