Chap 2.3 Application: Superdense Coding p 97-98

1. Chap 2.3 Application: Superdense Coding p 97-98 Dr. Charles Tappert The information presented here, although greatly condensed, comes almost entirely from the course textbook: Quantum Computation and Quantum Information by Nielsen & Chuang

2. 2.3 Application: Superdense Coding • Superdense coding is a simple yet surprising application of elementary quantum mechanics • Ideal example application of quantum mechanics • Involves two parties known as ‘Alice’ and ‘Bob’ who are a long distance from each other • Goal: Alice sends 2 bits to Bob using one qubit • Alice & Bob share pair of prepared entangled qubits • Alice initially possesses the 1st qubit and Bob the 2nd

3. 2.3 Application: Superdense Coding • To make the story of Alice and Bob interesting • Alice and Bob are married • Alice is going high-altitude mountain climbing • Alice agrees to send daily information in two bits • 1st bit = temperature: 0 if below 00C, 1 otherwise • 2nd bit = cloud cover: 0 if no clouds, 1 if clouds • Four possible messages: • 00 = Below 00C and no clouds. Love, Alice • 01 = Below 00C and clouds. Love, Alice • 10 = Above 00C and no clouds. Love, Alice • 11 = Above 00C and clouds. Love, Alice

4. 2.3 Application: Superdense Coding • Initial setup: Alice and Bob each possess half of an entangled pair of qubits • Alice can send 2 classical bits of info to Bob using a single qubit of communication

5. 2.3 Application: Superdense Coding • Alice sends her qubit to Bob after doing this • To send bits ‘00’, she does nothing to her qubit • To send bits ‘01’, she applies phase flip Z gate • To send bits ‘10’, she applies NOT gate X • To send bits ‘11’, she applies iY gate

6. 2.3 Application: Superdense Coding • The resulting states are the Bell states

7. 2.3 Application: Superdense Coding • The Bell states form an orthonormal basis that can be distinguished by quantum measurement • Alice sends her qubit to Bob so he has both of the qubits • By doing a measurement in the Bell basis, Bob can determine which of the four bit strings Alice sent • More detailed descriptions of this process follow • First, from another textbook • Second, from a Michael Nielsen video

8. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

9. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing • The two entangled qubits are in the state • Alice transforms her qubit using Pauli operators as shown on the next slide

10. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

11. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing • The Pauli gates are one-qubit gates but we are dealing with an entangled qubit in 4D • Transform of the first qubit comes from the tensor product of Alice’s qubit gate and the identity matrix • The 1st qubit is transformed and the 2nd untouched • The transfer matrices and the outputs are as shown in the next four slides

12. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

13. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

14. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

15. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

16. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing • After getting Alice’s qubit, Bob has both qubits of the entangled pair and decodes the message • He applies a CNOT gate to the pair and gets the results for each of the four cases as shown on the next four slides

17. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

18. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

19. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

20. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

21. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing • Bob can now measure the second qubit without affecting the state of the entangled pair • If 2nd qubit is , it means Alice sent 00 or 01 • If 2nd qubit is , it means Alice sent 10 or 11 • Bob now applies Hadamard gate to 1st qubit to get the results shown on the next slide

22. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing

23. 2.3 Application: Superdense Codingfrom another book Approaching Quantum Computing • Bob now knows which message was sent based on the value of the 2nd qubit and the result of applying the Hadamard gate to the 1st • The following table summarizes the process

24. 2.3 Application: Superdense CodingNielsen Video

25. 2.3 Application: Superdense Coding • Summary: Alice, interacting with only one qubit, can transmit two bits of info to Bob • Two entangled qubits are involved in the protocol • But Alice interacts with only one • This task is not possible classically (see chap 12) • This protocol received partial verification in the lab • Entangled photon pairs in a variant of superdense coding • Information is physical, and surprising physical theories such as quantum mechanics may predict surprising information processing abilities • Famous Einstein’s quote "Spooky Action at a Distance"