Efficient many-party controlled teleportation of multi-qubit quantum information via entanglement

1. Efficient many-party controlled teleportation of multi-qubit quantum information via entanglement Chui-Ping Yang, Shih-I Chu, Siyuan Han Physical Review A, 2004 Presenting: Victoria Tchoudakov

2. Motivation • Teleportation via the control of agents is a way to create a teleportation network. • Can be used for quantum secret sharing. • Multi-qubit teleportation allows to teleport (complicated) states. • Teleport a whole system (e.g. quantum computer).

3. Outline • Introduction • Single qubit teleportation using Bell states • Single qubit teleportation using GHZ • Previous work • Single qubit teleportation via the control of n agents (using GHZ) • Extension to multi-qubit teleportation via the control of n agents (using GHZ) • Presenting a more efficient method • Single qubit teleportation via the control of one agent (using Bell states) • Extension to multi-qubit teleportation via the control of one agent (using Bell states) • Extension to multi-qubit teleportation via the control of n agents (using Bell states and GHZ)

4. Remarks • All normalization factors are omitted for simplicity. • Throughout the presentation I will use the following to represent the Bell states: • All one qubit measurements are performed in the computational basis. • I will refer to unitary rotation by respectively, as “simple rotations”.

5. Teleportation using two-particle entanglement • Suppose Alice wants to send the (unknown) quantum state to Bob. • She prepares an entangled Bell state , and shares it with Bob. • The state of the system now can be rewritten as: • Then she measures (in the Bell measurement base) the two particles she possesses and gets one of the states • She sends Bob two classical bits, according to the state she measured, and he performs a simple rotation to retrieve the original state .

6. Teleportation using three-particle entanglement (via the control of one agent) • Suppose Alice wants to send Cliff the (unknown) state via the control of Bob. • Alice uses a three-particle entangled GHZ state , which she divides between herself (2) Cliff (4) and Bob (3). • The initial state of the system can be rewritten as:

7. Teleportation using three-particle entanglement (via the control of one agent) - 2 The algorithm: • Alice performs a Bell- state measurement on her qubits (1,2) and gets one of the states . Then she sends Cliff a 2-bit classical message indicating which of the Bell states she measured. • Bob performs a Hadamard transformation on his qubit (3), and then measures it and sends the result (one classical bit) to Cliff. • Once Cliff has got all the information, he can reconstruct the original state by performing a simple rotation on his qubit.

8. Teleportation using three-particle entanglement (via the control of one agent) - 3 For example: • If Alice measured , then Bob and Cliff are left sharing • After Bob performs Hadamard transformation their shared state becomes • When Bob measures his qubit and sends the result to Cliff, the latter knows in what state his qubit is - or , and whether he should perform a simple rotation on his qubit or not, respectively.

9. Teleportation using three-particle entanglement (via the control of one agent) - 4 • Note that without Bob’s cooperation Cliff cannot fully restore the original state . • The density matrix of Cliff’s particle without Bob’s information is: or (depending on Alice’s measurement outcome). • Hence Cliff has amplitude information about Alice’s qubit, but knows nothing about its phase.

10. Single qubit teleportation via the control of n agents (using GHZ state) • It is possible to use (n+2)-qubit GHZ state to teleport Alice’s state to Bob. • The GHZ state is divided between Alice (a), Bob (b) and the n agents. • The initial state of the system can be rewritten as:

11. Single qubit teleportation via the control of n agentsusing GHZ state - 2 The algorithm: • Alice performs a Bell-state measurement on her qubits (A, a), gets one of the states , and sends the result (2-bit classical message) to Bob. • Each of the agents performs a Hadamard transformation on his qubit, measures it, and sends one classical bit to Bob. • Bob can reconstruct the original state by performing a simple rotation according to Alice’s and the agents’ results.

12. Single qubit teleportation via the control of n agentsusing GHZ state - 3 Example for n = 2: For example: • If Alice measured , then Bob and the 2 agents are left sharing . • After the agents perform Hadamard transformation the shared state becomes • When the 2 agents measure their qubits and send the result to Bob, he knows in what state his qubit is - or , and whether he should perform a simple rotation on his qubit or not, respectively.

13. Single qubit teleportation via the control of n agents using GHZ state - 4 • After Alice’s Bell state measurement, Bob and the agents share a (n+1)-qubit state if the form: or depending on Alice’s measurement outcome. • Hence even if only one of the agents doesn’t cooperate (and the rest do), after tracing out all the agents’ qubits, Bob’s qubit density will be or - insufficient to reconstruct the original state . (No information about the phase).

14. Multi-qubit teleportation via the control of n agents using GHZ state – inefficient! • It is possible to extend the above method to teleport m qubits, by preparing m copies of the (n+2)-qubit GHZ state and then performing the above protocol for each of the original m qubits. • Such a procedure requires for each agent: • m “GHZ qubits” • m Hadamard transformations • m single-qubit measurements • m-bits classical message sent to Bob (by each agent) • Thus, the described algorithm requires considerable resources and classical communication for teleportation of a large number of qubits (large m). • Note: Any agent can be chosen to be the receiver in this algorithm. (There is noting special about Bob).

15. Multi-qubit teleportation via the control of n agents – efficient method • The article [1] presents a more efficient way to teleport m qubits via the control of n agents. For each agent it will require: • 1 “GHZ qubit” • 1 Hadamard transformation • 1 single-qubit measurement • 1 bit classical message to Bob • How is that achieved? • Two-qubit entanglement (Bell states) is used for communication between Alice and Bob, • One copy of the (n+1)-qubit entangled state (Bell for n=1, GHZ for n>1) is distributed among Alice and the n agents for control. • Thus, preventing copying the controlling GHZ state for each teleported qubit, as it was in the method described before.

16. “Entangling entanglement” • Suppose you have two systems A and B, each has four states: and . • One can build an entangled state (for instance for we will get . • Now, if are Bell states, the state will be entangled “twice” – we are entangling the already entangled Bell states, “entangling entanglement”.

17. Single qubit teleportation via the control of one agent (using Bell states) • Suppose Alice wants to send Bob the unknown state via the control of Carol. • Alice prepares the following entangled state: which is divided between herself (2,4), Bob (3) and Carol (5). • Bits (2,3) are used for the communication, and bits (4,5) are used for control. • Notice that this is an “entangling entanglement” state, where the communication and control Bell pairs are entangled with each other. • The entire system state can be rewritten as

18. Single qubit teleportation via the control of one agent (using Bell states) - 2 The algorithm: • Alice performs a Bell state measurement on qubits (1,2) and sends the results to Bob – like in simple teleportation. Then the system state becomes where is the state of Bob’s qubit (3). • If Alice measured  • If Alice measured  • In order to know in which of his qubit (3) is, Bob needs information about qubits (4,5).

19. Single qubit teleportation via the control of one agent (using Bell states) - 3 • Alice and Carol perform Hadamard transformation on qubits (4,5) respectively, then . They measure their respective qubits and send the result to Bob. He can determine now in which state his qubit is: • If he got 0 (1) from both Alice and Carol then his qubit is in state • If he got 0 (1) from Alice, and 1 (0) from Carol then his qubit is in state . • Now Bob can reconstruct Alice’s original state by performing a simple rotation on his qubit.

20. Multi-qubit teleportation via the control of one agent (using Bell states) • Suppose Alice wants to send Bob m qubits, via the control of one agent (Carol). • Alice prepares the following entangled state: which is divided between herself (a,i’) Bob (i”) and Carol (c). • Then the whole system state can be rewritten as

21. Multi-qubit teleportation via the control of one agent (using Bell states) - 2 The algorithm: • Alice performs Bell-state measurements for qubits . Then the system state is where and , while are the states of the Bob’s qubits. Alice measured  Bob gets Alice measured  Bob gets

22. Multi-qubit teleportation via the control of one agent (using Bell states) - 3  Bob can recover the original state by performing a simple rotation on his qubits. But in order to know which rotation to perform, he needs information about the phase of Alice’s original state. • To provide Bob with that information, Alice and Carol both perform a Hadamard transformation on qubits (a,c) respectively. The system state now is . • Alice and Carol measure the qubits (a, c) respectively and send the results (1-bit classical message) to Bob. Now he has enough information to recover Alice’s original state: • If both Alice and Carol sent him 0 (or 1) then he knows his qubits are in the state . • If Alice sent 0 (1) and Carol 1 (0), then Bob knows his qubits are in the state .

23. Multi-qubit teleportation via the control of one agent (using Bell states) - 4 Let us show, that without Carol’s collaboration Bob cannot recover Alice’s original state: • If only Alice performs the Hadamard transformation the system’s state becomes • After tracing out qubit c, Bob’s m qubits’ density operator is:

24. Multi-qubit teleportation via the control of one agent (using Bell states) - 5 • Then, after some tedious math, one can show that the density operator for any qubit i” (belonging to Bob), after tracing out the other m-1 qubits is: , if Alice measured and , if Alice measured  Without Carol’s cooperation Bob only has the amplitude information about each qubit in Alice’s original state, but knows nothing about it’s phase.

25. Yang, Chu, and Han GHZ - only method Comparing the methods – multi-qubit teleportation via the control of one agent Alice Bob Alice Bob Carol Carol message controller ancilla entanglement “twice” entanglement target

26. Comparing the methods – multi-qubit teleportation via the control of one agent - 2 • Yang, Chu and Han’s method requires • 2(m+1) qubits to prepare the Bell states • 1 qubit for the agent • 1 single-qubit Hadamard transformation and 1 single-qubit measurement performed by the agent • 1 bit classical message sent by the agent to the receiver • Using only GHZ entanglement (as described earlier) requires: • 3m qubit to prepare the entangled GHZ state • m qubits for the agent • m single-qubit Hadamard transformations and m single-qubit measurements performed by the agent • m bit classical message sent by the agent to the receiver Yang et al method is more effective for m ≥ 2

27. We will expand the previous method to n>1 agents control, by dividing a (n+1)-qubit entangled GHZ state between Alice and the n agents. This way Bob’s ability to fully reconstruct Alice’s qubits will depend on the collaboration of all n agents, yet the reconstruction process will remain very similar to one-agent controlled teleportation n = 3 Multi-qubit teleportation via the control of many agents (by Yang, Chu, and Han) Bob Alice Eve Carol Diana message controller ancilla entanglement “twice” entanglement target

28. Multi-qubit teleportation via the control of many agents (by Yang, Chu, and Han) - 2 Decomposition of GHZ states: • When performing a Hadamard transform on each of the GHZ state’s qubits, we get: Where and , And is a sum over all possible basis states each containing an even (odd) number of “1”s. • For example, n=4:

29. Multi-qubit teleportation via the control of many agents (by Yang, Chu, and Han) - 3 • Suppose Alice wants to send Bob m qubits, via the control ofn agents . • Alice prepares the following entangled state: which is divided between herself (i’), Bob (i”) and the agents (the (n+1)-qubit GHZ states are divided between Alice and the agents). • The state of the whole system now is:

30. Multi-qubit teleportation via the control of many agents (by Yang, Chu, and Han) - 4 The algorithm: • Alice performs two-qubit Bell state measurements on her m qubit pairs . Then the system state is: where and are the states of Bob’s m qubits (i”). • Alice and the n agents perform a Hadamard transformation on their GHZ qubits. Then, the state of the system becomes: • Alice and the n agents measure their GHZ qubits and send the results (1-bit classical message each) to Bob.He can reconstruct the original state from state (or ) using a simple rotation.

31. Multi-qubit teleportation via the control of many agents (by Yang, Chu, and Han) - 5 • Bob will determine in which of the states or his qubits are by the results of Alice’s and the agents’ measurement on their GHZ qubits. • If the n agents’ results contain an even (odd) number of “1”s and Alice measured 0 (1), then Bob’s qubits are in state . • If the n agents’ results contain an odd (even) number of “1”s and Alice measured 0 (1), then Bob’s qubits are in state . • If Alice and all the agents collaborate (perform Hadamard and measure), Bob can reconstruct the original state, and the teleportation succeeds. • It is possible to show, with more tedious math, that even if one agent does not collaborate, Bob’s density matrix will not be I  he will not be able to reconstruct Alice’s original state.

32. References [1] Efficient many-party controlled teleportation of multiqubit quantum information via entanglement. C.P. Yang, S.I. Chu, and S. Han, Physical Review A 70, 022329 (2004). [2] Quantum teleportation using three-particle entanglement. A. Karlsson and M. Bourennane, Physical Review A 58, 4394. [3] Quantum Secret Sharing. M. Hillery, V. Buzek, and A. Berthiaume, Physical Review A 59, 1829 (1999).