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. 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. 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. 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. 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. 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. 7 Teleportation using three-particle entanglement (via the control of one agent) - 2 The algorithm: i.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. ii.Bob performs a Hadamard transformation on his qubit (3), and then measures it and sends the result (one classical bit) to Cliff. iii.Once Cliff has got all the information, he can reconstruct the original state by performing a simple rotation on his qubit.

8. 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. 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. 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. 11 Single qubit teleportation via the control of n agents using GHZ state - 2 The algorithm: i.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. ii.Each of the agents performs a Hadamard transformation on his qubit, measures it, and sends one classical bit to Bob. iii.Bob can reconstruct the original state by performing a simple rotation according to Alice’s and the agents’ results.

12. 12 Single qubit teleportation via the control of n agents using 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. 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. 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. 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. 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. 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. 18 Single qubit teleportation via the control of one agent (using Bell states) - 2 The algorithm: I.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  In order to know in which of his qubit (3) is, Bob needs information about qubits (4,5).

19. 19 Single qubit teleportation via the control of one agent (using Bell states) - 3 II.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. III.Now Bob can reconstruct Alice’s original state by performing a simple rotation on his qubit.

20. 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. 21 Multi-qubit teleportation via the control of one agent (using Bell states) - 2 The algorithm: I.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. 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. II.To provide Bob with that information, Alice and Carol both perform a Hadamard transformation on qubits (a,c) respectively. The system state now is. III.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. 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. 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. 25 Comparing the methods – multi-qubit teleportation via the control of one agent Yang, Chu, and Han GHZ - only method controllerancilla Alice Carol Bob entanglement Alice Carol Bob message target“twice” entanglement

26. 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. 27 Multi-qubit teleportation via the control of many agents (by Yang, Chu, and Han) 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 Alice Bob Diana Carol Eve controllerancillaentanglementmessage “twice” entanglementtarget

28. 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. 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 of n 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. 30 Multi-qubit teleportation via the control of many agents (by Yang, Chu, and Han) - 4 The algorithm: I.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”). II.Alice and the n agents perform a Hadamard transformation on their GHZ qubits. Then, the state of the system becomes: II.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. 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. 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).