Transmitting and Hiding Quantum Information

Transcription

1 4th KIAS WORKSHOP on Quantum Information and Thermodynamics Transmitting and Hiding Quantum Information Seung-Woo Lee Quantum Universe Center Korea Institute for Advanced Study (KIAS)

2 Contents 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information

3 Information The amount of uncertainty before we learn (measure) quantifying the resource needed to store information Bit Qubit random variable X with probability distribution Shannon Entropy quantum state von Neumann Entropy

4 Quantum Measurement Quantum to classical transition of information magnetic moment General quantum measurement can be described by a set of operators satisfying the completeness relation (probability sum = 1) Quantum measurement process = quantum system to be measured + measurement apparatus (probe) The probability that the outcome is r The post measurement state Each operator can be written by singular-value decomposition unitary operator is a diagonal matrix, Singular values

5 Information gain by Measurement How much information has gained by measurement? Y Y (C-C) (C-Q-C) X Mutual Information i Mutual Information X (Q-C) Y Y i (Q-C-Q) Y i QC Mutual Information F. Buscemi et al. PRL (2008); T. Sagawa et al. PRL (2008)? i

6 Information Gain and Disturbance The relation between information gain and disturbance by measurement? estimated state The amount of information gain and disturbance? Estimation e ri - closeness of the states by fidelity (or distance) measurement outcome r for pure state Measurement i r i input state post measurement state

7 Information Gain and Disturbance The relation between information gain and disturbance by measurement? Estimation estimated state e ri Information Gain by averaging the estimation fidelity by using the optimal estimation strategy: we guess that the state is the singular basis of measurement operators with maximal value measurement outcome r Measurement i r i Disturbance by averaging the operation fidelity h r i 2 input state post measurement state

8 Information Gain and Disturbance The relation between information gain and disturbance by measurement? estimated state e ri Information Gain by averaging the estimation fidelity Trade-off between info-gain and disturbance K. Banaszek, PRL 86, 1366 (2001) by using the optimal estimation strategy: Estimation measurement outcome r we guess that the state is the singular basis of The more information measurement is obtained operators from measurement, with maximal value the more its state is disturbed. We gain information by measurement Measurement + Measurement disturbs a quantum state i r i Disturbance by averaging the operation fidelity h r i 2 input state post measurement state

9 No-Cloning Theorem William Wootters and Wojciech Zurek (1982). A Single Quantum Cannot be Cloned. Nature It is impossible to copy an unknown quantum state. Proof Due to the linearity of quantum theory. - If it were possible, superluminal signaling (communication faster than light) would be also possible - Fundamental resources for Quantum Cryptography (QKD)

10 No-Deleting Theorem Arun K. Pati and Samuel L. Braunstein (2000). Impossibility of deleting an unknown quantum state. Nature Given two copies of arbitrary quantum state, it is impossible to delete one of the copies. Proof (1) to satisfy (1) Not deleted! In a closed system, one cannot destroy quantum information.

11 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information

12 Information conservation and no-go theorems Entropy of entanglement: for M. Horodecki et al. (2003) No-Cloning theorem No-Deleting theorem copied by Bob deleted by Bob S( A ) < S( 0 A) Entanglement increases. S( A ) > S( 0 A) Entanglement decreases. Entanglement is invariant by local operation and classical communication (LOCC) No-Cloning & No- Deleting theorem Conservation of information (no change of entanglement) 2nd law of thermodynamics (entropy cannot be decreased in a closed system)

13 Information conservation in Quantum measurement Can we reverse quantum measurement? Estimation estimated state e ri measurement outcome r Common belief Quantum measurement is irreversible? : true for ideal (projection) measurements In fact, It is possible to reverse (undo) the quantum weak measurement!! : non-zero success probability to retrieve the arbitrary input state after the measurement Measurement i r i Reversal i input state post measurement state input state

14 Information conservation in Quantum measurement Can we reverse quantum measurement? Estimation estimated state e ri measurement outcome r Common belief Quantum measurement is irreversible? Non-ideal measurement : true for ideal (projection) measurements process (weak measurement) In fact, a 0i + b 1i Partial collapse It is possible to reverse (undo) the quantum weak measurement!! 0i 0 1 : non-zero success probability to retrieve the arbitrary Strength input state of after 1/2 the measurement apple apple 1 measurement (a p 0i + b p 1 1i) 0i +(a p 1 0i + b p 1i) 1i Measurement i r i Reversal i input state post measurement state input state

15 Information conservation in Quantum measurement Can we reverse quantum measurement? Estimation estimated state e ri measurement outcome r Measurement i r i Common belief Quantum measurement is irreversible? Non-ideal measurement : true for ideal (projection) measurements process (weak measurement) In fact, 1st a 0i + measurement b 1i It is possible to reverse (undo) the quantum weak measurement Reversal!! 0i 0 1 : non-zero success probability to retrieve the arbitrary Strength input state of after 1/2 the measurement apple apple 1 measurement Measurement and Reversal condition Reversal i 2nd measurement Partial collapse (a p 0i + b p 1 1i) 0i +(a p 1 0i + b p 1i) 1i r= 1, 2,, N selective process input state post measurement state input state

16 Information conservation in Quantum measurement Can we reverse quantum measurement? Estimation estimated state e ri measurement outcome r Measurement i r i Common belief Quantum measurement is irreversible? Non-ideal measurement : true for ideal (projection) measurements process (weak measurement) In fact, 1st a 0i + measurement b 1i It is possible Reversibility to reverse (undo) the quantum weak measurement Reversal!! 0i 0 1 : non-zero We define success the reversibility probability as to the retrieve maximal the total arbitrary reversal Strength input probability state of after over 1/2 the all measurement the measurement outcomes, apple apple 1 measurement r Measurement and Reversal given as condition a function of measurement sets Reversal (independent on the input state) i 2nd measurement Partial collapse (a p 0i + b p 1 1i) 0i +(a p 1 0i + b p 1i) 1i r= 1, 2,, N selective process input state post measurement state input state

17 Result 1 Information balance in quantum measurement [1] Y. W. Cheong and SWL*, Phys. Rev. Lett. 109, (2012). [2] H.-T. Lim*, Y.-S. Ra, K.-H, Hong, SWL*, and Y.-H. Kim*, Phys. Rev. Lett. 113, (2014) After reversing quantum measurement, where is the already obtained information? Reversal succeeds. Information Erasure! The same amount of information is erased by measurement reversal!! Information gain Quantitative bound of information gain and reversibility of quantum measurement For qubit

18 Result 1 Information balance in quantum measurement [1] Y. W. Cheong and SWL*, Phys. Rev. Lett. 109, (2012). [2] H.-T. Lim*, Y.-S. Ra, K.-H, Hong, SWL*, and Y.-H. Kim*, Phys. Rev. Lett. 113, (2014) After reversing quantum measurement, where is the already obtained information? Information gain Trade-off between info-gain and reversibility Reversal succeeds. The more information is obtained from quantum measurement, the less possible it is to undo the measurement. Information Erasure! The same amount of information is erased by measurement reversal!! Information balance (conservation of quantum information) Quantitative bound of information gain and reversibility of quantum measurement For qubit

19 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information

20 Information transmission and no-go theorems Entanglement is invariant by local operation and classical communication (LOCC) No-Cloning & No- Deleting theorem Conservation of information (no change of entanglement) 2nd law of thermodynamics Information transfer is possible, but it should be still located in a closed quantum system. i LO Entangled channel LO

21 Quantum Teleportation A quantum task to transfer an arbitrary quantum state to remote place i M quantum channel R i % & Sender classical channel Receiver! " # $ Teleportation Protocol Entanglement is distributed between the sender and receiver Measurement is performed on the input and a part of the quantum channel Measurement result is shared through the classical channel Reversing operation is performed on the out part of the quantum channel mode a mode b mode c unknown state i joint measurements entangled quantum channel M (projection) R Reversing operation

22 Quantum Teleportation After sender s joint measurement (for an outcome r) where Reversing operation Overall teleportation process effective overall measurement Quantum teleportation can be regarded as a quantum measurement and reversal process In general, effective measurement operator can be defined as : local joint measurement basis : entangled quantum channel with its optimal reversing operation

23 Result II Information balance determines optimal quantum teleportation [3] SWL, to be submitted. Non-ideal quantum channel or measurement Information gain by the sender Maximal success prob. of teleportation = Reversibility No info gain All info transferred Info gain Partial info transferred i M maximal entanglement R i i M non- maximal entanglement R i % & Sender Receiver % & Sender Receiver Success prob. = 1 Success prob. < 1 Conditions for optimal quantum communications minimize the information gain by the (effective nonlocal) measurement maximize the reversibility of the measurement

24 (Example) quantum channel where joint measurement : : product state : maximal entanglement Bell basis effective measurement optimal reversing operators

25 (Example) quantum channel where joint measurement : : product state : maximal entanglement Bell basis Reversibility = the highest success probability of the teleportation effective measurement (standard teleportation) when higher than the ones by previously known protocols optimal reversing operators

26 Multiparty Quantum Teleportation! Entanglement is shared between senders, intermediators and receivers M R i % & Sender Receiver " Measurements are performed by senders and intermediators effective measurement : measurement basis by intermediator ' M Intermediator $ Reversing operations are performed on the # receivers parties Measurement results are sent from senders and intermediators to receivers through classical channels

27 Quantum communications in arbitrary quantum network M M M M M M R M M M M M Information balance determines optimal protocols of any quantum communications!

28 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information

29 Hiding Classical Information Hiding information Destroying information from macroscopic objects by irreversibility of dissipative process (e.g. destroying the media itself) Classical information can be effectively hidden by encrypting the original message e.g. Vernam Cipher M : original message encoding message C = M K K : random n-bit key decoding message C K = M Original information is neither in the encoded message nor the random key, but in the correlations between the two strings. Can we hide quantum information into correlations between two subsystems, such that any subsystem has no information?

30 Perfect hiding of quantum information Perfect hiding process It maps an arbitrary quantum state to a fixed state. ih =! 0 Encodes an arbitrary input quantum state into a larger Hilbert-space. If i I! i OA 0 =tr A ( i OA h ) is independent of the input state, the process is a hiding process.

31 No-Hiding Theorem Samuel L. Braunstein and Arun K. Pati (2007). Quantum Information cannot be completely hidden in correlations. PRL Quantum information cannot be hidden in the correlations between a pair of systems. - If the original information is missing at some parts, it must move to somewhere else. Conservation of information i I Ai! X k p pk ki O A k ( )i A where 0 = X { ki} are the (orthonormal) eigenvectors for p k kihk k { A k i} are the (orthonormal) eigenvectors for the ancilla. due to the linearity and unitarity of physical process A k (a i + b? i)i = a A k ( )i + b A k (? )i Information is here!

32 Imperfect hiding approaches Randomization Decoupling - An unknown quantum state chosen from a set of orthogonal states can be perfectly hidden. - Set of input states Q assuming an input system - Reformulation of no-hiding - Achievable bound

33 Teleportation? argument from No-Hiding theorem Information gain by Alice =0?

34 Result III Revision of the quantum information conservation [4] SWL, in preparation 2nd law of thermodynamics Quantum information No-Cloning, No-Deleting, No-Hiding theorem Classical Information Landauer s principle Theory of relativity i Entangled channel i Conservation of quantum information?

35 Result III Revision of the quantum information conservation [4] SWL, in preparation 2nd law of thermodynamics Quantum information No-Cloning, No-Deleting, No-Hiding theorem Quantum measurement Encoding, feedforwarding Classical Information Landauer s principle Theory of relativity i i M R memory Entangled channel i i M memory R Conservation of quantum information!

36 Summary 1. Our result explains the conservation of quantum information quantitatively in quantum measurement. 2. It determines optimal quantum communication protocols. 3. Complete hiding of quantum information is possible by spatially (non-locally) separating classical and quantum information.