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Transmitting and Hiding Quantum Information 2018/12/20 @ 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) Contents 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information Information “The amount of uncertainty before we learn (measure)” quantifying the resource needed to store information Bit Qubit random variable X quantum state with probability distribution Shannon Entropy von Neumann Entropy 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, withSingular singular values values Information gain by Measurement “How much information has gained by measurement ?” Y Y (C-C) (C-Q-C) X | i X Mutual Information Mutual Information Y | Y i Y (Q-C) (Q-C-Q) | i | i QC Mutual Information ? F. Buscemi et al. PRL (2008); T. Sagawa et al. PRL (2008) Information Gain and Disturbance “The relation between information gain and disturbance by measurement ?” The amount of information estimated state gain and disturbance ? | ri - closeness of the states Estimation e by fidelity (or distance) measurement outcome r for pure state Measurement r | i | i input post state measurement state Information Gain and Disturbance “The relation between information gain and disturbance by measurement ?” Information Gain estimated state by averaging the estimation fidelity r | i by using the optimal estimation strategy: e we guess that the state is the singular basis of Estimation measurement operators with maximal value measurement outcome r Disturbance Measurement r by averaging the operation 2 | i | i fidelity |h | ri| input post state measurement state Information Gain and Disturbance “The relation between information gain and disturbance by measurement ?” Information Gain estimated state by averaging the estimation fidelity Trade-off between info-gain and disturbance r K. Banaszek, PRL 86, 1366 (2001) | i by using the optimal estimation strategy: e we guess that the state is the singular basis of Estimation “The more informationmeasurement is obtained operators from measurement, with maximal value the more its state is disturbed.” measurement outcome r “We gain information by measurement” Disturbance Measurement+ “Measurement disturbs a quantum r state” by averaging the operation 2 | i | i fidelity |h | ri| input post state measurement state No-Cloning Theorem William Wootters and Wojciech Zurek (1982). “A Single Quantum Cannot be Cloned”. Nature 299 802–803. 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) No-Deleting Theorem Arun K. Pati and Samuel L. Braunstein (2000). “Impossibility of deleting an unknown quantum state”. Nature 404164–165. Given two copies of arbitrary quantum state, it is impossible to delete one of the copies. … (1) Proof to satisfy (1) Not deleted ! In a closed system, one cannot destroy quantum information. 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information Information conservation and no-go theorems M. Horodecki et al. (2003) Entropy of entanglement: for No-Cloning theorem No-Deleting theorem copied by Bob deleted by Bob (⇢ ) > (⇢0 ) (⇢ ) < (⇢0 ) A A S A S A S S Entanglement increases. Entanglement decreases. Entanglement is invariant by local operation and classical communication (LOCC) No-Cloning & No- Conservation of information Deleting theorem (no change of entanglement) 2nd law of thermodynamics (entropy cannot be decreased in a closed system) Information conservation in Quantum measurement “Can we reverse quantum measurement ?” Common belief estimated state Quantum measurement is irreversible ? : true for ideal (projection) measurements In fact, | ri It is possible to reverse (undo) Estimation e the quantum weak measurement !! : non-zero success probability to retrieve the arbitrary input state after the measurement measurement outcome r Measurement Reversal r | i | i | i input post input state measurement state state Information conservation in Quantum measurement “Can we reverse quantum measurement ?” Common belief estimated state Quantum measurement is irreversible ? Non-ideal measurement : trueprocess for ideal ( weak(projection) measurement measurements) r In fact, | i a 0 + b 1 Partial collapse It |isi possible| i to reverse (undo) e Estimation the quantum0 weak measurement0 !!1 : non-zero| i success probability to retrieve the arbitraryStrength input state of after 1the/2 measurement⌘ 1 measurement measurement outcome r (ap⌘ 0 + b 1 ⌘ 1 ) 0 +(a 1 ⌘ 0 + bp⌘ 1 ) 1 | i − | i | i − | i | i | i p p Measurement Reversal r | i | i | i input post input state measurement state state Information conservation in Quantum measurement “Can we reverse quantum measurement ?” Common belief estimated state Quantum measurement is irreversible ? Non-ideal measurement : trueprocess for ideal ( weak(projection) measurement measurements) r In fact, 1st 2nd | i a 0 +measurementb 1 measurementPartial collapse It |isi possible| i to reverse (undo) e Estimation the quantum0 weak measurementReversal0 !!1 : non-zero| i success probability to retrieve the arbitraryStrength input state of after 1the/2 measurement⌘ 1 measurement measurement outcome r (ap⌘ 0 + b 1 ⌘ 1 ) 0 +(a 1 ⌘ 0 + bp⌘ 1 ) 1 | ir= 1, 2, −…, N| i | i selective− | processi | i | i p p Measurement and Reversal condition Measurement Reversal r | i | i | i input post input state measurement state state Information conservation in Quantum measurement “Can we reverse quantum measurement ?” Common belief estimated state Quantum measurement is irreversible ? Non-ideal measurement : trueprocess for ideal ( weak(projection) measurement measurements) r In fact, 1st 2nd | i a 0 +measurementb 1 measurementPartial collapse It |isi possible| i Reversibility to reverse (undo) e Estimation the quantum0 weak measurementReversal0 !!1 : non-zeroWe define| i success the reversibility probability asto theretrieve maximal the total reversal probability over all the measurement arbitraryStrength input state of after 1the/2 measurement⌘ 1 measurement outcomes,measurement r outcome r (ap⌘ 0 + b 1 ⌘ 1 ) 0 +(a 1 ⌘ 0 + bp⌘ 1 ) 1 | ir= 1, 2, −…, N| i | i selective− | processi | i | i p p Measurement and Reversalgiven as condition a function of measurement sets Measurement Reversal(independent on the input state) r | i | i | i input post input state measurement state state Result 1 Information balance in quantum measurement [1] Y. W. Cheong and SWL*, Phys. Rev. Lett. 109, 150402 (2012). [2] H.-T. Lim*, Y.-S. Ra, K.-H, Hong, SWL*, and Y.-H. Kim*, Phys. Rev. Lett. 113, 020504 (2014) After reversing quantum measurement, where is the already obtained information ? Reversal succeeds. The same amount of information is erased Information Erasure ! by measurement reversal !! Information gain Quantitative bound of information gain and reversibility of quantum measurement For qubit Result 1 Information balance in quantum measurement [1] Y. W. Cheong and SWL*, Phys. Rev. Lett. 109, 150402 (2012). [2] H.-T. Lim*, Y.-S. Ra, K.-H, Hong, SWL*, and Y.-H. Kim*, Phys. Rev. Lett. 113, 020504 (2014) After reversing quantum measurement, where is the already obtained information ? Trade-off between info-gain and reversibility Reversal succeeds. “The more information is obtained from quantum measurement,The same amount of the less possible it is to undo the measurement.”information is erased Information Erasure ! by measurement reversal !! InformationInformation gain balance (conservation of quantum information) Quantitative bound of information gain and reversibility of quantum measurement For qubit 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information Information transmission and no-go theorems Entanglement is invariant by local operation and classical communication (LOCC) No-Cloning & No- Conservation of information Deleting theorem (no change of entanglement) 2nd law of thermodynamics Information transfer is possible, but it should be still located in a closed quantum system. Entangled channel LO LO | i Quantum Teleportation “A quantum task to transfer an arbitrary quantum state to remote place” quantum channel | i M R | i % & Sender Receiver classical channel Tele portation mode a Protocol mode b mode c ! Entanglement is distributed unknown state entangled quantum channel between the sender and receiver " Measurement is performed on the | i input and a part of the quantum channel joint measurements M # Measurement result is shared through the classical channel (projection) R $ Reversing operation is performed Reversing on the out part of the quantum channel operation Quantum Teleportation • After sender’s joint measurement (for an outcome r) where effective overall measurement • Reversing operation • Overall teleportation process “Quantum teleportation can be regarded as a quantum measurement and reversal process” In
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