Transmitting and Hiding Quantum Information

2018/12/20 @ 4th KIAS WORKSHOP on Quantum Information and Thermodynamics

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 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 general, effective measurement operator can be deﬁned as

: local joint measurement basis : entangled quantum channel

with its optimal reversing operation Result II Information balance determines optimal quantum teleportation [3] SWL, to be submitted.

Maximal success prob. of Non-ideal quantum Information gain teleportation = channel or measurement by the sender Reversibility

No info gain All info transferred Info gain Partial info transferred M maximal R non- maximal | i entanglement | i | i M 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 (Example) • quantum channel where : product state : maximal entanglement • joint measurement : Bell basis

• effective measurement

• optimal reversing operators (Example) • quantum channel where : product state : maximal entanglement • joint measurement : 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 Multiparty Quantum ! Entanglement is shared between Teleportation senders, intermediators and receivers

M R | i % & Sender Receiver

" Measurements are ' M $ performed by senders Intermediator Reversing and intermediators operations are performed on the effective measurement # receivers’ parties Measurement results are sent from senders and intermediators to receivers through classical : measurement basis by channels intermediator 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 ! 1. Basic concepts 2. Conservation of quantum information 3. Transmitting quantum information 4. Hiding quantum information Hiding information • Destroying information from macroscopic objects by irreversibility of dissipative process (e.g. destroying the media itself) • Hiding Classical Information Classical information can be effectively hidden by encrypting the original message e.g. Vernam Cipher M : original message K : random n-bit key encoding message decoding message C = M K 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 ? 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.

| iI ! | iOA If ⇢ =tr ( ) 0 A | iOAh | is independent of the input state, the process is a hiding process. No-Hiding Theorem Samuel L. Braunstein and Arun K. Pati (2007). “Quantum Information cannot be completely hidden in correlations”. PRL 98 080502

- 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

A ppk k O Ak( ) A | iI ⌦ | i ! | i ⌦ | i Xk where k are the (orthonormal) eigenvectors for ⇢0 = pk k k {| i} | ih | k A are the (orthonormal) eigenvectors for the ancilla.X {| ki} due to the linearity and unitarity of physical process

Ak(a + b ) = a Ak( ) + b Ak( ) Information is here ! | | i | ?i i | i | ? i 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 Teleportation ? • argument from No-Hiding theorem Information gain by Alice =0

? Result III Revision of the quantum information conservation [4] SWL, in preparation

2nd law of thermodynamics

Quantum information Classical Information

No-Cloning, Landauer’s principle No-Deleting, Theory of relativity No-Hiding theorem

| i

Entangled channel

| i

Conservation of quantum information ? Result III Revision of the quantum information conservation [4] SWL, in preparation

2nd law of thermodynamics

Quantum information Classical Information

Landauer’s principle No-Cloning, Quantum measurement No-Deleting, Theory of relativity No-Hiding theorem Encoding, feedforwarding

memory | i M | i R 0101110

Entangled channel

memory M R | i | i 0101110

Conservation of quantum information ! 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.