Transmitting and Hiding Quantum Informationevents.kias.re.kr/ckfinder/userfiles/201812/files... · 2018-12-28 · Reversibility We deﬁne the reversibility as the maximal total reversal

Transmitting and Hiding Quantum Information

Seung-Woo Lee

Quantum Universe Center Korea Institute for Advanced Study (KIAS)

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

1. Basic concepts

2. Conservation of quantum information

3. Transmitting quantum information

4. Hiding quantum information

Contents

Information“The amount of uncertainty before we learn (measure)”

quantifying the resource needed to store information

Bit

Shannon Entropy

random variable Xwith probability distribution

Qubit

quantum state

von Neumann Entropy

Quantum Measurement

magnetic moment

Quantum measurement process =quantum system to be measured + measurement apparatus (probe)

General quantum measurement can be described by a set of operators

satisfying the completeness relation (probability sum = 1)

“Quantum to classical transition of information”

The probability that the outcome is r

Each operator can be written by singular-value decomposition

unitary operator is a diagonal matrix,

with singular values Singular values

The post measurement state

Information gain by Measurement “How much information has gained by measurement ?”

(C-C) (C-Q-C)

Y

XMutual Information

Y

X| i

Mutual Information

(Q-C-Q) Y

| i

| Y i

?

(Q-C)

Y

| iQC Mutual Information

F. Buscemi et al. PRL (2008); T. Sagawa et al. PRL (2008)

post measurement

state

Measurement

measurement outcome r

| i | ri

| e ri

Estimation

estimated state

input state

The amount of information gain and disturbance ?

- closeness of the states

by fidelity (or distance)

“The relation between information gain and disturbance by measurement ?”

Information Gain and Disturbance

for pure state

post measurement

state

Measurement


| i | ri

| e ri

Estimation

estimated state

input state

Information Gainby 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

Disturbanceby averaging the operation fidelity |h | ri|2



post measurement

state

Measurement


| i | ri

| e ri

Estimation

estimated state

input state

Information Gainby 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

Disturbanceby averaging the operation fidelity |h | ri|2


“Measurement disturbs a quantum state”+

“We gain information by measurement”

K. Banaszek, PRL 86, 1366 (2001)

Trade-off between info-gain and disturbance

“The more information is obtained from measurement,the more its state is disturbed.”


No-Cloning TheoremWilliam Wootters and Wojciech Zurek (1982).

“A Single Quantum Cannot be Cloned”. Nature 299 802–803.

Proof

Due to the linearity of quantum theory.

It is impossible to copy an unknown quantum state.

- If it were possible, superluminal signaling (communication faster than light) would be also possible…- Fundamental resources for Quantum Cryptography (QKD)

http://en.wikipedia.org/wiki/Nature_(journal)

No-Deleting TheoremArun K. Pati and Samuel L. Braunstein (2000).

“Impossibility of deleting an unknown quantum state”. Nature 404164–165.

Proof

In a closed system, one cannot destroy quantum information.

Given two copies of arbitrary quantum state, it is impossible to delete one of the copies.

… (1)

to satisfy (1)

Not deleted !

http://en.wikipedia.org/wiki/Nature_(journal)

1. Basic concepts




Information conservation and no-go theorems

Entropy of entanglement: for

Entanglement is invariant by local operation and classical communication (LOCC)

No-Deleting theorem

deleted by Bob

S(⇢A) > S(⇢0A)

Entanglement decreases.

No-Cloning theorem

copied by Bob

S(⇢A) < S(⇢0A)

Entanglement increases.

No-Cloning & No-Deleting theorem

Conservation of information (no change of entanglement)

2nd law of thermodynamics

M. Horodecki et al. (2003)

(entropy cannot be decreased in a closed system)

| i

input state

Reversal

post measurement

state

Measurement


| i | ri

| e ri

Estimation

estimated state

input state

Quantum measurement is irreversible ?

It is possible to reverse (undo) the quantum weak measurement !!

Common belief

: true for ideal (projection) measurements

: non-zero success probability to retrieve the arbitrary input state after the measurement

In fact,

“Can we reverse quantum measurement ?”

Information conservation in Quantum measurement

| i

input state

Reversal

post measurement

state

Measurement


| i | ri

| e ri

Estimation

estimated state

input state



Common belief



In fact,a|0i+ b|1i

|0i 0 1

Partial collapse

(ap⌘|0i+ b

p1� ⌘|1i)|0i+ (a

p1� ⌘|0i+ b

p⌘|1i)|1i

Non-ideal measurement process (weak measurement)

1/2 ⌘ 1Strength of measurement



| i

input state

Reversal

post measurement

state

Measurement


| i | ri

| e ri

Estimation

estimated state

input state



Common belief



In fact,a|0i+ b|1i

|0i 0 1

Partial collapse

(ap⌘|0i+ b

p1� ⌘|1i)|0i+ (a

p1� ⌘|0i+ b

p⌘|1i)|1i



selective process r= 1, 2, …, N

1st measurement

Reversal

2nd measurement

Measurement and Reversal condition



| i

input state

Reversal

post measurement

state

Measurement


| i | ri

| e ri

Estimation

estimated state

input state



Common belief



In fact,a|0i+ b|1i

|0i 0 1

Partial collapse

(ap⌘|0i+ b

p1� ⌘|1i)|0i+ (a

p1� ⌘|0i+ b

p⌘|1i)|1i



selective process r= 1, 2, …, N

1st measurement

Reversal

2nd measurement

Measurement and Reversal condition

ReversibilityWe define the reversibility as the maximal total reversal probability over all the measurement outcomes, r

given as a function of measurement sets (independent on the input state)



[1] Y. W. Cheong and SWL*, Phys. Rev. Lett. 109, 150402 (2012).

Information balance in quantum measurementResult 1

For qubit

Information Erasure !

Reversal succeeds.

Information gain

Quantitative bound of information gain and reversibility of quantum measurement

The same amount of information is erased by measurement reversal !!

After reversing quantum measurement, where is the already obtained information ?

[2] H.-T. Lim*, Y.-S. Ra, K.-H, Hong, SWL*, and Y.-H. Kim*, Phys. Rev. Lett. 113, 020504 (2014)

[1] Y. W. Cheong and SWL*, Phys. Rev. Lett. 109, 150402 (2012).

Information balance in quantum measurementResult 1

For qubit

Information Erasure !

Reversal succeeds.

Information gain

Quantitative bound of information gain and reversibility of quantum measurement

The same amount of information is erased by measurement reversal !!

After reversing quantum measurement, where is the already obtained information ?

Trade-off between info-gain and reversibility

“The more information is obtained from quantum measurement,the less possible it is to undo the measurement.”

Information balance (conservation of quantum information)

[2] H.-T. Lim*, Y.-S. Ra, K.-H, Hong, SWL*, and Y.-H. Kim*, Phys. Rev. Lett. 113, 020504 (2014)

1. Basic concepts




Information transmission and no-go theorems

Information transfer is possible, but it should be still located in a closed quantum system.

Entanglement is invariant by local operation and classical communication (LOCC)

No-Cloning & No-Deleting theorem

Conservation of information (no change of entanglement)


Entangled channel

| i

LO LO

Quantum Teleportation

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

!

"

#

$

Sender Receiver

quantum channel

classical channel

| i

% &

M R | i

“A quantum task to transfer an arbitrary quantum state to remote place”

unknown state

| i

mode a mode b mode c

entangled quantum channel

joint measurements

(projection)

Reversing operation

MR

Teleportation Protocol

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

Conditions for optimal quantum communications

✓ minimize the information gain by the (effective nonlocal) measurement

✓ maximize the reversibility of the measurement

Sender Receiver

maximal entanglement

| i

% &

M R | i

No info gain All info transferred

Success prob. = 1

Sender Receiver

non- maximal entanglement

| i

% &

M R | i

Info gain Partial info transferred

Success prob. < 1

by the senderNon-ideal quantum channel or measurement

Maximal success prob. of teleportation =

Reversibility

Information gain

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

(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

• effective measurement

• optimal reversing operators

Reversibility = the highest success probability of the teleportation

(standard teleportation)

when

higher than the ones by previously known protocols

Multiparty Quantum Teleportation

Sender Receiver % &

Entanglement is shared between senders, intermediators and receivers

!

Measurements are performed by senders and intermediators

"

Measurement results are sent from senders and intermediators to receivers through classical channels

#

Reversing operations are performed on the receivers’ parties

$

M R | i

Intermediator' M

effective measurement

: measurement basis by intermediator

RM

M

M

M

M

M

M

M

M

MM

Quantum communications in arbitrary quantum network

Information balance determines optimal protocols of any quantum communications !

1. Basic concepts




Hiding information

• Hiding Classical 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 K : random n-bit key

C = M �Kencoding message

C �K = Mdecoding message

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

| ih | = ⇢! ⇢0

| iI ! | iOA

⇢0 = trA(| iOAh |)

It maps an arbitrary quantum state to a fixed state.

Encodes an arbitrary input quantum state into a larger Hilbert-space.

If

is independent of the input state, the process is a hiding process.

No-Hiding TheoremSamuel 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

| iI ⌦ |Ai !X

k

ppk|kiO ⌦ |Ak( )iA

due to the linearity and unitarity of physical process

|Ak(a| i+ b| ?i)i = a|Ak( )i+ b|Ak( ?)i

⇢0 =X

k

pk|kihk|where are the (orthonormal) eigenvectors for{|ki}

{|Aki} are the (orthonormal) eigenvectors for the ancilla.

Information is here !

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


Quantum information

No-Cloning, No-Deleting, No-Hiding theorem

Entangled channel

| i

| i

Classical Information

Landauer’s principleTheory of relativity

Conservation of quantum information ?

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


Quantum information

No-Cloning, No-Deleting, No-Hiding theorem

Entangled channel

| i

| i

Classical Information

Conservation of quantum information !

Quantum measurement Landauer’s principleTheory of relativity

Encoding, feedforwarding

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

| imemoryM0101110R

| i M memory R0101110

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.

Transmitting and Hiding Quantum Informationevents.kias.re.kr/ckfinder/userfiles/201812/files... · 2018-12-28 · Reversibility We deﬁne the reversibility as the maximal total reversal

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