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Randomizing Quantum States. Charles Bennett Patrick Hayden Debbie Leung Peter Shor Andreas Winter RSP: quant-ph/0307100 RAND: quant-ph/0307104. 1 bit of key per bit of message necessary and sufficient [Shannon]. One-time pad. Message. Shared key. Eavesdropper learns nothing:.

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Randomizing quantum states

Randomizing Quantum States

Charles Bennett

Patrick Hayden

Debbie Leung

Peter Shor

Andreas Winter

RSP: quant-ph/0307100 RAND: quant-ph/0307104


One time pad

1 bit of key per bit of message necessary and sufficient [Shannon]

One-time pad

Message

Shared key


Private quantum channels

Eavesdropper learns nothing: [Shannon]

Lower bound on key length:

[BR,AMTW 2000]

Private quantum channels


Relax security criterion

MAIN RESULT: [Shannon]

Given >0, there exists a choice of unitaries

{Uk}, k=1,…,n such that the map

is ε-randomizing, with

Relax security criterion

A CPTP map is ε-randomizing if for all states ,


Approximate pqc

Security: [Shannon]

1 bit of key/qubit

(1+2α) bits of key/qubit

Approximate PQC

Can encrypt a quantum state using 1 secret

random bit per encrypted qubit asymptotically.


Remote state preparation non oblivious teleportation
Remote state preparation: [Shannon]Non-oblivious teleportation

A circuit that needs no introduction:


Remote state preparation non oblivious teleportation1
Remote state preparation: [Shannon]Non-oblivious teleportation

Allow Alice to perform an arbitrary φ-dependent measurement


From randomization to rsp
From randomization to RSP [Shannon]

Circuit for teleportation:

l qubits

k: 2l bits

Before receiving k Bob knows nothing:

(Private quantum channel!)


From randomization to rsp1
From randomization to RSP [Shannon]

Circuit for remote state preparation:

l qubits

k:l+o(l) bits

Before receiving k, Bob knows nothing:

(Approximate private quantum channel!)


Knowledge is power
Knowledge is power [Shannon]

All you need in this life is ignorance and confidence,

and then success is sure. -Mark Twain

Oblivious Alice (Teleportation):

1 ebit + 2 cbits per 1 qubit sent

There is no knowledge that is not power.

-Ralph Waldo Emerson

Non-Oblivious Alice (RSP):

1 ebit + 1 cbit per 1 qubit sent

Knowledge is power, if you know it about the right person.

-Ethel Watts Mumford


Randomization and nonlocality
Randomization and nonlocality [Shannon]

Does an ε-randomizing R destroy all

correlations with the environment?

True for separable ρ (easy).

Not true for entangled inputs!


Rank argument

Fidelity with maximally mixed state small: [Shannon]

Rank argument

Recall ε-randomizing map:

Act on half of a maximally entangled state:

has rank no more than n~d log d

~


Characterizing leftover correlations

X [Shannon]

Y

Alice and Bob implement an LOCC measurement.

Characterizing leftover correlations

What does randomization map do to entangled inputs?

R

Charlie prepares maximally entangled state k then randomizes it.


Characterizing leftover correlations1

Separable state! [Shannon]

k info randomized

Conclusion: LOCC cannot distinguish strong and weak randomization

Characterizing leftover correlations

What does randomization map do to entangled inputs?

R

X

Y

Charlie prepares maximally entangled state k then randomizes it.

Alice and Bob implement an LOCC measurement.


Detecting entanglement

(conjectured) [Shannon]

Separable

states

Distillable

states

Product

states

Non-distillable

states

Detecting entanglement


Randomization proof ideas

Strategy: [Shannon]

• Select the unitaries {Uk} at random from Haar measure.

• Show they are ε-randomizing with probability >0.

Ingredients:

• Large deviation estimate

• Discretization

Randomization proof ideas


Large deviation estimate

Will be concerned with distribution of: [Shannon]

where

In particular,

(Exponential convergence to the mean)

Large deviation estimate


Discretization

Will allow replacement of “all states” by “all states in M”

in definition of ε-randomization.

Discretization

Find anε-net M:

For all pure states φ, there exists a state

φ’ in M such that

Cd


Combine not as bad at it looks

defn in

ε-net

union bound

large deviations

Combine (Not as bad at it looks)


Quantum data hiding

? ? in

Quantum data hiding

GOAL: Charlie hides a bit from Alice and Bob, secure against LOCC

RESULT: There exist bipartite n-qubit states hiding a bit with security 2-(n-1).

[DLT, 2001]


Hiding qubits

Security: in

Decodability:

Hiding qubits

GOAL: Charlie hides qubits from Alice and Bob, secure against LOCC

[dV H T, 2002]




Security

Similar calculation: in

Choose

Security

Hinges on showing

(Rank one projectors)



Data hiding summary

It is possible to hide a space of dimension in

in a bipartite system where each of the subsystem has

dimension d.

Asymptotically: 1 hidden qubit per 2 physical qubits.

Previous best: ~ 1 hidden qubit per ~75 physical qubits.

Data hiding summary


Conclusion
Conclusion in

  • Encryption of quantum states using 1 bit of shared key per encrypted qubit

  • Remote state preparation using 1 bit of communication and 1 ebit per qubit sent

  • Physical operation that destroys classical correlations but not quantum

  • Data hiding of 1 qubit in 2 physical qubits



Overview

Act I in

LOCC data hiding for quantum states

(quant-ph/0207147)

Act II

From RSP to PQC to data hiding

Overview


A few years ago in in

a lab moderately

far away...


Nonlocality without entanglement

? ? in

0 1 2

0

1

2

Not always possible:

Nonlocality without entanglement

[BDFMRSSW, 1999]


Hiding a qubit first attempt

Prepare superpositions of well hidden states? in

Hiding a qubit: First attempt

TASK: Hide an arbitrary quantum state


Hiding a qubit first attempt1
Hiding a qubit: First attempt in

TASK: Hide an arbitrary quantum state

PROBLEM: Data hiding with pure states is impossible!

(So much for superpositions.)

[WSHV,2000]


2nd simplest idea
2nd simplest idea in

THE PLAN: Use classical hidden bits as key to randomize a qubit

PROPERTIES: 1) can be recovered using quantum communication

2) Naïve attacks fail (AB1 to find key then rotate B2)

PROBLEM: Alice and Bob can attack AB1B2


Actually not a problem
Actually, in not a problem

Any method to learn about by LOCC will provide a

method to defeat the original cbit hiding scheme.

Will argue the contrapositive:

Assume there is an LOCC operation L (with output on

Bob’s system alone) and two input states to the hiding

map E such that


Minor algebra

CLAIM: Not all in Li can be the same TPCP map. If they were, then

by linearity:

This says that L would never reveal any information about

the input state, violating the hypothesis that L defeats the

qubit hiding scheme.

Minor algebra


Defeating the cbit hiding

The attack: 1) Bob prepares a local maximally entangled state on B2B3

2) Alice and Bob apply L to AB1B2

3) Bob performs a measurement on B2B3

By Choi, there is a measurement that can partially distinguish L0 andLk

Defeating the cbit hiding

Conclusion from previous slide: there is a k such that L0Lk


Imperfect hiding
Imperfect hiding state on

Wish to limit distinguishability through LOCC:

For all input states and attacks.

If the original 2n bit hiding scheme has security ,

then  <2n+1 .

Not so bad: security of classical hiding schemes appears

to improve exponentially with number of qubits used.


Multipartite cbit hiding

state on Authorized sets can recover the secret using quantum communication

Multipartite cbit hiding

A

E

B

D

C

 With LOCC alone the five parties cannot learn i

All monotonic access structures are possible: [Eggeling, Werner 2002]


Multipartite qubit hiding

This authorization structure state on

is impossible due to no-cloning!

Multipartite qubit hiding

A

E

B

D

C

 With LOCC alone the five parties cannot learn

 Authorized sets can recover the secret using quantum communication


Multipartite qubit hiding1

Problem for generalizing construction: B must be in all authorized sets

Multipartite qubit hiding

A

E

B

D

C

 With LOCC alone the five parties cannot learn

 Authorized sets can recover the secret using quantum communication


Quantum secret sharing
Quantum secret sharing authorized sets

A

e.g. ((2,3)) threshold scheme

E

B

D

C

Secure against quantum communication in unauthorized sets but

secret can be recovered by quantum communication in authorized sets.

All monotonic threshold schemes not violating no-cloning [CGL,1999]

All monotonic schemes not violating no-cloning [G,2000]


Hiding distributed quantum data
Hiding distributed authorized setsquantum data

A

Logical Pauli operators

Multipartite cbit

hiding states

E

B

D

C

Resulting state provides strengthening of quantum secret sharing:

 Secure against classical communication between all parties

 Secure against quantum communication in unauthorized sets

 Secret can be recovered only by quantum communication in authorized sets.

All monotonic schemes not violating no-cloning


Act ii
Act II authorized sets

Fig. 1: Glimpse of a master magician’s workshop


On the meaning of
On the meaning of “ “ authorized sets

For any probability density P() on states in Cd and >0 there exists

a choice of unitaries {Us}, s=1,…,S such that

and

Compare to the perfect private quantum channel:

To achieve =0 requires log M = 2 log d.


Another version
Another version authorized sets

There exists a choice of unitaries {Ups}, p=1,…,P, s=1,…,S such

that for all states  in Cd

and

Can randomize every n-qubit state using 1 secret

random bit and 1 public random bit per qubit.


Consequences
Consequences authorized sets

  • Universal remote state preparation with only 1 ebit + 1 cbit per qubit

    • (No shared random bits necessary)

  • Weakly randomized maximally entangled states indistinguishable from maximally mixed states using LOCC

    • (Not just 1-way LOCC as sketched earlier)


Application to data hiding
Application to data hiding authorized sets

R

Consider ensemble of randomized states with V chosen using Haar measure

Rank bound on entropy

So we can do coding to get about n hidden bits using nxn bipartite states!


Glyph collection
Glyph collection authorized sets


Competing visions

Faction 1 authorized sets

Destroying classical correlations requires only 1 rbit per qubit

Destroying quantum correlations requires 2 rbits per qubit

Faction 2

Randomizing an arbitrary pure quantum state requires 1 public rbit and 1 secret rbit per qubit

Competing visions


Summary
Summary authorized sets

  • Described a method for hiding qubits given one for hiding bits (construction and proof not restricted to data hiding)

  • Outlined a connection between LOCC data hiding, private quantum channels and remote state preparation


Building the povm
Building the POVM authorized sets

Alice wishes to send φ. By the definition of ε-randomization:


From rsp to randomization
From RSP to randomization authorized sets

Circuit for teleportation:

n qubits

i: 2n bits

Before receiving i, Bob knows nothing:

(“Private quantum channel”, “Quantum one-time pad”, etc.)


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