Randomizing Quantum States

1
Randomizing Quantum States

• Charles Bennett
• Patrick Hayden
• Debbie Leung
• Peter Shor
• Andreas Winter
• RSP quant-ph/0307100 RAND quant-ph/0307104

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One-time pad
Message
Shared key
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Private quantum channels
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Relax security criterion
A CPTP map is e-randomizing if for all
states ,
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Approximate PQC
Can encrypt a quantum state using 1 secret random
bit per encrypted qubit asymptotically.
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Remote state preparationNon-oblivious
teleportation
A circuit that needs no introduction
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Remote state preparationNon-oblivious
teleportation
Allow Alice to perform an arbitrary f-dependent
measurement
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From randomization to RSP
Circuit for teleportation
l qubits
k 2l bits
Before receiving k Bob knows nothing
(Private quantum channel!)
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From randomization to RSP
Circuit for remote state preparation
l qubits
k lo(l) bits
Before receiving k, Bob knows nothing
(Approximate private quantum channel!)
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Knowledge is power
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
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Randomization and nonlocality
Does an e-randomizing R destroy all correlations
with the environment?
True for separable ? (easy). Not true for
entangled inputs!
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Rank argument
Recall e-randomizing map
Act on half of a maximally entangled state
has rank no more than nd log d

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Characterizing leftover correlations
What does randomization map do to entangled
inputs?
R
?Charlie prepares maximally entangled state k
then randomizes it.
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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.
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Detecting entanglement
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Randomization proof ideas
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Large deviation estimate
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Discretization
Find an e-net M For all pure states f, there
exists a state f in M such that
Cd
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Combine (Not as bad at it looks)
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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
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Hiding qubits
GOAL Charlie hides qubits from Alice and Bob,
secure against LOCC
dV H T, 2002
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Strategy
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Strategy
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Security
Hinges on showing
(Rank one projectors)
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Decodability
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Data hiding summary
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Conclusion

• 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

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The End
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Overview

• Act I
• LOCC data hiding for quantum states
• (quant-ph/0207147)

• Act II
• From RSP to PQC to data hiding

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A few years ago in a lab moderately far away...
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Nonlocality without entanglement
BDFMRSSW, 1999
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Hiding a qubit First attempt
TASK Hide an arbitrary quantum state
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Hiding a qubit First attempt
TASK Hide an arbitrary quantum state
PROBLEM Data hiding with pure states is
impossible! (So much for
superpositions.)
WSHV,2000
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2nd simplest idea
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
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Actually, 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 As
sume there is an LOCC operation L (with output
on Bobs system alone) and two input states to
the hiding map E such that
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Minor algebra
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Defeating the cbit hiding
Conclusion from previous slide there is a k such
that L0?Lk
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Imperfect hiding
Wish to limit distinguishability through LOCC
For all input states and attacks.
If the original 2n bit hiding scheme has security
?, then ? lt2n1 ?.
Not so bad security of classical hiding schemes
appears to improve exponentially with number of
qubits used.
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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
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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
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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
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Quantum secret sharing
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
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Hiding distributed quantum 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
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Act II
Fig. 1 Glimpse of a master magicians workshop
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On the meaning of
For any probability density P(?) on states in Cd
and ?gt0 there exists a choice of unitaries Us,
s1,,S such that
and
Compare to the perfect private quantum channel
To achieve ?0 requires log M 2 log d.
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Another version
There exists a choice of unitaries Ups,
p1,,P, s1,,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.
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Consequences

• 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)

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Application to data hiding
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!
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Glyph collection
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Competing visions

• Faction 1
• 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

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Summary

• 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

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Building the POVM
Alice wishes to send f. By the definition of
e-randomization
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From RSP to randomization
Circuit for teleportation
n qubits
i 2n bits
Before receiving i, Bob knows nothing
(Private quantum channel, Quantum one-time
pad, etc.)