On complementarity in qec and quantum cryptography

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion

On Complementarity In QECAnd Quantum Cryptography

David Kribs

Professor & ChairDepartment of Mathematics & Statistics

University of Guelph

Associate MemberInstitute for Quantum Computing

University of Waterloo

QEC II — USC — December 2011


Outline

1 IntroductionNotation

2 Stinespring Dilation TheoremHeisenberg & Schrodinger PicturesPurification of Mixed StatesConjugate/Complementary Channels

3 Private Quantum CodesDefinitionSingle Qubit Private Channels

4 Connection with QEC and BeyondComplementarity of Quantum CodesFrom QEC to QCrypto?

5 ConclusionSummary


Notation

HA, HB will denote Hilbert spaces for systems A and B.

B(H) will denote the set of (bounded) linear operators on H;B(H)t will denote the trace class operators on H. In finitedimensions these sets coincide and so we’ll simply write L(H)

B(HA,HB) will denote the set of linear transformations fromHA to HB .

We’ll write X ,Y for operators in B(H), and ρ, σ for densityoperators in B(H)t . (And we’ll just refer to L(H) whenappropriate.)

Given a linear map Φ : B(HA)t → B(HB)t , its dual mapΦ† : B(HB)→ B(HA) is defined via the Hilbert-Schmidt innerproduct: Tr(ρΦ†(X )) = Tr(Φ(ρ)X ).


Heisenberg Picture

Suppose that Φ† : B(HB)→ B(HA) is a completely positive(CP) unital (Φ†(IB) = IA) linear map.

Then there is a Hilbert space K (of dimension at mostdim(A) dim(B)) and a co-isometry V ∈ B(HB ⊗K,HA)(VV † = IA) such that

Φ†(X ) = V (X ⊗ IK)V † ∀X .

V is unique up to a unitary on K. Here the Kraus operatorsfor Φ† can be read off as the “coordinate operators” of V †:

V † =

V †1...

V †AB


Heisenberg Picture



Φ†(X ) = V (X ⊗ IK)V † ∀X .


V † =

V †1...

V †AB


Heisenberg Picture



Φ†(X ) = V (X ⊗ IK)V † ∀X .


V † =

V †1...

V †AB


Schrodinger Picture

Suppose that Φ : B(HA)t → B(HB)t is a CP trace preserving(CPTP) linear map.

Then there is a Hilbert space K (of dimension at mostdim(A) dim(B)), an isometry U ∈ B(HA ⊗K,HB ⊗K), and apure state |ψ〉 ∈ K such that

Φ(ρ) = TrK(U(ρ⊗ |ψ〉〈ψ|)U†) ∀ρ.

The two pictures are connected via V †|φ〉 := U(|φ〉 ⊗ |ψ〉),which gives Φ(ρ) = TrK(V †ρV ).

The Kraus operators for Φ are the coordinate operators Vi

from above, and the general form for U is

U =[V † ∗

]


Schrodinger Picture



Φ(ρ) = TrK(U(ρ⊗ |ψ〉〈ψ|)U†) ∀ρ.




U =[V † ∗

]


Schrodinger Picture



Φ(ρ) = TrK(U(ρ⊗ |ψ〉〈ψ|)U†) ∀ρ.




U =[V † ∗

]


Schrodinger Picture



Φ(ρ) = TrK(U(ρ⊗ |ψ〉〈ψ|)U†) ∀ρ.




U =[V † ∗

]


Purification of Mixed States

Fix a density operator ρ0 ∈ B(H)t , and consider the CPTPmap Φ : C→ B(H)t defined by

Φ(c · 1) = c ρ0 ∀c ∈ C.

Then the Stinespring Theorem gives (here K = C⊗H = H):

ρ0 = Φ(1) = TrK(U(1⊗ |ψ〉〈ψ|)U†) = TrK(|ψ′〉〈ψ′|),

where |ψ′〉 ∈ H ⊗H is a purification of ρ0 – and the unitaryfreedom in the theorem captures all purifications.


Purification of Mixed States

Fix a density operator ρ0 ∈ B(H)t , and consider the CPTPmap Φ : C→ B(H)t defined by

Φ(c · 1) = c ρ0 ∀c ∈ C.

Then the Stinespring Theorem gives (here K = C⊗H = H):

ρ0 = Φ(1) = TrK(U(1⊗ |ψ〉〈ψ|)U†) = TrK(|ψ′〉〈ψ′|),

where |ψ′〉 ∈ H ⊗H is a purification of ρ0 – and the unitaryfreedom in the theorem captures all purifications.


Conjugate/Complementary Channels

Definition

(King, et al.; Holevo) Given a CPTP map Φ : L(HA)→ L(HB),consider V ∈ L(HB ⊗K,HA) and K above for which

Φ(ρ) = TrK(V †ρV ).

Then the corresponding conjugate (or complementary) channelis the CPTP map Φ : L(HA)→ L(K) given by

Φ(ρ) = TrB(V †ρV ).

Fact: Any two conjugates Φ, Φ′ obtained in this way are relatedby a partial isometry W such that Φ(·) = W Φ′(·)W †. We talk of“the” conjugate channel for Φ with this understanding.


Computing Kraus Operators for Conjugates

Suppose that Vi ∈ L(HA,HB) are the Kraus operators forΦ : L(HA)→ L(HB). Then we can obtain Kraus operators{Rµ} for Φ as follows.

Fix a basis {|ei 〉} for K and define for ρ ∈ L(HA),

F (ρ) =∑i ,j

|ei 〉〈ej | ⊗ ViρV†i ∈ L(K ⊗HB).

Then Φ(ρ) = TrKF (ρ) and

Φ(ρ) = TrBF (ρ) =∑i ,j

Tr(ViρV†j )|ei 〉〈ej | =

∑µ

RµρR†µ,

where R†µ = [V †

1 |fµ〉V†2 |fµ〉 · · · ] and {|fµ〉} is a basis for HB .





F (ρ) =∑i ,j





∑µ

RµρR†µ,







F (ρ) =∑i ,j





∑µ

RµρR†µ,




Private Quantum Codes

Definition

(Ambainis, et al.) Let S ⊆ H be a set of pure states, letΦ : L(HA)→ L(HB) be a CPTP map, and let ρ0 ∈ L(H). Then[S,Φ, ρ0] is a private quantum channel if we have

Φ(|ψ〉〈ψ|) = ρ0 ∀|ψ〉 ∈ S.

Motivating class of examples: random unitary channels, whereΦ(ρ) =

∑i piUiρU

†i .


Single Qubit Private Codes

Recall a single qubit pure state |ψ〉 can be written

|ψ〉 = cosθ

2

(10

)+ e iϕ sin

θ

2

(01

).

We associate |ψ〉 with the point (θ, ϕ), in spherical coordinates, onthe Bloch sphere via α = cos

(θ2

)and β = e iϕ sin

(θ2

). The

associated Bloch vector is ~r = (cosϕ sin θ, sinϕ sin θ, cos θ).Using the Bloch sphere representation, we can associate to anysingle qubit density operator ρ a Bloch vector ~r ∈ R3 satisfying‖~r‖ ≤ 1, where

ρ =I +~r · ~σ

2.

We use ~σ to denote the Pauli vector; that is, ~σ = (σx , σy , σz)T .



Every unital qubit channel Φ can be represented as

Φ

(1

2[I +~r · ~σ]

)=

1

2[I + (T~r) · ~σ] ,

where T is a 3× 3 real matrix that represents a deformation of theBloch sphere.

We are interested in cases where S is nonempty. So we considerthe cases in which T has non-trivial nullspace; that is, the subspaceof vectors ~r such that T~r = 0 is one, two, or three-dimensional.



Theorem

Let Φ : M2 →M2 be a unital qubit channel, with T the mappinginduced by Φ as above. Then there are three possibilities for aprivate quantum channel [S,Φ, 1

2 I ] with S nonempty:

1 If the nullspace of T is 1-dimensional, then S consists of apair of orthonormal states.

2 If the nullspace of T is 2-dimensional, then the set S is theset of all trace vectors (see below) of the subalgebra U†∆2Uof 2× 2 diagonal matrices up to a unitary equivalence.

3 If the nullspace of T is 3-dimensional, then Φ is thecompletely depolarizing channel and S is the set of all unitvectors. In other words, S is the set of all trace vectors ofC · I2.


Nullspace of T is 1-dimensional

Figure: Case (1)



Figure: Case (2)



Figure: Case (3)


Private Code Side-Bar

Note: There are interesting connections between private quantumcodes and the notions of conditional expectations and trace vectorsin the theory of operator algebras – maybe as far back as eightyyears ago. For more on this see:

A. Church, D.W. Kribs, R. Pereira, S. Plosker, Private QuantumChannels, Conditional Expectations, and Trace Vectors. QuantumInformation & Computation, 11 (2011), no. 9 & 10, 774 - 783.


Complementarity of Quantum Codes

Theorem

(Kretschmann-K.-Spekkens) Given a conjugate pair of CPTP mapsΦ, Φ, a code is an error-correcting code for one if and only if it is aprivate code for the other.

The extreme example of this phenomena is given by a unitarychannel paired with the completely depolarizing channel – wherethe entire Hilbert space is the code.


Example 4.1

Consider the 2-qubit swap channel Φ(σ ⊗ ρ) = ρ⊗ σ, whichhas a single Kraus operator, the swap unitary U.

The conjugate map Φ : L(C2 ⊗ C2)→ C is implemented withfour Kraus operators, which are

R1 =[1 0 0 0

]R2 =

[0 0 1 0

]R3 =

[0 1 0 0

]R4 =

[0 0 0 1

],

and one can easily see that Φ(ρ) = 1 for all ρ ∈ L(C2).


Example 4.2

Consider the 2-qubit phase flip channel Φ with (equally weighted)Kraus operators {I ,Z1}. The dilation Hilbert space here is 3-qubitsin size, and the conjugate channel Φ : L(C2 ⊗ C2)→ L(C2) isimplemented with the following Kraus operators:

R1 =1√2

[1 0 0 01 0 0 0

]R2 =

1√2

[0 1 0 00 1 0 0

]R3 =

1√2

[0 0 1 00 0 −1 0

]R4 =

1√2

[0 0 0 10 0 0 −1

]


Example 4.2

We have Φ(ρ) = 12(ρ+ Z1ρZ1) and Φ(ρ) =

∑4i=1 RiρR

†i for

all 2-qubit ρ.

It is clear that the code {|00〉, |01〉} is correctable for Φ; infact it is noiseless/decoherence-free. And thus we know it isprivate for the conjugate channel Φ.

Indeed, one can check directly that every density operator ρsupported on {|00〉, |01〉}, satisfies

Φ(ρ) = |+〉〈+| =1

2(|0〉+ |1〉)(〈0|+ 〈1|).


From QEC to QCrypto?

General Program: Using the “algebraic bridge” given by thenotion of conjugate channels, investigate what results, techniques,special types of codes, applications, etc, etc, from QEC haveanalogues or versions in the world of private quantum codes andquantum cryptography. At the least, we can use work from QEC asmotivation for studies in this different setting...


“Knill-Laflamme” Type Conditions for Private Codes

Theorem

(K.-Plosker) A projection P is a private code for a CPTP map Φwith output state ρ0; i.e., Φ(ρ) = ρ0 for all ρ ∈ L(PH)

if and only if

∀X ∃λX ∈ C : PΦ†(X )P = λXP.

In this case, λX = Tr(ρ0X ).


“Knill-Laflamme” Type Conditions for Private Codes

QEC Motivation: If P is correctable for Φ then ∃λµν ∈ Csuch that PR†

µRνP = λµνP. But

PR†µRνP = PΦ†(|fµ〉〈fν |)P.

Consequence: For instance in the case thatρ0 ∝ Q =

∑k |ψk〉〈ψk | and P =

∑l |φl〉〈φl |, it follows that

there are scalars uikl such that for all i

ViP =∑k,l

uiklAkl where Akl =1√

rank(Q)|ψk〉〈φl |.


Conclusion

The Stinespring Dilation Theorem is a foundationalmathematical result for quantum information, and it naturallygives rise to the notion of conjugate channels.

Private quantum codes are a basic tool in quantumcryptography, and can also be viewed from an operatortheoretic perspective.

Quantum error correcting codes are complementary to privatequantum codes, with conjugate channels providing a cleanalgebraic bridge between the two studies.

It should be possible to develop analogues of many results,techniques, applications, etc, from QEC for use in quantumcryptography. We have discussed some here, but there seemto be many other natural questions...


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On complementarity in qec and quantum cryptography