On Complementarity In QEC And Quantum Cryptography David Kribs - - PowerPoint PPT Presentation

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

On Complementarity In QEC And Quantum Cryptography

David Kribs

Professor & Chair Department of Mathematics & Statistics University of Guelph Associate Member Institute for Quantum Computing University of Waterloo

QEC II — USC — December 2011

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

Outline

1 Introduction

Notation

2 Stinespring Dilation Theorem

Heisenberg & Schr¨

• dinger Pictures

Purification of Mixed States Conjugate/Complementary Channels

3 Private Quantum Codes

Definition Single Qubit Private Channels

4 Connection with QEC and Beyond

Complementarity of Quantum Codes From QEC to QCrypto?

5 Conclusion

Summary

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

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 finite dimensions these sets coincide and so we’ll simply write L(H) B(HA, HB) will denote the set of linear transformations from HA to HB. We’ll write X, Y for operators in B(H), and ρ, σ for density

• perators in B(H)t. (And we’ll just refer to L(H) when

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

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

Heisenberg Picture

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

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

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 most dim(A) dim(B)) and a co-isometry V ∈ B(HB ⊗ K, HA) (VV † = IA) such that Φ†(X) = V (X ⊗ IK)V † ∀X.

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

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 most dim(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 operators for Φ† can be read off as the “coordinate operators” of V †: V † =    V †

1

. . . V †

AB

  

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

Schr¨

• dinger Picture

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

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

Schr¨

• dinger 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 most dim(A) dim(B)), an isometry U ∈ B(HA ⊗ K, HB ⊗ K), and a pure state |ψ ∈ K such that Φ(ρ) = TrK(U(ρ ⊗ |ψψ|)U†) ∀ρ.

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

Schr¨

• dinger 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 most dim(A) dim(B)), an isometry U ∈ B(HA ⊗ K, HB ⊗ K), and a pure state |ψ ∈ K such that Φ(ρ) = TrK(U(ρ ⊗ |ψψ|)U†) ∀ρ. The two pictures are connected via V †|φ := U(|φ ⊗ |ψ), which gives Φ(ρ) = TrK(V †ρV ).

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

Schr¨

• dinger 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 most dim(A) dim(B)), an isometry U ∈ B(HA ⊗ K, HB ⊗ K), and a pure state |ψ ∈ K such that Φ(ρ) = TrK(U(ρ ⊗ |ψψ|)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 †

∗

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

Purification of Mixed States

Fix a density operator ρ0 ∈ B(H)t, and consider the CPTP map Φ : C → B(H)t defined by Φ(c · 1) = c ρ0 ∀c ∈ C.

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

Purification of Mixed States

Fix a density operator ρ0 ∈ B(H)t, and consider the CPTP map 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 ⊗ |ψψ|)U†) = TrK(|ψ′ψ′|), where |ψ′ ∈ H ⊗ H is a purification of ρ0 – and the unitary freedom in the theorem captures all purifications.

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

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) channel is 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 related by a partial isometry W such that Φ(·) = W Φ′(·)W †. We talk of “the” conjugate channel for Φ with this understanding.

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

Computing Kraus Operators for Conjugates

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

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

Computing Kraus Operators for Conjugates

Suppose that Vi ∈ L(HA, HB) are the Kraus operators for Φ : 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

|eiej| ⊗ ViρV †

i ∈ L(K ⊗ HB).

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

Computing Kraus Operators for Conjugates

Suppose that Vi ∈ L(HA, HB) are the Kraus operators for Φ : 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

|eiej| ⊗ ViρV †

i ∈ L(K ⊗ HB).

Then Φ(ρ) = TrKF(ρ) and

• Φ(ρ) = TrBF(ρ) =
• i,j

Tr(ViρV †

j )|eiej| =

• µ

RµρR†

µ,

where R†

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

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

Private Quantum Codes

Definition (Ambainis, et al.) Let S ⊆ H be a set of pure states, let Φ : 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(|ψψ|) = ρ0 ∀|ψ ∈ S. Motivating class of examples: random unitary channels, where Φ(ρ) =

i piUiρU† i .

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

Single Qubit Private Codes

Recall a single qubit pure state |ψ can be written |ψ = cos θ 2 1

• + eiϕ sin θ

2 1

• .

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

2

• and β = eiϕ sin

θ

2

• . The

associated Bloch vector is r = (cos ϕ sin θ, sin ϕ sin θ, cos θ). Using the Bloch sphere representation, we can associate to any single 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.

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

Single Qubit Private Codes

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 the Bloch sphere. We are interested in cases where S is nonempty. So we consider the cases in which T has non-trivial nullspace; that is, the subspace

• f vectors

r such that T r = 0 is one, two, or three-dimensional.

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

Single Qubit Private Codes

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

2I] with S nonempty:

1 If the nullspace of T is 1-dimensional, then S consists of a

pair of orthonormal states.

2 If the nullspace of T is 2-dimensional, then the set S is the

set of all trace vectors (see below) of the subalgebra U†∆2U

• f 2 × 2 diagonal matrices up to a unitary equivalence.

3 If the nullspace of T is 3-dimensional, then Φ is the

completely depolarizing channel and S is the set of all unit

• vectors. In other words, S is the set of all trace vectors of

C · I2.

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

Nullspace of T is 1-dimensional

Figure: Case (1)

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

Nullspace of T is 2-dimensional

Figure: Case (2)

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

Nullspace of T is 3-dimensional

Figure: Case (3)

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

Private Code Side-Bar

Note: There are interesting connections between private quantum codes and the notions of conditional expectations and trace vectors in the theory of operator algebras – maybe as far back as eighty years ago. For more on this see:

• A. Church, D.W. Kribs, R. Pereira, S. Plosker, Private Quantum

Channels, Conditional Expectations, and Trace Vectors. Quantum Information & Computation, 11 (2011), no. 9 & 10, 774 - 783.

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

Complementarity of Quantum Codes

Theorem (Kretschmann-K.-Spekkens) Given a conjugate pair of CPTP maps Φ, Φ, a code is an error-correcting code for one if and only if it is a private code for the other. The extreme example of this phenomena is given by a unitary channel paired with the completely depolarizing channel – where the entire Hilbert space is the code.

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

Example 4.1

Consider the 2-qubit swap channel Φ(σ ⊗ ρ) = ρ ⊗ σ, which has a single Kraus operator, the swap unitary U. The conjugate map Φ : L(C2 ⊗ C2) → C is implemented with four Kraus operators, which are R1 =

• 1
• R2 =
• 1
• R3 =
• 1
• R4 =
• 1
• ,

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

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

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-qubits in size, and the conjugate channel Φ : L(C2 ⊗ C2) → L(C2) is implemented with the following Kraus operators: R1 = 1 √ 2 1 1

• R2 =

1 √ 2 1 1

• R3 =

1 √ 2 1 −1

• R4 =

1 √ 2 1 −1

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

Example 4.2

We have Φ(ρ) = 1

2(ρ + Z1ρZ1) and

Φ(ρ) = 4

i=1 RiρR† i for

all 2-qubit ρ. It is clear that the code {|00, |01} is correctable for Φ; in fact it is noiseless/decoherence-free. And thus we know it is private for the conjugate channel Φ. Indeed, one can check directly that every density operator ρ supported on {|00, |01}, satisfies

• Φ(ρ) = |++| = 1

2(|0 + |1)(0| + 1|).

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

From QEC to QCrypto?

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

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

“Knill-Laflamme” Type Conditions for Private Codes

Theorem (K.-Plosker) A projection P is a private code for a CPTP map Φ with 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).

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

“Knill-Laflamme” Type Conditions for Private Codes

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

µRνP = λµνP. But

PR†

µRνP = PΦ†(|fµfν|)P.

Consequence: For instance in the case that ρ0 ∝ Q =

k |ψkψk| and P = l |φlφl|, it follows that

there are scalars uikl such that for all i ViP =

• k,l

uiklAkl where Akl = 1

• rank(Q)

|ψkφl|.

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

Conclusion

The Stinespring Dilation Theorem is a foundational mathematical result for quantum information, and it naturally gives rise to the notion of conjugate channels. Private quantum codes are a basic tool in quantum cryptography, and can also be viewed from an operator theoretic perspective. Quantum error correcting codes are complementary to private quantum codes, with conjugate channels providing a clean algebraic bridge between the two studies. It should be possible to develop analogues of many results, techniques, applications, etc, from QEC for use in quantum

• cryptography. We have discussed some here, but there seem

to be many other natural questions...

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

THANK YOU !