Introduction to quantum information processing Fidelity and other distance metrics Brad Lackey 3 November 2016 FIDELITY AND OTHER DISTANCE METRICS 1 of 17
OUTLINE 1 Noise channels 2 Fidelity 3 Trace distance FIDELITY AND OTHER DISTANCE METRICS 2 of 17
LAST TIME... Mixed states on the Bloch sphere: ρ = 1 2 (1 + r xx + r y Y + r z Z) (r x, r y, r z ). We can purify a mixed state to have ρ = tr B ( ψ ψ ). Quantum channels are generally given in Kraus form: C(ρ) = j E j ρe j. FIDELITY AND OTHER DISTANCE METRICS 3 of 17
OUTLINE 1 Noise channels 2 Fidelity 3 Trace distance FIDELITY AND OTHER DISTANCE METRICS 4 of 17
BIT FLIP AND PHASE FLIP CHANNELS The bit flip channel: flips the qubit state, 0 1, with probability p, and leaves the qubit alone with probability (1 p). Therefore B p (ρ) = (1 p)ρ + pxρx. On the Bloch sphere: (r x, r y, r z ) (r x, (1 2p)r y, (1 2p)r z ). The phase flip channel: flips the phase, 0 0 and 1 1, with probability p, and leaves the qubit alone with probability (1 p). Therefore P p (ρ) = (1 p)ρ + pzρz. On the Bloch sphere: (r x, r y, r z ) ((1 2p)r x, (1 2p)r y, r z ). FIDELITY AND OTHER DISTANCE METRICS 5 of 17
THE DEPOLARIZING CHANNEL The depolarizing channel is a very population error channel to study: with probability p it applies one of the Pauli operators {X, Y, Z} (each equally likely), and with probability p it leaves the qubit alone. Therefore D p (ρ) = (1 p)ρ + p 3 (XρX + YρY + ZρZ). However we showed last time that ρ + XρX + YρY + ZρZ = 2 1. So: ( D p(ρ) = 1 4p ) ρ + p3 ( 3 (ρ + XρX + YρY + ZρZ) = 1 4p ) ρ + 2p 3 3 1. On the Bloch sphere, this is easy to understand: ( ) (r x, r y, r z ) 1 4p 3 (r x, r y, r z ). So it simply shrinks the Bloch sphere vector, well at least when p < 3 4. FIDELITY AND OTHER DISTANCE METRICS 6 of 17
THE AMPLITUDE AND PHASE DAMPING CHANNELS The amplitude damping channel models loss of energy to the environment. It has two Kraus operators, which in basis ( 0, 1 ) are ( ) ( ) 1 0 E 0 = 0 γ and E 0 1 γ 1 =. 0 0 And: (r x, r y, r z ) (r x 1 γ, ry 1 γ, γ + rz (1 γ)). The phase damping channel models coherence loss (quantum information). It also has two Kraus operators, which in basis ( 0, 1 ) are ( ) ( ) 1 0 E 0 = 0 0 and E 0 1 λ 1 =. 0 λ And: (r x, r y, r z ) (r x 1 γ, ry 1 γ, rz ). FIDELITY AND OTHER DISTANCE METRICS 7 of 17
OUTLINE 1 Noise channels 2 Fidelity 3 Trace distance FIDELITY AND OTHER DISTANCE METRICS 8 of 17
FIDELITY BETWEEN STATES The classical fidelity, or Bhattacharyya coefficient, of p and q is F(p, q) = i pi q i. A high fidelity indicates the distributions are close. It is related to the so-called cosine metric for distributions. Then to make a notion of quantum fidelity F(ρ, σ) it makes sense to: note that for any POVM {E i }, we have p i = tr(e i ρ) and q i = (E i σ), choose the POVM that distinguishes these distributions best: F(ρ, σ) = min F({tr(E iρ)}, {tr(e i σ)}). {E i} POVM This has a simple formula F(ρ, σ) = tr ρ 1/2 σρ 1/2 (we won t prove this). This isn t too bad if one state is pure: F(ρ, ψ ψ ) = ψ ρ ψ. It s even better if both are pure: F( φ φ, ψ ψ ) = ψ φ. FIDELITY AND OTHER DISTANCE METRICS 9 of 17
UHLMANN S THEOREM Theorem (Uhlmann (1976)) Let ρ and σ be densities on H, with dim H = n. Then F(ρ, σ) = max ψ φ ψ, φ where the maximum is over all purifications ψ of ρ and φ of σ on H C n. This really can t be used to compute the fidelity, but we can derive nice facts: Since always 0 ψ φ 1, we must have 0 F(ρ, σ) 1. If F(ρ, σ) = 1 then ρ = σ (a purification has ψ φ ). If F(ρ, σ) = 0 then ρ and σ are distinguishable (from ψ φ = 0). FIDELITY AND OTHER DISTANCE METRICS 10 of 17
MONOTONICITY OF FIDELITY A unitary transformation, U, does not change the fidelity. From the spectral theorem ρ 1/2 = j λj φ j φ j. So (UρU ) 1/2 = j λj U φ j φ j U = U(ρ 1/2 )U. Therefore, F(UρU, UσU ) = tr Uρ 1/2 U UσU Uρ 1/2 U = tr(u ρ 1/2 σρ 1/2 U ) = F(ρ, σ). A quantum channel, C, never shrinks the fidelity (but may enlarge it). We need some facts about what we can do: We can jointly purify ρ = tr B( ψ ψ ) and σ = tr B( φ φ ). We can add more ancilla to dilate C(ρ) = tr E(U(ρ 0 0 )U ). We can combine these C(ρ) = tr BE(U( ψ ψ 0 0 )U ) Then U ψ 0 is a purification of C(ρ). Same for U( φ 0 ) of C(σ). Therefore, using Uhlman s theorem F(ρ, σ) = ψ φ = ( ψ 0 )(UU )( φ 0 ) F(C(ρ), C(σ)). FIDELITY AND OTHER DISTANCE METRICS 11 of 17
HOW WELL DO CHANNELS PRESERVE INFORMATION? E.g. consider the fidelity of the depolarizing channel on a pure state: ( ) ) F( ψ ψ, D p ψ ψ ) = ψ 2p 3 (1 1 + 4p 3 ψ ψ ψ 2p = 3 + (1 4p 3 1 ) = 2p 3. Here the loss of fidelity is the same for any state. However for the phase damping channel: F( ψ ψ, P p ψ ψ ) = = where ψ = cos θ 2 0 + eiϕ sin θ 2 1. ( ) E 0 ψ ψ E 0 + E 1 ψ ψ E 1 ψ ψ 1 1 2 ( 1 λ 1) sin 2 θ Loss of fidelity depends on the coherence 1 2 sin2 θ = cos θ 2 sin θ 2. FIDELITY AND OTHER DISTANCE METRICS 12 of 17
OUTLINE 1 Noise channels 2 Fidelity 3 Trace distance FIDELITY AND OTHER DISTANCE METRICS 13 of 17
TRACE DISTANCE Another popular metric is the statistical distance: D(p, q) = 1 2 i p i q i. The quantum analogue of this is the trace distance: D(ρ, σ) = 1 2tr( ρ σ ). Like fidelity, one can optimize statistical distance over POVMs: D(ρ, σ) = max D({tr(E i ρ)}, {tr(e i σ)}). {E i} POVM Here s a sketch of the proof: Using the spectral decomposition 1 ρ σ = A + A where A +, A 0 and A A + = A + A. 2 The A ± commute so ρ σ = A + A = A + + A. 3 Then D({tr(E i ρ)}, {tr(e i σ)}) = i tr(e i(ρ σ)) and this equals tr(e i (A + A )) i i = i tr(e i (A + + A )) (since x y x + y ) tr(e i ρ σ ) = D(ρ, σ). 4 To achieve the maximum take the projections onto supp(a ± ). FIDELITY AND OTHER DISTANCE METRICS 14 of 17
CONTRACTIVITY OF CHANNELS Also like the fidelity, the trace distance is preserved under unitaries: Easily U ρ σ U = U(ρ σ)u. Quantum channels contract trace distance: D(C(ρ), C(σ)) D(ρ, σ). We won t prove this, but it s not too difficult. Note ρ tr B (ρ) is a channel, so D(tr B (ρ), tr B (σ)) D(ρ, σ). A useful metric is the gate error E(U, C) = max ρ D(UρU 1, C(ρ)). This bounds errors from approximating to U with the channel C. It has the nice identity E(VU, D C) E(U, C) + E(V, D). This follows from the triangle inequality and contractivity. FIDELITY AND OTHER DISTANCE METRICS 15 of 17
TRACE DISTANCE VERSUS FIDELITY For pure states ψ and φ (with angle θ between them): D( ψ ψ, φ φ ) = sin θ, and F( ψ ψ, φ φ ) = cos θ. Therefore D( ψ ψ, φ φ ) = 1 F( ψ ψ, φ φ ) 2. Otherwise, by Uhlmann s theorem F(ρ, σ) = ψ φ = F( ψ ψ, φ φ ). Then D(ρ, σ) = D(tr B ( ψ ψ ), tr B ( φ φ )) D( ψ ψ, φ φ ) Therefore, D(ρ, σ) 1 F(ρ, σ) 2. On the other hand using a POVM with F(ρ, σ) = F({tr(E j ρ)}, {tr(e j σ)}). For distributions: i ( p i q i ) 2 = 2 2 i pi q 1. Also: i ( p i q i ) 2 i p i q i. Therefore, 1 F(ρ, σ) 1 2 i p i q i D(ρ, σ). FIDELITY AND OTHER DISTANCE METRICS 16 of 17
NEXT TIME... Classical and quantum codes. FIDELITY AND OTHER DISTANCE METRICS 17 of 17