VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ The Bloch Sphere Ian Glendinning February 16, 2005 & % QIA Meeting, TechGate 1 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Outline • Introduction • Definition of the Bloch sphere • Derivation of the Bloch sphere • Properties of the Bloch sphere • Future Topics & % QIA Meeting, TechGate 2 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Introduction The Bloch sphere is a geometric representation of qubit states as points on the surface of a unit sphere. Many operations on single qubits that are commonly used in quantum information processing can be neatly described within the Bloch sphere picture. & % QIA Meeting, TechGate 3 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Definition of the Bloch sphere It turns out that an arbitrary single qubit state can be written: µ ¶ θ θ |ψi = eiγ cos |0i + eiφ sin |1i 2 2 where θ, φ and γ are real numbers. The numbers 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π define a point on a unit three-dimensional sphere. This is the Bloch sphere. Qubit states with arbitrary values of γ are all represented by the same point on the Bloch sphere because the factor of eiγ has no observable effects, and we can therefore effectively write: θ θ |ψi = cos |0i + eiφ sin |1i 2 2 & % QIA Meeting, TechGate 4 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ The Bloch Sphere & % QIA Meeting, TechGate 5 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Derivation of the Bloch Sphere The Bloch Sphere is is a generalisation of the representation of a complex number z with |z|2 = 1 as a point on the unit circle in the complex plane. If z = x + iy, where x and y are real, then: |z|2 = z∗z = (x − iy)(x + iy) = x2 + y2 and x2 + y2 = 1 is the equation of a circle of radius one, centered on the origin. & % QIA Meeting, TechGate 6 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ The Unit Circle & % QIA Meeting, TechGate 7 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Polar Coordinates & % QIA Meeting, TechGate 8 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Polar Coordinates For arbitrary z = x + iy we can write x = r cos θ, y = r sin θ, so z = r(cos θ + i sin θ) and using Euler’s identity: eiθ = cos θ + i sin θ we have z = reiθ and the unit circle (r = 1) can be written in the compact form: z = eiθ Notice that the constraint |z|2 = 1 has left just one degree of freedom. & % QIA Meeting, TechGate 9 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Qubit States A general qubit state can be written |ψi = α|0i + β|1i with complex numbers α and β, and the normalization constraint hψ|ψi = 1 requires that: |α|2 + |β|2 = 1 We can express the state in polar coordinates as: iφα iφβ |ψi = rαe |0i + rβe |1i with four real parameters rα, φα, rβ and φβ. & % QIA Meeting, TechGate 10 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Global Phase Invariance However, the only measurable quantities are the probabilities |α|2 and |β|2, so multiplying the state by an arbitrary factor eiγ (a global phase) has no observable consequences, because: |eiγ α|2 = (eiγ α)∗(eiγ α) = (e−iγ α∗)(eiγ α) = α∗α = |α|2 and similarly for |β|2. So, we are free to multiply our state by e−iφα , giving: 0 i(φβ −φα) iφ |ψ i = rα|0i + rβe |1i = rα|0i + rβe |1i which now has only three real parameters, rα, rβ, and φ = φβ − φα. & % QIA Meeting, TechGate 11 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ The Normalization Constraint In addition we have the normalization constraint hψ0|ψ0i = 1 Switching back to cartesian representation for the coefficient of |1i 0 iφ |ψ i = rα|0i + rβe |1i = rα|0i + (x + iy)|1i and the normalization constraint is: 2 2 2 ∗ |rα| + |x + iy| = rα + (x + iy) (x + iy) 2 = rα + (x − iy)(x + iy) 2 2 2 = rα + x + y = 1 which is the equation of a unit sphere in real 3D space with cartesian coordinates (x, y, rα)! & % QIA Meeting, TechGate 12 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Spherical Polar Coordinates & % QIA Meeting, TechGate 13 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Spherical Polar Coordinates Cartesian coordinates are related to polar coordinates by: x = r sin θ cos φ y = r sin θ sin φ z = r cos θ so renaming rα to z, and remembering that r = 1, we can write: |ψ0i = z|0i + (x + iy)|1i = cos θ|0i + sin θ(cos φ + i sin φ)|1i = cos θ|0i + eiφ sin θ|1i Now we have just two parameters defining points on a unit sphere. & % QIA Meeting, TechGate 14 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Half Angles But this is still not the Bloch sphere. What about the half angles? Let |ψi = cos θ0|0i + eiφ sin θ0|1i 0 0 π iφ and notice that θ = 0 ⇒ |ψi = |0i and θ = 2 ⇒ |ψi = e |1i 0 π which suggests that 0 ≤ θ ≤ 2 may generate all points on the Bloch sphere. & % QIA Meeting, TechGate 15 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Half Angles Consider a state |ψ0i corresponding to the opposite point on the sphere, which has polar coordinates (1, π − θ0, φ + π) |ψ0i = cos(π − θ0)|0i + ei(φ+π) sin(π − θ0)|1i = − cos(θ0)|0i + eiφeiπ sin(θ0)|1i = − cos(θ0)|0i − eiφ sin(θ0)|1i |ψ0i = −|ψi 0 π So it is only necessary to consider the upper hemisphere 0 ≤ θ ≤ 2 , as opposite points in the lower hemisphere differ only by a phase factor of −1 and so are equivalent in the Bloch sphere representation. & % QIA Meeting, TechGate 16 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ The Bloch Sphere We can map points on the upper hemisphere onto points on a sphere by defining θ θ = 2θ0 ⇒ θ0 = 2 and we now have θ θ |ψi = cos |0i + eiφ sin |1i 2 2 where 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π are the coordinates of points on the Bloch sphere. & % QIA Meeting, TechGate 17 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ The Bloch Sphere 0 0 π • Notice that θ = 2θ is a one-to-one mapping except at θ = 2 , where all the points on the θ0 ‘equator’ are mapped to the single point θ = π, the ‘south pole’ on the Bloch sphere • This is okay, since at the south pole |ψi = eiφ|1i and φ is a global phase with no significance. (Longitude is meaningless at a pole!) • Rotations in a 2D complex vector space contain a double representation of rotations in 3D real space • Formally, there is a 2 to 1 homomorphism of SU(2) on SO(3) • Notice also that as we cross the θ0-equator going south, we effectively start going north again on the other side of the Bloch sphere, because opposite points are equivalent on the θ0 sphere. & % QIA Meeting, TechGate 18 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Properties of the Bloch Sphere • Orthogonality of Opposite Points • Rotations on the Bloch Sphere & % QIA Meeting, TechGate 19 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Orthogonality of Opposite Points Consider a general qubit state |ψi θ θ |ψi = cos |0i + eiφ sin |1i 2 2 and |χi corresponding to the opposite point on the Bloch sphere µ ¶ µ ¶ π − θ π − θ |χi = cos |0i + ei(φ+π) sin |1i 2 2 µ ¶ µ ¶ π − θ π − θ = cos |0i − eiφ sin |1i 2 2 So µ ¶ µ ¶ µ ¶ µ ¶ θ π − θ θ π − θ hχ|ψi = cos cos − sin sin 2 2 2 2 & % QIA Meeting, TechGate 20 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Orthogonality of Opposite Points µ ¶ µ ¶ µ ¶ µ ¶ θ π − θ θ π − θ hχ|ψi = cos cos − sin sin 2 2 2 2 But cos(a + b) = cos a cos b − sin a sin b, so π hχ|ψi = cos = 0 2 and opposite points correspond to orthogonal qubit states. Note that in the coordinate system we used in the derivation of the Bloch sphere, with θ0 = θ/2, the two points are also orthogonal - 90◦ apart. & % QIA Meeting, TechGate 21 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Rotations on the Bloch Sphere The Pauli X, Y and Z matrices are so-called because when they are exponentiated, they give rise to the rotation operators, which rotate the Bloch vector (sin θ cos φ, sin θ sin φ, cos θ) about thex ˆ,y ˆ andz ˆ axes: −iθX/2 Rx(θ) ≡ e −iθY/2 Ry(θ) ≡ e −iθZ/2 Rz(θ) ≡ e In order to evaluate these exponentials, let’s take a look at operator functions. & % QIA Meeting, TechGate 22 Ian Glendinning / February 16, 2005 VCPC EUROPEAN CENTRE FOR PARALLEL COMPUTING AT VIENNA ' $ Operator Functions If A is a normal operator (A†A = AA†) with spectral decomposition P A = a a|aiha| then we can define X f(A) ≡ f(a)|aiha| a It is possible to use this approach to evaluate the exponentials of the Pauli matrices, but it turns out to be simpler to use the equivalent power series definition of an operator function.