Open & Time-Evolving Quantum Systems

The common description of time-evolution in a quantum system is the Schr¨odinger equation, given as

i~_∂t∂ |ψ(t)i = H |ψ(t)i (2.14)

for a quantum pure state |ψ(t)i. Here H is the Hamiltonian, the observable operator corresponding to the system’s energy. The effect of this time-evolution for a fixed amount of time t is to apply the unitary matrix Ut = exp(−iHt) to the state by left multiplication |ψi → U |ψi. For a density ρ, ρ → U ρU†. One may

equivalently apply the adjoint matrix to observables, O → U†OU in the Heisenberg picture. By the Stone von Neumann theorem, every finite-dimensional unitary is generated by some Hamiltonian in finite time.

The Schr¨odinger equation describes the evolution of closed quantum systems, so pure states say pure, and all unitaries are invertible. In fields like condensed matter and high-energy, this is often a reasonable assumption. In quantum information, however, exposure to environment is arguably the most important challenge. The mathematical formulation of an open quantum process (when there is no initial system- environment correlation) is the quantum channel : a completely positive, trace-preserving map Φ : S(HA) →

S(HB), for input system A and output B. The physical intuition for quantum channels as open processes

comes from the Stinespring dilation: any channel Φ has the form

Φ(ρ) = trE(U ρU†) ,

where U : A → BE is an isometry. We may think of a channel as attaching an initial environment, time- evolving unitarily, and then tracing out the final environment (which need not always be the same system as the initial environment). In finite-dimension, it is always sufficient to take a pure initial environment, and the size of environment needed to implement any quantum channel is at most |A||B|. Furthermore, all finite-dimensional Stinespring dilations are equivalent up to partial isometries, which are norm-preserving maps, and all minimal Stinespring dilations are equivalent up to unitaries on the environment. I illustrate the Stinespring dilation in Figure 2.2. Infinite-dimensional analogs exist but may involve different constructions.

Analogous to the Stinespring dilation, any finite-dimensional density is the marginal of a pure state on a larger system. For a density ρA _{on system A, ρ = tr}

A0(|ρ1/2ihρ1/2|AA 0

Subsystem (A) Environment (E)

Figure 2.2: A diagram of the Stinespring dilation of finite-dimensional quantum systems. For simplicity, we assume that the channel, Φ, maps densities in system A to other densities in A.

explicitly as a convex combination of pure states,

X i ρi|iihi|A= trA0   X i √ ρi|ii A ⊗ |iiA0× h.c.  .

Purification and Stinespring dilation allow formulations of open processes on mixed states in terms of unitary evolution on pure states, if desired.

2.6.1

Quantum Markov Semigroups

Though the quantum channel often describes a process occurring over time, it hides the passing of time in its internals. Given a quantum channel Φ occurring over a time t ∈ R+_{, one does not necessarily know the}

channel for another time length s 6= t. Physically, this is because the system may have some backaction on its environment. The process that occurs in time 2t, for instance, is not assured to be equivalent to Φ · Φ, since the environment in the later half may store pre-existing correlations with the input system. In many cases, however, physicists assume that the environment immediately dissipates such correlations and resets itself to an initial state. Mathematically, this allows us to more explicitly model the channel in terms of time, parameterizing it as Φt_{. Furthermore, we obtain the crucial semigroup property, Φ}t_{· Φ}s _{= Φ}t+s _for

all t, s ∈ R+_{. A family of channels Φ}t

for t ∈ R+ _{with the semigroup property is called a quantum Markov}

semigroup (QMS).

A QMS will have a Lindbladian generator L [54], so that Φt_{(ρ) = ∂ρ/∂t = −ad}

exp(−tL)(ρ) is the solution

to a differential equation given by

∂

∂t(ρ) = −adL(ρ) .

generates a semigroup of usually non-invertible quantum channels. A usual physical scenario is coupling to a large bath of interacting particles. I describe results on quantum Markov semigroups in Chapter 7. To simplify the notation, I will often write L(ρ) ≡ adL(ρ).

2.6.2

Adjoints and Recovery

In Hilbert space, the adjoint of a unitary U is given by the inner product formula hφ|U ψi = hφU†|ψi and in finite dimension is equal to its Hermitian conjugate. Because matrices are a Hilbert space under ha|bi = tr(a†_{b), we can similarly define the adjoint Φ}† _{of a quantum channel Φ by ha|Φ(b)i = hΦ}†_(a)|bi.

In general, the adjoint of a quantum channel Φ : S(HA) → S(HB) is a unital (maps the identity to the

identity), completely positive map Φ†_{: B(H}B) → B(HA), reversing the direction.

The adjoint of a unitary matrix is its inverse. In contrast, quantum channels generally lack inverses. While a self-adjoint unitary is the identity, there are many non-trivial examples of self-adjoint channels. Conditional expectations are a special case of self-adjoint quantum channels. Self-adjointness is a mathe- matical property that we might deem analogous to having no coherent or rotational part. Other examples of self-adjoint channels include depolarization, which replaces a state by complete mixture, and dephasing, which reduces the off-diagonal components of a density (a completely dephasing channel is a pinching con- ditional expectation). Nonetheless, the fact that the adjoint of a quantum channel switches the input with output space, as well as its role on unitaries, suggests that it has some markings of an inverse-like operation. While non-unitary channels need not have inverses, it is still possible and fruitful to study operations that partially reconstruct the input state from the output, known as recovery maps. Probably the most canonical recovery map is the Petz recovery map Rω,Φ: S(HB) → S(HA), given by

Rω,Φ(ρ) = ω1/2Φ† (Φ(ω)−1/2)ρ(Φ(ω)−1/2)
ω1/2.

In addition to the original channel, the Petz recovery map contains the extra parameter ω, a density in S(HA). Rω,Φ perfectly recover ω. Furthermore,

D(ρkω) = D(Φ(ρ)kΦ(ω)) ⇐⇒ Rω,Φ· Φ(ρ) = ρ . (2.15)

More recently, approximate versions of Equation (2.15) have appeared for modifications of the Petz map [55, 56, 57].