Unifying Decoherence and the Heisenberg Principle

We exhibit three inequalities involving quantum measurement, all of which are sharp and state independent. The first inequality bounds the performance of joint measurement. The second quantifies the trade-off between the measurement quality and the disturbance caused on the measured system. Finally, the third inequality provides a sharp lower bound on the amount of decoherence in terms of the measurement quality. This gives a unified discription of both the Heisenberg principle and the collapse of the wave function.


Introduction
In quantum mechanics, observables are modelled by self-adjoint operators A, and states by normalized trace-class operators ρ. A state ρ induces a probability measure on an observable A. It is the objective of a quantum measurement to portray this probability measure as faithfully as possible.
According to the uncertainty relation σXσY ≥ 1 2 |tr(ρ[X, Y ])| , (see [He, Ke, Ro]), there is an inherent variance in the quantum state. Furthermore, quantum theory puts severe restrictions on the performance of measurement. These restrictions, which come on top of the measurement restrictions implied by the above uncertainty relation, fall into three distinct classes.
I The impossibility of perfect joint measurement. It is not possible to perform a simultaneous measurement of two noncommuting observables in such a way that both measurements have perfect quality.
II The Heisenberg principle, (see [He]). This states that quantum information cannot be extracted from a system without disturbing that system.
III The collapse of the wave function. When information is extracted from a quantum system, a so-called decoherence is experimentally known to occur on this system.
We will see that this collapse of the wave function is a mathematical consequence of information extraction. In the process, II and III will be clearly exhibited as two sides of the same coin. The subject of uncertainty relations in quantum measurement is already endowed with an extensive literature. For example, the Heisenberg principle and the impossibility of joint measurement are quantitatively illustrated in [AK, Oz, Is, Ha].
However, the inequalities in these papers depend on the state ρ, which somewhat limits their practical use. Indeed, the bound on the measurement quality can only be calculated if the state ρ is known, in which case there is no need for a measurement in the first place.
Our state-independent figures of merit (sections 2 and 3) will lead us quite naturally to stateindependent bounds on the performance of measurement. In order to illustrate their practical use, we will give some applications. We investigate the beamsplitter, resonance fluorescence and nondestructive qubit measurement.
In section 4, we will prove a sharp, state independent bound on the performance of jointly unbiased measurement. This generalizes the impossibility of perfect joint measurement.
In section 5, we will prove a sharp, state independent bound on the performance of a measurement in terms of the maximal disturbance that it causes. This generalizes the Heisenberg principle.
In contrast with the Heisenberg principle and its abundance of inequalities, the phenomenon of decoherence has mainly been investigated in specific examples (see e.g. [Hp, Zu, JZ]). Although there are some bounds on the remaining coherence in terms of the measurement quality (see [JM, Se]), a sharp, information-theoretic inequality does not yet appear to exist.
We will provide such an inequality in section 6, where we will prove a sharp upper bound on the amount of coherence which can survive information transfer. Not only does this generalize the collapse of the wave function, it also shows that no information can be extracted if all coherence is left perfectly intact. It is therefore a unified description of both the Heisenberg principle and the collapse of the wave function.

Information Transfer
In quantum mechanics, a system is described by a von Neumann algebra A of bounded operators on a Hilbert space H. (Usually the algebra B(H) of all bounded operators.) Its state space is formed by the normalized density matrices S(A) = {ρ ∈ A ; ρ ≥ 0, tr(ρ) = 1}. With the system in state ρ ∈ S(A), observation of a (Hermitean) observable A ∈ A is postulated to yield the average value tr(ρA).
Definition. Let A and B be von Neumann algebras. A map T : B → A is called Completely Positive (or CP for short) if it is linear, normalized (i.e. T (1) = 1), positive (i.e. T (X † X) ≥ 0 for all X ∈ B) and if moreover the extension Idn ⊗ T : Mn ⊗ B → Mn ⊗ A is positive for all n ∈ N, where Mn is the algebra of complex n × n-matrices. In this article, we will require CP-maps to be weakly continuous unless specified otherwise.
Its dual T * : S(A) → S(B), defined by the requirement tr(T * (ρ)X) = tr(ρT (X)) ∀ X ∈ B, has a direct physical interpretation as an operation between quantum systems. First of all, due to positivity and normalization of T , each state ρ ∈ S(A) is again mapped to a state T * (ρ) ∈ S(B). Secondly, linearity implies that T * satisfies pT * (ρ1) + (1 − p)T * (ρ2) = T * (pρ1 + (1 − p)ρ2) for all p ∈ [0, 1], ρ1, ρ2 ∈ S(A). This expresses the stochastic equivalence principle: a system which is in state ρ1 with probability p and in state ρ2 with probability (1 − p) cannot be distinguished from a system in state pρ1 + (1 − p)ρ2. Finally, it is possible to extend the systems A and B under consideration with another system Mn, on which the operation acts trivially. Due to complete positivity, states in S(Mn ⊗ A) are once again mapped to states in S(Mn ⊗ B). Incidentally, any CP-map T automatically satisfies T (X † ) = T (X) † and T (X) ≤ X for all X ∈ B.

General, Unbiased and Perfect Information Transfer
Suppose that we are interested in the distribution of the observable A ∈ A, with the system A in some unknown state ρ. We perform the operation T * : S(A) → S(B), and then observe the 'pointer' B in B in order to obtain information on A. One may (see [Ha]) take the position that any CP-map T : B → A is an information transfer from any observable A ∈ A to any pointer B ∈ B. The following is a figure of demerit for the quality of such an information transfer.
Definition. Let T : B → A be a CP-map. Its measurement infidelity δ in transferring information from A to the pointer B is defined as δ := sup S 1S (A) − T (1S(B)) , where S runs over the Borel subsets of R.
It measures how accurately probability distributions on the measured observable A are copied to the pointer B.
The initial state ρ defines a probability distribution Pi on the spectrum of A by Pi(S) := tr(ρ1S(A)), where 1S(A) denotes the spectral projection of A associated to the set S. Similarly, the final state T * (ρ) defines a probability distribution P f on the spectrum of B. δ is now the maximum distance between Pi and P f , where the maximum is taken over all initial states ρ. That is, δ = sup ρ D(Pi, P f ).
The trace distance (a.k.a. variational distance or Kolmogorov distance) is defined as D(P f , Pi) := sup S {|Pi(S) − P f (S)|}, the difference between the probability that the event S occurs in the distribution Pi and the probability that it occurs in the distribution P f , for the worst case Borel set S. Writing out this definition, we see that indeed sup ρ D(Pi, P f ) = sup ρ,S |tr(ρ1S(A)) − tr(ρT (1S(B)))| = sup S 1S(A) − T (1S(B)) = δ. The measurement infidelity δ is precisely the worst case difference between input and output probabilities.
In this article, we will devote considerable attention to the class of unbiased information transfers.

Definition. A CP-map T : B → A is called an unbiased information transfer from the Hermitean observable
Recall that we are interested in the distribution of A, with the system A in some unknown state ρ. We perform the operation T * : S(A) → S(B), and then observe the 'pointer' B in B. Since tr(T * (ρ)B) = tr(ρT (B)) by definition of the dual, and tr(ρT (B)) = tr(ρA) by definition of unbiased information transfer, the expectation value of B in the final state T * (ρ) is the same as that of A in the initial state ρ. We conclude that the expectation of A was transferred to B.
Definition. An information transfer T : B → A from A ∈ A to B ∈ B is called perfect if T (B) = A and if the restriction of T to B ′′ , the von Neumann algebra generated by B, is a * -homomorphism B ′′ → A ′′ .
The entire probability distribution of A is then transferred to B, rather than merely its average value. Indeed, for all moments tr(ρA n ), we have tr(T * (ρ)B n ) = tr(ρT (B n )) = tr(ρT (B) n ) = tr(ρA n ). Everything there is to know about A in the initial state ρ can be obtained by observing the 'pointer' B in the final state T * (ρ). Note that the transfer is perfect if and only if δ = 0.

Example: von Neumann Qubit Measurement
Let Ω := {+1, −1}. Denote by C(Ω) the (commutative) algebra of C-valued random variables on Ω. A state on C(Ω) is precisely a probability distribution P on Ω, and tr(Pf ) should be read as E(f ). Define the probability distributions P± to assign probability 1 to ±1.
A CP-map can thus be seen as an extension of a POVM that keeps track of the system output as well as the measurement output. Since we will be interested in disturbance of the system, it is imperative that we consider the full CP-map rather than merely its POVM.

Maximal Added Variance
For unbiased information transfer, there exists a figure of demerit more attractive than δ. Consider the variance Var(B, T * (ρ)) of the output. (The variance is defined as Var(X, ρ) := tr(ρX † X) − tr(ρX) * tr(ρX).) The output variance can be split in two parts. One part Var(A, ρ) is the variance of the input, which is intrinsic to the quantum state ρ. The other part Var(B, T * (ρ)) − Var(A, ρ) ≥ 0 is added by the measurement procedure. This second part determines how well the measurement performs.
The maximal added variance (where the maximum is taken over the input states ρ) will be our figure of demerit. For example, perfect information transfer from A to B satisfies Var(B, T * (ρ)) = Var(A, ρ), so that the maximal added variance is 0. There is uncertainty in the measurement outcome, but all uncertainty 'comes from' the quantum state, and none is added by the measurement procedure.
Definition. The maximal added variance of an unbiased information transfer T is defined as

It is straightforward to verify
. This inspires the following definition.
Definition. Let T : B → A be a CP-map. We define the operator-valued sesquilinear form It satisfies (X, Y ) † = (Y, X) and is positive semi-definite: (B, B) ≥ 0 for all B ∈ B. This 'length' has the physical interpretation (B, B) = Σ 2 , and there is even a Cauchy-Schwarz inequality: Proof: By the Stinespring-theorem (see [Ta]), we may assume without loss of generality that T is of the form T (X) = V † XV for some contraction V . Writing this out, we obtain (X, If an information transfer is perfect, then of course Σ 2 = (B, B) = 0. (No variance is added.) We will now show that the converse also holds: if Σ 2 = 0, then T is a * -homomorphism on B ′′ . (Compare this with the fact that probability distributions of 0 variance are concentrated in a single point.) Theorem 2 Let T : B → A be a CP-map, let B ∈ B be Hermitean. Then among 1 (B, B) = 0.
2 The restriction of T to B ′′ , the von Neumann algebra generated by B, is a * -homomorphism for all measurable functions f on the spectrum of B.

T maps the relative commutant
the following relations hold: ) for all polynomials f . Thus T is a * -homomorphism from the algebra of polynomials on the spectrum of B to that on T (B). Since T is weakly continuous, this statement extends to the algebras of measurable functions on the spectra of B and T (B), isomorphic to B ′′ and T (B) ′′ respectively. For (2) We see that the maximal added variance Σ 2 equals 0 if and only if T is a perfect information transfer. We shall take Σ to parametrize the imperfection of unbiased information transfer.

Joint Measurement
In a jointly unbiased measurement, information on two observables A andÃ is transferred to two commuting pointers B andB. If A andÃ do not commute, then it is not possible for both information transfers to be perfect. (See [Ne], [We2].) Indeed, the degree of imperfection is determined by the amount of noncommutativity: (1) . By Cauchy-Schwarz, the latter is at most 2ΣB ΣB in norm.
We now show that this bound is sharp in the sense that for all S,S > 0, there exist T , B,B such that (1) attains equality with ΣB = S, ΣB =S.

Application: the Beamsplitter as a Joint Measurement
A beamsplitter is a device which takes two beams of light as input. A certain fraction of each incident beam is refracted and the rest is reflected, in such a way that the refracted part of the first beam coincides with the reflected part of the second and vice versa. We will show that the beamsplitter serves as an optimal joint unbiased measurement.
In cavity QED, a single mode in the field is described by a Hilbert space H of a harmonic oscillator, with creation and annihilation operators a † and a satisfying [a, a † ] = 1, as well as n! |n are dense in H, and satisfy a|α = α|α .
Quantummechanically, a beamsplitter is described by the unitary operator U on H ⊗ H, (This can be seen by sandwiching both sides between coherent vectors.) Since the map Y → U † Y U respects +, · and † , we readily calculate We are now interested in the map ρ → U ρ⊗|0 0|U † , from S(H) to S(H)⊗S(H). In other words, we feed the beamsplitter only one beam of light in a state ρ, the other input being the vacuum. The dual of this is the CP-map T : Apparently, splitting a beam of light in two parts, measuring x ⊗ 1 in the first beam and 1 ⊗ p in the second, and then compensating for the loss of intensity provides a simultaneous unbiased measurement of x and p in the original beam. Since [x, p] = i, we must 1 have ΣBΣB ≥ 1 2 . We now calculate ΣB and ΣB explicitly. From 0|x 2 |0 = 1 2 , we see that T (B 2 ) = x 2 + 1 2 tan 2 (θ)1. Thus Σ 2 B = (B, B) = 1 2 tan 2 (θ). Similarly Σ 2 B = 1 2 tan −2 (θ). We see that ΣBΣB = 1 2 , so that the beamsplitter is indeed an optimal jointly unbiased measurement. By scaling B, optimal joint measurements can be found for arbitrary values of ΣB and ΣB , which shows the bound in theorem 3 to be sharp. It may therefore be used to evaluate joint measurement procedures. For example, it was shown in [JB] that homodyne detection of the spontaneous decay of a two-level atom constitutes a joint measurement with ΣΣ ′ = 1.056, slightly above the bound ΣΣ ′ ≥ 1 provided by theorem 3.
The beamsplitter is an optimal joint measurement in the sense that it minimizes ΣΣ ′ . It also performs well with other figures of merit. For example, if the quality of joint measurement is judged by the state-dependent cost R(T ) := Var(B, T * (ρ)) + Var(B, T * (ρ)), then at least for Gaussian ρ, the optimal measurement is again the above beamsplitter with θ = π/4. (See [Ho].)

The Heisenberg Principle
The Heisenberg Principle may be stated as follows: If all states are left intact, no quantum-information can be extracted from a system. This alludes to an information transfer from an initial system A to a final system consisting of two parts: the system A and an ancilla B, containing the pointer B. We thus have an information transfer T : A ⊗ B → A from A to 1 ⊗ B.
An initial state ρ ∈ S(A) gives rise to a final state T * (ρ) ∈ S(A ⊗ B). Restricting this final state to the system A (i.e. taking the partial trace over B) yields a 'residual' state R * (ρ) ∈ S(A), whereas taking the partial trace over A yields the final state Q * (ρ) ∈ S(B) of the ancilla. We define the CP-maps R : A → A by R(A) := T (A ⊗ 1) and Q : B → A by Q(B) := T (1 ⊗ B). The map R describes what happens to A if we forget about the ancilla B, and Q describes the ancilla, neglecting the original system A.
We wish to find a quantitative version of the Heisenberg principle, i.e. we want to relate the imperfection of the extracted quantum-information to the amount of state disturbance.
Definition. The maximal disturbance ∆ of a map R : A → A is given by ∆ := sup{ R(P ) − P ; P ∈ A, P 2 = P † = P }.

Heisenberg Principle for Unbiased Information Transfer
We first turn our attention to unbiased information transfer. The imperfection of the information is then captured in the maximal added variance Σ 2 .
The Heisenberg principle only holds for quantum-information. Classical observables are contained in the centre Z = {A ∈ A ; [A, X] = 0 ∀X ∈ A}, whereas quantum observables are not. The degree in which an observable A is 'quantum' is given by its distance to the centre d(A, Z) = infZ∈Z A − Z . In the following, we will take the algebra of observables to be B(H) for some Hilbert space H. The centre is then simply C1.
This bound is sharp in the sense that for all ∆ ∈ [0, 1 2 ], there exist T and A for which (2) attains equality.
In the case of no disturbance, ∆ = 0, we see that Σ → ∞. No information transfer from A is allowed if all states on A are left intact. This is Werner's (see [We2]) formulation of the Heisenberg principle. In the opposite case of perfect information transfer, Σ = 0, inequality 2 shows that ∆ must equal at least one half. We shall see in section 6 that this corresponds with a so-called 'collapse of the wave function'. These two extreme situations are connected by theorem 4 in a continuous fashion, as indicated in the graph below: The upper left corner of the curve illustrates the Heisenberg principle, whereas in the the lower right corner, we can see the collapse of the wave function.

Heisenberg Principle for General Information Transfer
We now prove a version of the Heisenberg Principle for general information transfer.

Corollary 5 Let T : B(H) ⊗ B → B(H) be a CP-map, let A ∈ B(H) and B ∈ B be Hermitean,
This bound is sharp in the sense that for all ∆ ∈ [0, 1 2 ], there exists a T for which (4) attains equality. A measurement which does not disturb any state (∆ = 0) cannot yield information (δ ≥ 1 2 ). This is the Heisenberg principle. On the other hand, perfect information (δ = 0) implies full disturbance (∆ ≥ 1 2 ), corresponding to the collapse of the wave function. Both extremes are connected in a continuous fashion, as depicted above.

Application: Resonance Fluorescence
Corollary 5 may be used to determine the minimum amount of disturbance if the quality of the measurement is known. Alternatively, if the system is only mildly disturbed, one may find a bound on the attainable measurement quality. Let us concentrate on the latter option.
We investigate the radiation emission of a laser-driven two-level atom. The emitted EMradiation yields information on the atom. A two-level atom (i.e. a qubit) only has three independent observables: σx, σy and σz. There are various ways to probe the EM field: photon counting, homodyne detection, heterodyne detection, etcetera. For a strong (Ω ≫ 1) resonant (ω laser = ωatom) laser, we will use corollary 5 to prove that any EM-measurement of σx, σy or σz will have a measurement infidelity of at least with λ the coupling constant. For a measurement with two outcomes, δ is the maximal probability of getting the wrong outcome.

Unitary Evolution on the Closed System
The atom is modelled by the Hilbert-space C 2 (only two energy-levels are deemed relevant). In the field, we discern a forward and a side channel, each described by a bosonic Fock-space F. The laser is put on the forward channel, which is thus initially in the state φΩ, the coherent state with frequency ω and strength Ω. (The field strength is parametrized by the frequency of the induced Rabi-oscillations). The side channel starts in the vacuum state φ0. The state at time t is then given by We now investigateTt instead of Tt. Indeed, we are looking for a bound on the measurement infidelity δ = sup S { T (1 ⊗ 1S(B)) − 1S(A) } of T . Yet ifB := U † 2 BU2, thenT (1S(B)) = T (1S(B)), so thatδ = sup S { T (1 ⊗ 1S(B)) − 1S(A) } = δ. If we find the interaction-picture disturbance∆, corollary 5 will yield a bound onδ, and thus on δ.
In the weak coupling limit λ ↓ 0,Tt is given byT * t (ρ) =Û (t/λ 2 )(ρ ⊗ φΩ ⊗ φ0)Û † (t/λ 2 ), where the evolution of the unitary cocycle t →Ût is described (see [AFL]) by a Quantum Stochastic Differential Equation or QSDE. Explicitly calculating the maximal added variances Σ 2 by solving the QSDE is in general rather nontrivial, if possible at all. ( See [JB] for the case of spontaneous decay, i.e. Ω = 0, with the mapTt restricted to the commutative algebra of homodyne measurement results.)

Master Equation for the Open System
Fortunately, in contrast to the somewhat complicated time evolutionTt of the combined system, the evolution restricted to the two-level system is both well-known and uncomplicated. If we use λ −2 as a unit of time, then the restricted evolutionR * t (ρ) := trF⊗FT * t (ρ) of the two-level system is known (see [BGM]) to satisfy the Master equation with the Liouvillian L(ρ) : In this expression, E is the energy-spacing of the two-level atom and V † = σ+, V = σ− are its raising and lowering operators. In the case ω = E of resonance fluorescence, we obtain If we parametrize a state by its Bloch-vectorR * t (ρ) = 1 2 (1 + xσx + yσy + zσz), then equation 5 is simply the following differential equation on R 3 : This can be solved explicitly. For Ω ≫ 1, the solution approaches If we move to the interaction picture once more to counteract the Rabi-oscillations, i.e. with U1(t) = e i 2 Ωtσx and U2 = 1, we see that the time evolution is transformed to Since the trace distance D(ρ, τ ) is exactly half the Euclidean distance between the Bloch vectors of ρ and τ , (see [NC]), we see that ∆ = 1 2 (1 − e − 3 4 t ). For any measurement of σx, σy or σz, we therefore have δ ≥ 1 2 − 1 2 1 − e − 3 2 t by corollary 5 (remember that t is in units of λ −2 ).

Collapse of the Wave function
The 'collapse of the wave function' may be seen as the flip side of the Heisenberg principle. It states that if information is extracted from a system, then its states undergo a very specific kind of perturbation, called decoherence.

Collapse for Unbiased Information Transfer
We start out by investigating unbiased Information Transfer. We prove a sharp upper bound on the amount of remaining coherence in terms of the measurement quality.
Consider the ideal case of perfect (Σ = 0) information transfer. Suppose that the system A is initially in the coherent state |αψx + βψy αψx + βψy|. Then theorem 6 says that, after the information transfer to the ancilla B, the system A cannot be distinguished from one that started out in the 'incoherent' state |α| 2 |ψx ψx| + |β| 2 |ψy ψy|. As far as the behaviour of A is concerned, it is therefore completely harmless to assume that a collapse |αψx+βψy αψx+βψy| → |α| 2 |ψx ψx| + |β| 2 |ψy ψy| has occurred at the start of the procedure. Now consider the other extreme of a measurement which leaves all states intact, i.e. R * (ρ) = ρ for all ρ. Then there exist states for which the l.h.s. of equation (6)  This agrees with physical intuition: decoherence between ψx and ψy is expected to occur in case the information transfer is able to distinguish between the two. This is the case if the variance is small w.r.t. the differences in mean.

Collapse of the Wave function for General Measurement
We will prove a sharp bound on the remaining coherence in general information transfer. For technical convenience, we will focus attention on nondestructive measurements. A measurement of A is called 'nondestructive' (or 'conserving' or 'quantum nondemolition') if it leaves the eigenstates of A intact, so that repetition of the measurement will yield the same result. For example, the measurement in section 2.2 is nondestructive, the one in section 5.3 is destructive. Restriction to nondestructive measurements is quite common in quantum measurement theory (see [Per]).
This bound is sharp in the sense that for all δ ∈ [0, 1 2 ], there exist T , ψi, ψj , α and β for which (8) attains equality. Proof: For sharpness, see section 6.5. Choose a set S such that ai ∈ S and aj / ∈ S. T is an unbiased measurement of T (1 ⊗ 1S(B)) with pointer 1S(B) and maximal added variance Σ 2 ≤ δ(1 − δ) (cf. the proof of corollary 5). We will prove that ψi and ψj are eigenvectors of T (1 ⊗ 1S(B)) with eigenvalues x and y which differ at least 1 − 2δ.

Application: Nondestructive Qubit-Measurement
In quantum information theory, a σz-measurement is often taken to yield output +1 or −1, according to whether the input was | ↑ or | ↓ . It is nondestructive if it leaves the states | ↑ and | ↓ intact, yet it is only unbiased if it is perfect. Corollary 7 shows that in the nondestructive case, the Bloch-sphere collapses to the cigar-shaped region depicted below: z x y Fig. 8: Collapse on the Bloch-sphere with δ = 0.01. Current single-qubit readout technology is just now moving into the regime where the bound (8) becomes significant. in [Lu], a nondestructive measurement of a SQUID-qubit was described, with experimentally determined measurement infidelity δ = 0.13. The bound then equals 0.336.

Conclusion
Our investigation of joint measurement, the Heisenberg principle and decoherence has yielded the following results.
I Theorem 3 provides a sharp, state independent bound on the performance of unbiased joint measurement of noncommuting observables. In the case of perfect (Σ = 0) measurement of one observable, it implies that no information whatsoever (Σ ′ = ∞) can be gained on the other.
II Theorem 4 (for unbiased information transfer) and corollary 5 (for general information transfer) provide a sharp, state independent bound on the performance of a measurement in terms of the maximal disturbance that it causes. In the case of zero disturbance, when all states are left intact, it follows that no information can be obtained. This is the Heisenberg principle.
III Theorem 6 (for unbiased information transfer) and corollary 7 (for general information transfer) provide a sharp upper bound on the amount of coherence which can survive information transfer. For perfect information transfer, all coherence vanishes. This clearly proves that decoherence on a system is a mathematical consequence of information transfer out of this system. If, on the other hand, all states are left intact, then it follows that no information can be obtained. This is the Heisenberg principle. Theorem 6 and corollary 7 connect these two extremes in a continuous fashion; they form a unified description of the Heisenberg principle and the collapse of the wave function.