One Interaction, Different Measurement Records

A POVM predicts outcome probabilities, an instrument also predicts the state after each outcome, and a channel predicts the state after outcomes are averaged. An instrument determines both its POVM and its averaged channel: $E_r=\mathcal I_r^\dagger(I)$ and $\Phi=\sum_r\mathcal I_r$, where the adjoint is defined by the trace pairing. The reverse inference is generally not unique: neither a POVM nor an averaged channel determines the instrument. A master equation adds evolution in time but does not uniquely specify the monitored record. One ancilla interaction makes these distinctions concrete.

Required background. The volume introduction distinguishes recorded and ignored outcomes. Use Density Operators and Partial Trace to follow the example below.

One interaction, several descriptions

Let a system qubit begin in $\alpha|0\rangle+\beta|1\rangle$, with $|\alpha|^2+|\beta|^2=1$, and an ancilla begin in $|0\rangle$. A controlled-NOT gate, with the system as control, produces

$$(\alpha|0\rangle+\beta|1\rangle)|0\rangle \longmapsto \alpha|0\rangle|0\rangle+\beta|1\rangle|1\rangle.$$

This is a unitary operation on both qubits. The system alone has a different description depending on what is done with the ancilla.

Read the ancilla in the Z basis. Outcome $0$ occurs with probability $|\alpha|^2$ and leaves the system in $|0\rangle$; outcome $1$ similarly leaves it in $|1\rangle$. The system instrument is $\mathcal I_r(\rho)=P_r\rho P_r$, with $P_r=|r\rangle\langle r|$. Its POVM consists of $P_0,P_1$.

Read the ancilla in the X basis. The two system Kraus operators are now $I/\sqrt2$ and $Z/\sqrt2$. Each outcome has probability $1/2$, independent of the input. The conditional system state is either the original state or its phase-flipped version. This detector basis reveals no information about the initial system state, yet the outcome specifies a correctable phase change.

Ignore the ancilla. Either choice of ancilla measurement gives the same averaged system channel:

$$\Phi(\rho)=P_0\rho P_0+P_1\rho P_1 =\frac12(\rho+Z\rho Z).$$

The equality holds for every input density operator. One channel therefore admits instruments with very different records and conditional states. Tracing out the ancilla without measuring it gives this same channel. See Quantum Instruments and Kraus Representation for the general statements.

What each mathematical object supplies

Object	What it predicts	What it does not specify by itself
POVM	Probabilities $p_r=\operatorname{Tr}(\rho E_r)$	The output state or detector mechanism
Instrument	Probabilities and conditional output states	A unique physical realization
Channel	The averaged output for any input	Which outcomes were observed, or how the environment evolved
System–bath model	Joint evolution once the initial state is supplied	Which bath observables the detector monitors
Master equation	Reduced evolution over time in its stated regime	A unique monitored record or trajectory
Trajectory model	Conditional states for a specified monitoring scheme	A unique path that exists regardless of how it is monitored

A sharp PVM fixes probabilities just as a POVM does. The familiar Lüders update is an additional specification of its instrument; it is especially consequential for degenerate outcomes. See Degenerate Measurements.

Where decoherence enters

In the ancilla example, the two system alternatives become correlated with orthogonal ancilla states. The system coherence disappears after the ancilla is discarded. More generally, an interaction can correlate the alternatives with normalized states $|e_0\rangle,|e_1\rangle$ whose overlap is nonzero. Then

$$\rho_{01}\longmapsto \rho_{01}\langle e_1|e_0\rangle.$$

The overlap quantifies how much local interference remains. This connects Environment-Induced Decoherence to measurement: a distinguishable environmental record can suppress interference even if nobody reads it. The joint state still contains correlations, and the reduced mixture alone does not select one outcome.

The $X$-basis readout above also illustrates why reduced decoherence need not mean that information is irretrievably destroyed. Given the record and access to a phase correction, the original system state can be restored in this ideal model. That possibility depends on control of the ancilla and record; it is rarely available for a large uncontrolled environment.

Adding time and memory

A single channel says nothing about rates. A family $\Phi_t$ specifies evolution from an initial time, and a master equation may provide a convenient differential representation. A time-independent GKLS generator produces a completely positive semigroup under the usual finite-dimensional assumptions. A time-dependent equation can describe more general evolution, including some dynamics with memory.

Do not identify “time local” with “memoryless.” When $\Phi_t$ is invertible and differentiable, its generator can formally be written $\dot\Phi_t\Phi_t^{-1}$ even for non-CP-divisible evolution. Singular times require separate treatment. Time-Convolutionless Master Equations develops this distinction; Non-Markovian Dynamics explains why several notions of memory coexist.

Monitoring a time-dependent interaction adds another choice: which emitted field or bath observable is read out. Continuous Measurement and Quantum Trajectories shows how averaging those conditional states recovers the corresponding unconditioned evolution.

Exercises

Same channel, different knowledge. In the ancilla example, take the input $|+\rangle$. List the conditional system states for ancilla $Z$ readout and for ancilla $X$ readout, and average each pair.

Solution

The $Z$ readout prepares $|0\rangle$ or $|1\rangle$. The $X$ readout prepares $|+\rangle$ or $|-\rangle$. Every outcome here has probability $1/2$, and both averages equal $I/2$. Equal density operators need not correspond to the same available classical record.

Partial distinguishability. If $\langle e_1|e_0\rangle=1/2$, what happens to the $X$ expectation for an initial $|+\rangle$?

Solution

The initial off-diagonal element is $1/2$ and becomes $1/4$. Thus $\langle X\rangle=2\operatorname{Re}\rho_{01}$ falls from $1$ to $1/2$. The populations remain $1/2$, while interference is partially suppressed.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Reduced maps and master equations.
• Nielsen, M. A., and I. L. Chuang. Quantum Computation and Quantum Information. 10th anniversary edition, Cambridge University Press, 2010. Ancilla implementations of measurements and channels.
• Watrous, J. The Theory of Quantum Information. Cambridge University Press, 2018. Chapter 2, measurements, channels, and representations. Author manuscript.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Conditional evolution and distinct monitoring schemes.