Completing the generalized-state description of a quantum system between two measurements: weak-vector constraints, mixed generalized states, the weak–strong measurement converse for general observables, and definite values of non-commuting observables

Aharonov and Vaidman (J. Phys. A 24, 2315 (1991)) introduced the generalized state — an arbitrary superposition of two-state vectors describing a quantum system at a time between a pre-selection and a post-selection measurement — and proved the first theorems linking weak and strong measurement outcomes for such systems. The paper explicitly left central components of the formalism undeveloped: the restriction on weak vectors of product-type (two-state-vector) states, stated as $|\sigma_w|=1$ in a footnote without derivation; the density-matrix analogue for mixed generalized states, only outlined; the classification of sets of non-commuting observables that can be assigned simultaneous definite values, completed only for spin-1/2; and the converse of the theorem 'a strong measurement with a definite outcome forces the weak value to equal that outcome', which was proved only for dichotomic (two-eigenvalue) variables. Later work developed parts of this program: the density-matrix analogue has been largely covered by the multiple-time-state formalism, and the definite-value classification has been advanced through the mean king's problem and mutually unbiased bases, but the weak-vector restriction derivation, the general weak–strong converse, and the complete classification beyond spin-1/2 remain without a published resolution. The problem asks to complete these components of the formalism.

Background

In the standard formulation of quantum mechanics, the most complete description of a system at a given time is its state vector, or its density matrix when the system is entangled with other systems; this description is built only from the results of measurements performed in the past. Aharonov, Bergmann and Lebowitz (1964) observed that if a system is pre-selected at time $t_1$ and later post-selected at time $t_2$, the probabilities of outcomes of a measurement at an intermediate time $t$ are given by a time-symmetric expression (the ABL rule) that uses information from both boundaries, $\mathrm{Prob}(C=c_n) = |\langle\Phi|P_n|\Psi\rangle|^2 / \sum_m |\langle\Phi|P_m|\Psi\rangle|^2$, where $|\Psi\rangle$ is the forward-evolving state created at $t_1$, $\langle\Phi|$ is the backward-evolving state verified at $t_2$, and $P_n$ is the projector onto the eigenspace of $c_n$. The pair $\langle\Phi|,|\Psi\rangle$ became known as the two-state vector: a description of the system in the interval between two complete measurements that uses both past and future data.

A weak measurement is a standard von Neumann coupling of the system to a pointer that is made so imprecise (a pointer spread large compared with the possible shifts) that it does not appreciably disturb the system between the two boundary measurements. Aharonov, Albert and Vaidman (1988) showed that in the weak limit the pointer records not an eigenvalue and not the expectation value, but the weak value, $A_w = \langle\Phi|A|\Psi\rangle/\langle\Phi|\Psi\rangle$, a complex number that can lie far outside the spectrum of $A$ (a spin-1/2 component can weakly read 100). Weak values have since become a major tool of quantum foundations and metrology.

Aharonov and Vaidman (1991) generalized the two-state vector to the generalized state: an arbitrary superposition $\sum_i \alpha_i\,\langle\Phi_i|\otimes|\Psi_i\rangle$, i.e. an arbitrary vector in the tensor product of the space of backward-evolving bras and forward-evolving kets. The ordinary two-state vector $\langle\Phi|\,|\Psi\rangle$ is the special 'product' type of generalized state; every generalized state can be realized by entangling the system with an ancilla (an auxiliary quantum system) and performing complete measurements on the composite at both boundaries. The paper generalized the ABL probability rule and the weak-value formula to arbitrary generalized states, and introduced the three-box paradox (a pre- and post-selection puzzle in which a ball hidden in one of two boxes is later found, with probability one, in whichever box is opened) and the mean king's problem (a retrodiction puzzle, due to Vaidman, Aharonov and Albert (1987), in which one must determine after the fact the outcome of a measurement that was randomly chosen from several non-commuting observables).

For a spin-1/2 particle the weak values of the three Pauli operators $\sigma_x,\sigma_y,\sigma_z$ define the weak vector $\boldsymbol{\sigma}_w = \boldsymbol{\sigma}_R + i\,\boldsymbol{\sigma}_I$ in the three-dimensional space of observables, and the paper used its geometry to classify which spin components can be 'ascertained' (i.e. are known to yield a definite outcome with probability one if measured): for a product-type generalized state the ascertainable directions form a cone, and four common lines of such cones must satisfy $\sum_{i=1}^{4}\mathbf{n}_i = 0$ (a condition first derived by Ben-Menahem (1989)). A variable is called dichotomic if it has exactly two distinct eigenvalues (projectors and spin-1/2 components are examples).

The paper's section on weak and strong values proved two theorems: (i) if a strong (conventional, projective) measurement of any observable yields an outcome with probability one, then the weak value equals that outcome; and (ii) the converse — if the weak value equals one of the eigenvalues, then a strong measurement yields that eigenvalue with probability one — was proved only for dichotomic variables. The conclusions stated that the generalized state is the analogue of a pure state for systems about which information is available from both the future and the past, that the analogue of a density matrix 'can be defined easily' but was not developed, and that the formalism introduces no new physics beyond standard quantum mechanics while suggesting new experiments. Later work (Mermin (1990) and the mean-king literature) extended the definite-value constructions to higher-dimensional systems using mutually unbiased bases (MUB: orthonormal bases of a $d$-dimensional Hilbert space whose cross-basis overlaps all have magnitude $1/\sqrt{d}$).

Problem Statement

Complete the generalized-state formalism for the description of a quantum system in the time interval between a pre-selection and a post-selection measurement, by resolving the components that Aharonov and Vaidman (1991) explicitly left undeveloped. These components all serve one objective — making the generalized-state description as complete a theory as the ordinary state-vector description — and are linked by the weak-vector geometry and the weak–strong measurement theorems of the source paper:

1. Weak-vector constraints. Derive, from general principles of quantum mechanics, the restrictions satisfied by the weak vector $\boldsymbol{\sigma}_w$ (the vector of weak values of the spin components) of a product-type (two-state-vector) generalized state — stated in the source as $|\boldsymbol{\sigma}_w| = 1$ for the weak vector of the form $(1/\cos(\alpha/2), 0, i\tan(\alpha/2))$ — and, more generally, characterize which weak vectors arise from which generalized states, explaining why product-type states are constrained while arbitrary superpositions are not.

2. Mixed generalized states. Develop the systematic analogue of the density matrix for the generalized-state formalism: a complete description of a system correlated with external systems at one or both time boundaries, i.e. mixed generalized states, together with the corresponding probability and weak-value rules (the 1991 paper only outlined this extension and noted it could be handled as a mixture of generalized states). A resolution must state the mixed generalized state and its probability and weak-value rules explicitly within the generalized-state formalism and fix their relation to the multiple-time-state construction of ref. 9; citing that construction alone does not resolve the item.

3. Weak–strong converse for general observables. Prove or refute: if the weak value of an observable with more than two eigenvalues equals one of its eigenvalues (the weak value being well defined, i.e. the relevant scalar product non-vanishing), must a strong measurement of that observable on the same pre- and post-selected ensemble yield that eigenvalue with probability one? The forward implication (a strong measurement that is certain to yield an outcome forces the weak value to equal it) holds for all observables, and the converse is known only for dichotomic variables.

4. Classification of definite values of non-commuting observables. Extend the classification of sets of non-commuting observables that can be assigned simultaneous definite values between a pre-selection and a post-selection — i.e. whose measurement outcomes can each be ascertained with probability one — beyond the spin-1/2 results of the source paper (cones of directions; four common lines subject to $\sum_i \mathbf{n}_i = 0$): characterize the admissible sets for arbitrary numbers of observables, for higher-dimensional systems, and for arbitrary (not necessarily mutually unbiased or dichotomic) observables.

Resolve each of these four items within the generalized-state formalism (or an equivalent two-state-vector/multiple-time formalism), fixing the conventions the 1991 source left implicit, including the norm implicit in the $|\boldsymbol{\sigma}_w| = 1$ restriction.

References

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