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arXiv:2511.01559 · Open Systems & Information Dynamics

Security in a prepare-and-measure quantum key distribution protocol when the receiver uses weak values to guess the sender's bits

Rajendra Singh Bhati

quant-phSubmitted Nov 3, 2025• Announced Nov 4, 2025• 0 citations
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Abstract

The weak values and weak measurement formalism were initially limited to pure states, which were later extended to mixed states, leading to intriguing applications in quantum information processing tasks. Weak values are considered to be abstract properties of systems describing a complete picture between successive measurements in the two-state vector formalism (TSVF). The remarkable achievements of the weak value formalism in experimental quantum mechanics have persuaded most quantum physicists that it is impeccable. However, we explore a scenario where the formalism of weak values for mixed states is employed in a quantum communication protocol, but discover that it generates inaccurate outcomes. This reinforces our previous conclusion that the weak values may not be elements of the reality of weak measurements, contrary to what the proponents of weak values proposed.

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The weak values and weak measurement formalism were initially limited to pure states, which were later extended to mixed states, leading to intriguing applications in quantum information processing tasks. Weak values are considered to be abstract properties of systems describing a complete picture between successive measurements in the two-state vector formalism (TSVF). The remarkable achievements of the weak value formalism in experimental quantum mechanics have persuaded most quantum physicists that it is impeccable. However, we explore a scenario where the formalism of weak values for mixed states is employed in a quantum communication protocol, but discover that it generates inaccurate outcomes. This reinforces our previous conclusion that the weak values may not be elements of the reality of weak measurements, contrary to what the proponents of weak values proposed.
The novel physical quantity known as the weak value has attracted much attention for two reasons.First, from the perspective of applications, since the weak value can take arbitrarily large values, it is expected to be useful for detecting very small physical signals.Indeed, following an initial successful demonstration, numerous reports have supported the utility of this method, in which small signals are amplified through large weak values.Second, from the perspective of the foundations of quantum mechanics, the weak value is expected to shed light on certain quantum paradoxes.Its proponents argue that the weak value reveals the behavior of a quantum system prior to measurement.In this dissertation, we analyze the Two-State Vector Formalism (TSVF), which is the theoretical origin of the discovery of the weak value.The TSVF describes a quantum system that is conditioned by both pre-and post-selection.Therefore, the term "weak" does not necessarily imply that the formalism itself requires weak interactions.Rather, it refers to the fact that the measurement performed to detect the weak value is weak, typically characterized by a small coupling constant.While the weak value has attracted much attention, most studies have focused on the weak coupling regime, and the strong or intermediate coupling regions remain relatively unexplored.In this work, we numerically analyze systems described by the TSVF beyond the weak limit.Our findings suggest that the amount of information obtainable from the post-selected system has an upper bound.This observation motivates an information-theoretic analysis, through which we identify peculiar behavior in the information content associated with the quantum state in weak regime.
The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.
In this article we present a new prepare and measure quantum key distribution protocol that decouples the necessary quantum channel error estimation from its dependency on sifting, or otherwise post-selecting, the detection outcomes. Rather than estimating Eve's coupling to the quantum channel from the statistics of the sifted key, we infer this information from weak measurements made equally on all of the received photons immediately prior to post-selection by the photon detectors. We prove that the accuracy of the weak measurement parameter estimation is robust to reasonable device imperfections, even in an adversarial environment, and hence the asymptotic security of this protocol can be inferred from the security analysis of BB84. In addition to eliminating detector basis-dependent attacks, such as detector blinding, we demonstrate that this new prepared and measure QKD protocol is immune to a very powerful class of measurement-side device attacks that also allow an adversary control of the weak measurement outcomes given two modest requirements placed on the measurement-side devices. Finally, we compare the asymptotically achievable secure key rate of a decoy state version of the weak measurement protocol and show it is essentially equal to that of BB84 with decoy states and significantly higher than MDI-QKD for realistic system parameters.
In this thesis, the article “Can a Future Choice Affect a Past Measurement’s\nOutcome?” is used as a starting point to explore the peculiar\ncorrelations between subsequent weak (i.e. non-projective) and\nstrong measurements in a variation of the EPR-Bell experiment. Bell’s\ntheorem states that particles may not possess predetermined values.\nHowever, as the outcomes of the subsequent weak and strong measurements\ndo statistically agree, an apparent contridiction arises. The\nauthors claim that the correlations between the results are best explained\nusing the time-symmetric two-state vector formalism, stating\nthat the experimenter’s future choice has been encrypted in the weak\nmeasurement’s outcomes.\nAlternative explanations are analyzed and a numerical simulation was\nmade to invigorate the notion of a so called weak-bias. Meaning that\nthe weak measurements sufficiently influence the particles’ states to\naccount for the statistical agreement.
A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first pre-selected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measuring procedure that fulfills certain requirements of “weakness”. We call it weak measurement . Physical meaning and underlying mathematical structure of weak measurements are explored.
We investigate the sequential composition of weak values in the framework of time-symmetric quantum mechanics. Specifically, we consider a forward'' weak measurement from a preselected state ∣ψ⟩\ketψ∣ψ⟩ to a post-selected state ∣ϕ⟩\ketϕ∣ϕ⟩, followed by a reverse'' weak measurement. We show that the product of these two weak values corresponds to the normalized expectation value of a strong, state-conditioned observable B=APψAB = A P_ψAB=APψ​A, where Pψ=∣ψ⟩⟨ψ∣P_ψ= \ketψ\braψPψ​=∣ψ⟩⟨ψ∣ is the projector onto the preselected state. Analyzing the structure of BBB, we demonstrate how it encodes interference information, particularly when ∣ψ⟩\ketψ∣ψ⟩ is a superposition rather than an eigenstate of AAA. This formulation extends naturally to mixed states by replacing PψP_ψPψ​ with a generic density matrix ρρρ, linking the construction to the formalism of generalized quantum measurements. We illustrate practical applications in quantum information, including state-specific error witnessing in quantum computing, and show how the phase of a weak value can be inferred via strong measurements in the pure-state case.