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arXiv:2512.02477

Qudits offer no advantages over dits for sending random messages

Ronit Shah

quant-phcs.ITSubmitted Dec 2, 2025• Announced Dec 3, 2025• 0 citations
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Abstract

We consider the following simple scenario: Alice has one of many possible messages, drawn from a known distribution, and wants to maximize the probability that Bob guesses her message correctly. We prove that if Alice can send only a qudit to Bob, without preshared entanglement, there is never any advantage over sending him a classical dit. This result was previously known only for a uniform distribution. We also prove a mixed-state generalization of this result in the form of an upper bound on the success probability of discriminating between mixed quantum states with a single measurement. This bound is based solely on the dimension, probability distribution, and eigenvalues of the states and is sharp among such bounds.

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We consider the following simple scenario: Alice has one of many possible messages, drawn from a known distribution, and wants to maximize the probability that Bob guesses her message correctly. We prove that if Alice can send only a qudit to Bob, without preshared entanglement, there is never any advantage over sending him a classical dit. This result was previously known only for a uniform distribution.
We also prove a mixed-state generalization of this result in the form of an upper bound on the success probability of discriminating between mixed quantum states with a single measurement. This bound is based solely on the dimension, probability distribution, and eigenvalues of the states and is sharp among such bounds.
The capacity of quantum states to send information is a fundamental and intriguing issue in quantum communication theory. This study explores a scenario where Alice intends to transmit n bits of classical information to Bob by sending m qudits, using an arbitrary bipartite pure entangled state with E qudits as the shared resource. We employ the unambiguous discrimination strategy in decoding and establish an upper bound on the success probability, which extends the findings of a previous study (Li in Phys Rev A 85:052304, 2012).
We consider the scenario where Alice wants to send a secret (classical) nnn-bit message to Bob using a classical key, and where only one-way transmission from Alice to Bob is possible. In this case, quantum communication cannot help to obtain perfect secrecy with key length smaller then nnn. We study the question of whether there might still be fundamental differences between the case where quantum as opposed to classical communication is used. In this direction, we show that there exist ciphers with perfect security producing quantum ciphertext where, even if an adversary knows the plaintext and applies an optimal measurement on the ciphertext, his Shannon uncertainty about the key used is almost maximal. This is in contrast to the classical case where the adversary always learns nnn bits of information on the key in a known plaintext attack. We also show that there is a limit to how different the classical and quantum cases can be: the most probable key, given matching plain- and ciphertexts, has the same probability in both the quantum and the classical cases. We suggest an application of our results in the case where only a short secret key is available and the message is much longer.
Under two-party deterministic dense coding, Alice communicates (perfectly distinguishable) messages to Bob via a qudit from a pair of entangled qudits in pure state $|\ensuremath{\Psi}\ensuremath{\rangle}$. If $|\ensuremath{\Psi}\ensuremath{\rangle}$ represents a maximally entangled state (i.e., each of its Schmidt coefficients is 1/d\sqrt{1/d}1/d​), then Alice can convey to Bob one of d2{d}^{2}d2 distinct messages. If $|\ensuremath{\Psi}\ensuremath{\rangle}$ is not maximally entangled, then Ji et al. [Phys. Rev. A 73, 034307 (2006)] have shown that under the original deterministic dense-coding protocol, in which messages are encoded by unitary operations performed on Alice's qudit, it is impossible to encode ${d}^{2}\ensuremath{-}1$ messages. Encoding ${d}^{2}\ensuremath{-}2$ messages is possible; see, for example, the numerical studies by Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. Answering a question raised by Wu et al. [Phys. Rev. A 73, 042311 (2006)], we show that when $|\ensuremath{\Psi}\ensuremath{\rangle}$ is not maximally entangled, the communications limit of ${d}^{2}\ensuremath{-}2$ messages persists even when the requirement that Alice encode by unitary operations on her qudit is weakened to allow encoding by more general quantum operators. We then describe a dense-coding protocol that can overcome this limitation with high probability, assuming the largest Schmidt coefficient of $|\ensuremath{\Psi}\ensuremath{\rangle}$ is sufficiently close to 1/d\sqrt{1/d}1/d​. In this protocol, ${d}^{2}\ensuremath{-}2$ of the messages are encoded via unitary operations on Alice's qudit, and the final $({d}^{2}\ensuremath{-}1)$-th message is encoded via a non-trace-preserving quantum operation.
Quantum resources may provide an advantage over their classical counterparts in communication tasks. We refer to this as quantum advantage. Here, we consider a single communication task to study different approaches of observing quantum advantage. We call this setting a unified scenario. Our task is described as the following. There are three parties---the Manager, Alice, and Bob. The Manager sends a value of a random variable to Alice, and at the same time Bob receives some partial information regarding that value. Initially, neither Alice nor Bob knows the input of the other, which is received from the Manager. The goal of the task is achieved if and only if the value of the random variable, sent to Alice by the Manager, is identified by Bob with a success probability greater than half all the time. Here, nonzero error probability is allowed. However, to help Bob, Alice sends a limited amount of classical or quantum information to him (cbit or qubit). We show that the goal of the task can be achieved when Alice sends a qubit. On the other hand, a cbit communication is not sufficient for achieving the goal. Thus, it establishes quantum advantage. We further show that the optimal success probability in the overall process for a qubit communication can be higher than the optimal success probability for a cbit communication. In fact, this success probability is higher for qubit communication even if along with cbit communication Alice and Bob share randomness which allows mixing of classical strategies. Clearly, it not only establishes quantum advantage, it also demonstrates a more prominent nonclassical feature. Actually, we obtain higher success probability compared to other tasks. This also suggests the experiment-friendly nature of our task. For applications, we connect our task with semi-device-independence and show how our task can be useful to detect quantumness of a communication in a secure way. Moreover, within the same setting, we provide a way to detect universal coherence present in an ensemble. For a high-dimensional random variable, to achieve the goal, it may require a high-dimensional classical communication, while it can be achieved with only a qubit communication. This establishes a large separation between the quantum and the classical communication.
In this paper, we consider two related problems, unambiguous discrimination of quantum mixed states and unambiguously conveying classical messages using Einstein-Podolsky-Rosen (EPR) pairs. The main technical contributions are as follows. First, we present a general upper bound on the success probability for unambiguous discrimination among arbitrary NNN mixed quantum states with given a priori probability, which can be directly used to the unambiguous communication setting. Second, we propose a sufficient condition to show that a new upper bound can be achievable, and construct examples to clarify the relationship between the upper bound we derive and the existing upper bounds presented by other authors. Third, in the issue of conveying classical messages using EPR pairs, we also derive an upper bound on the success probability for unambiguously decoding, where Alice wants to convey nnn bit classical information to Bob by sending mmm quantum bits.
We investigate the quantum advantage that can arise in typical two-party communication scenarios, where the sender and the receiver are allowed to share prior correlations. Focusing on communication tasks constrained by the distinguishability of the sender's inputs, we demonstrate that entanglement-assisted communication with both classical and quantum message can outperform classical communication supplemented with shared randomness. We begin by developing a general framework for communication tasks with pre-shared correlations. Within this framework, we establish an equivalence among entanglement-assisted classical communication, entanglement-assisted quantum communication, and quantum communication, showing that no hierarchy exists between these three paradigms. We then investigate the scenario where the receiver has no input and prove that no advantage can arise in this case. However, an advantage in the entanglement-assisted setting emerges once additional constraints are imposed on the dimension of the communicated message. This further highlights the superiority of entanglement-assisted classical communication over standard quantum communication. Then we demonstrate several tasks where the entanglement-assisted protocol using one-bit communication proves to be advantageous over classical communication. Finally, by constructing an explicit class of communication tasks, we show that a non-maximally entangled states outperform the maximally entangled state as a pre-shared resource between the communicating parties.