Approach to optimal quantum transport via states over time

arXiv:2504.04856 · quant-ph, math-ph, math.MP · Submitted 2025-04-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Approach to optimal quantum transport via states over time".

Kai: The paper approaches quantum optimal transport by generalizing classical Monge's theory to quantum systems,

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So we're diving into the paper "Approach to optimal quantum transport via states over time" today. It sounds like they are tackling how to do the classical optimal transport idea in a quantum setting, which is pretty ambitious.

Mira: I agree, Kai, the title suggests they are trying to bridge that gap between classical mass distributions and quantum density matrices using this new "states over time" concept. It's interesting because it moves beyond just looking at state distinguishability.

Lev: From my perspective in error correction, I'm curious if this framework is practical for any sort of physical realization we have right now; we need to know if the mathematical structure actually maps onto something measurable on a quantum computer.

Kai: Exactly, Lev, and that’s what we’re going to probe today—what they actually built and measured. The paper starts by setting up this concept of a "state over time" as a Jordan product involving an initial density matrix and a completely positive map.

Mira: That structure is mathematically rich; it’s defining the cost functional as something bilinear in both the initial state and that channel map, which mirrors how classical transport works with probability densities. It attempts to keep that bilinear property in the quantum realm.

Lev: The way they define this coupling pi(x, y) as a stochastic matrix yielding an output nu for input mu, and then recovering the transformation using Bayes’ Theorem, that's the mechanism I need to look at for potential error propagation issues when we scale this up.

Kai: Right, and that leads us directly into how they define the transport cost itself as a linear function of this "state over time," which is what makes it an optimal transport problem.

Mira: The mathematical foundation they lay out using Choi and Jamiołkowski isomorphisms is key here because it clarifies the relationship between the quantum channels and these states, showing how you can check positivity in different but related ways.

Lev: I see why that distinction matters; if we are building error correction schemes, knowing which representation—Choi or Jamiołkowski—is more convenient for checking the positivity condition on E is a practical detail we'd need to worry about during implementation.

Kai: And they then define the specific quantum transport cost, kappa(rho, E), as the trace of this state over time multiplied by the channel cost matrix K. It’s that definition that sets up the minimization problem for finding the optimal transport plan E.

Mira: The core result they aim for is defining K(rho, sigma) as a minimum over all channels E within a specific set of states over time, using this cost kappa(rho, E). It’s an attempt to formalize that minimization in the quantum context.

Lev: So, if we were to try and run this on hardware today, determining that minimum requires efficiently characterizing the set Q(rho, sigma), which sounds like a computational bottleneck for any real-time process.

Title and authors: Kai: That’s a fair point, Lev; characterization of these sets is often computationally demanding in quantum information theory, and we need to see if they provide an efficient path forward for experimentalists.

Mira: The paper investigates several properties for the cost matrix K, like requiring the swap operator to have a zero trace involving S for the identity channel, which helps ensure certain consistency conditions are met.

Lev: Consistency checks on the cost matrix are vital because in error correction, we always need to ensure that our chosen metric respects physical constraints like positivity and bounds on errors.

Kai: They also look at the triangle inequality requirement for K, which involves a complex structure related to states over multiple times, which suggests they are building a partial order for these cost matrices.

Mira: That ordering of cost matrices is significant because it tells us when one transport plan is inherently "better" than another in terms of the physical resources it consumes.

Lev: A partial order helps define what we consider an admissible path for quantum evolution, which is something I can relate to the structure needed for robust quantum operations.

Kai: The paper then zeroes in on unitary invariant costs, which means the cost function doesn't change if you just rotate your input and output states by a unitary transformation. That’s a very strong requirement for robustness.

Mira: They found that only positive multiples of K zero = d one - S satisfy the condition of being in the dual cone of states over time and commuting with those specific unitaries, which is quite restrictive <ref:2504.04856#pg1>.

Lev: If we are designing universal quantum algorithms, having a cost function that is invariant under global unitary transformations would make our design much more flexible across different computational bases.

Kai: And when they look at the limit as the Hilbert space dimension d gets very large, they find an asymptotic cost K infinity(rho, sigma) related to entanglement fidelity and a classical transport problem involving total variation distance for commuting states.

Mira: That connection to the classical total variation distance in the large dimension limit is quite interesting because it suggests a deep underlying link between the quantum transport cost and simpler classical metrics when things are highly entangled.

Lev: So, if we look at that limit, it means for very complex systems, our physical transport effort starts behaving like a standard statistical measure of how different two states are in terms of their overall distribution overlap.

Kai: And they also point out a symmetry gap between the cost K(rho, sigma) and its reverse K(sigma, rho), which can become discontinuous when a channel maps a non-pure state to a pure one.

Mira: That discontinuity is something I think we need to keep in mind because it means the cost function isn't always behaving smoothly across different state transitions, which complicates our theoretical modeling of evolution.

Lev: Discontinuities in cost functions often signal points where the underlying physical assumptions break down, which is exactly where error correction protocols might fail unexpectedly.

Title and authors: Kai: So, to wrap up this discussion on "Approach to optimal quantum transport via states over time," we've seen how they formalize a cost based on the Jordan product of density matrices and channels.

Mira: The main implication is establishing a cost functional that respects the bilinear nature of classical transport while incorporating the physical constraints imposed by CPTP maps in a way that is fundamentally quantum.

Lev: For error correction research, it suggests we need to move beyond simple distance metrics and start quantifying the actual physical effort—the resource cost—required to move from one quantum state to another.

Kai: I think the paper gives us a solid mathematical language for defining what "effort" means in this context, which is something our experimentalists can use when designing new state preparation protocols.

Mira: It’s a step toward developing quantum-aware distinguishability metrics that aren't just based on fidelity or overlap, but on the cost of the transformation itself.

Lev: And for AI systems interacting with quantum hardware, this means we can build regularization techniques that minimize this transport cost to ensure the evolution stays physically constrained throughout the layers.

Kai: That ties back nicely to how we might use these concepts in developing generative models that aim to produce complex entangled states efficiently, minimizing the energy expended during that process.

Mira: We should keep an eye on those unitary invariant costs they studied; having a cost function stable under global rotations would be a huge win for creating generalizable quantum algorithms.

Lev: I think the most tangible impact right now is in developing more robust quantum machine learning algorithms that are inherently resilient to basis changes because they optimize against these invariant measures.

Kai: So, in summary, the paper lays out a novel way to define optimal transport cost using states over time, provides mathematical conditions for useful cost matrices, and explores the behavior of this cost in high-dimensional limits.

Mira: It’s a significant theoretical contribution because it formalizes how to integrate quantum channels into the classical structure of optimal transport theory without losing that essential bilinear relationship between state and plan.

Lev: My main thought is that if we can use this framework, error correction could transition from just checking if a state is close to target to calculating the actual minimum physical resources needed for the transition.

Kai: It’s fascinating work because it takes something very classical—optimal transport—and forces us to confront the specific constraints and structure of quantum mechanics in a rigorous way.

Mira: That rigor, combined with the explicit formulation of cost matrices like K zero gives us a much clearer roadmap for building new quantum information theories <ref:2504.04856#pg1>.

Lev: We have to see how these abstract concepts translate into something that can be simulated or implemented reliably on current noisy hardware, which is where the real test lies.

Kai: That’s what we'll keep an eye on as we look at the next steps in applying this work to actual quantum experiments and building systems.

The paper's summary: Kai: So, this paper is essentially setting up a way to calculate the most efficient way to move a quantum state from one place to another by defining what we call "states over time."

Mira: Exactly, Kai; they’re taking the classical idea of optimal transport and forcing it into a quantum structure where the cost depends on both the initial state and the actual channel used for transport.

Lev: From my side, I'm curious how this cost functional translates into something we can actually measure or simulate on current hardware; if it’s just a theoretical construct, it doesn't help me design better error correction protocols.

Kai: That’s the million-dollar question, Lev; the paper defines the cost as bilinear in those two elements, which is what makes it look like classical optimal transport but with quantum ingredients.

Mira: It achieves that by defining a "state over time" using a Jordan product, which mathematically links the initial density matrix with a completely positive map into one unified object. This formalism allows them to define the cost kappa in terms of this state over time, making it look like a standard optimization problem.

Lev: But what about the complexity of defining that set of "states over time," Q(rho, sigma) ? If characterizing all those maps is computationally hard, then finding the minimum cost K(rho, sigma) will be just as difficult to solve as classical optimal transport on a massive scale.

Kai: That’s where the paper focuses on the properties of the cost matrix K, showing how conditions like positivity and a triangle inequality give us constraints we need for it to represent a real physical effort.

Mira: The investigation into unitary invariant costs is particularly interesting because it identifies a very specific class of matrices, positive multiples of K zero = d one - S, that behave well under global rotations, which speaks to the stability you need in any quantum operation.

Lev: If we can find a way to constrain our error correction steps using these invariant costs, it would give us a much more robust framework for designing algorithms that don't break when the underlying basis is slightly rotated during execution.

Kai: And looking at their limit as the Hilbert space dimension grows, they connect this quantum cost directly to entanglement fidelity and classical total variation distance for commuting states, which suggests a deep structural relationship between quantum transport and simpler statistical measures in large systems.

Mira: That connection is telling; it implies that for highly complex states, the physical effort of transport starts mirroring how different those states are statistically, which is a profound statement about quantum correlations.

Lev: If the cost simplifies to something related to total variation distance in this limit, then maybe we can use classical optimization techniques to approximate these very difficult quantum transport problems when dealing with systems that are effectively "large."

Kai: It’s not about replacing the math; it’s about understanding what physical resources—like entanglement or coherence—are actually being consumed during a state transfer, which is what I need to measure in the lab.

Mira: The implications are huge for designing new quantum machine learning models; if we can define regularization based on this transport cost, we can ensure the model evolves in a physically meaningful way that respects the constraints of quantum mechanics.

Lev: That would be incredibly useful for developing those quantum generative models you mentioned earlier, allowing us to produce complex entangled states with a guaranteed minimum energy expenditure during creation.

Kai: So, we're moving toward a framework where we can quantify the "effort" in quantum computation itself, rather than just measuring the final state fidelity.

Mira: Precisely; they’ve provided the mathematical machinery to make that quantification explicit and grounded in physical constraints from the start.

Lev: I'm optimistic this moves us closer to designing error correction schemes where we don't just check for errors but actively minimize the transport cost required to fix them.

The paper's improvements: Kai: So, after laying out the core transport cost, they propose several ways to sharpen this framework for real application by focusing on specific properties of that cost matrix K.

Mira: Right, they suggest investigating conditions like requiring the swap operator to have a zero trace involving S for the identity channel; that’s a way to impose structural consistency on what we consider an admissible transport plan.

Lev: From an error correction standpoint, I think focusing on those structural constraints is useful because it gives us specific mathematical boundaries to work within when designing recovery maps.

Kai: They also highlight the requirement for positivity of K, meaning that the resulting transport cost must always be non-negative for any valid state transformation between two points.

Mira: That positivity condition ties the cost matrix directly into the dual cone of states over time, which is a fundamental structure in quantum information theory and helps us understand what transformations are physically possible.

Lev: If we can use that dual cone to prune our search space for optimal transport plans, it could actually make finding solutions tractable on hardware that has limited computational power.

Kai: Then they discuss the triangle inequality, which provides a partial order on cost matrices, essentially telling us when one transport path is inherently preferred over another in terms of resource consumption.

Mira: That partial order is key because it helps define a notion of "better" transport plan beyond just achieving the target state; it’s about minimizing some underlying physical resource that we're trying to formalize.

Lev: Having that partial order would be helpful if we want to build hierarchical error correction protocols where we can decide which type of transformation is most efficient for a given level of noise.

Kai: And they zero in on unitary invariant costs, which means the cost doesn't change even if you apply a global rotation to your input and output states; that’s a huge win for building stable quantum algorithms.

Mira: That invariance is important because it suggests we can build metrics that are robust against basis changes, which is what we need for generalizable models in any physical realization of quantum computation.

Lev: If the cost function remains invariant under those unitaries, it means our error correction logic won't be overly dependent on the specific qubit basis chosen during the process.

Kai: The paper also looks at what happens when you take the limit as the Hilbert space dimension gets really large; they show that this quantum transport cost starts behaving like a classical distance measure related to entanglement fidelity.

Mira: That link between high-dimensional quantum transport and classical total variation distance is a very telling result, suggesting a deep connection between quantum information and statistical mechanics in the thermodynamic limit.

Lev: If we can use that asymptotic behavior to guide our design of large-scale algorithms, it gives us a way to use classical tools to estimate the resource costs of massive quantum operations.

Kai: So these improvements seem aimed at making the theoretical framework more concrete by providing actionable constraints on the cost matrices themselves.

Mira: The overall implication is that we are moving toward defining physical transport not just as a sequence of steps, but as an optimization problem constrained by the fundamental geometry of quantum states.

Lev: This gives us a much stronger tool for designing fault-tolerant systems because we aren't just minimizing distance; we're minimizing the actual physical cost required to achieve that distance.

Conclusion: Kai: So, to wrap up our discussion on "Approach to optimal quantum transport via states over time," this paper essentially lays down a new mathematical language for defining how we measure physical effort in quantum state evolution.

Mira: It’s true; they’ve successfully integrated the concepts of classical optimal transport with the specific constraints of CPTP maps using this new "state over time" formalism, which is quite elegant.

Lev: For us in error correction, it means we have a way to quantify the minimum physical resource expenditure required for a state transfer between two points, which is something we haven't had before.

Kai: Exactly; it’s about moving beyond just checking if a final state is close to the target and instead calculating the actual cost of getting there, which I can see as a huge step toward experimental validation.

Mira: And their work on unitary invariant costs gives us specific, stable metrics that are independent of arbitrary basis choices, which is crucial for any practical quantum algorithm design.

Lev: That invariance is exactly what we need to ensure our error correction logic remains sound regardless of how the physical qubits are mapped or rotated during operation.

Kai: It really shows how foundational mathematical structures can guide the experimental setup, even when you’re dealing with complex density matrices and channels.

Mira: The implication is that we can start designing quantum machine learning models where we minimize a physically meaningful cost function rather than just relying on abstract overlap measures.

Lev: And if we can do that, it could lead to much more efficient quantum generative models capable of producing high-fidelity entangled states with minimal energy input.

Kai: It’s exciting because this gives us a rigorous foundation to start testing these ideas on actual quantum hardware, which is where the real validation happens.

Mira: This formalization of transport cost, as presented in "Approach to optimal quantum transport via states over time," sets a clear path for how we should think about resource allocation in quantum systems.

Lev: Moving forward, I see this framework leading directly into designing new regularization techniques for neural networks that are inherently constrained by physical laws rather than just statistical assumptions.

Kai: Next week, we'll be looking at how these cost concepts might apply to the compilation of quantum circuit states, which is a fascinating area that ties directly into what they've shown here.

Matt Hoogsteder-Riera, John Calsamiglia, Andreas Winter

Grup d’Informació Quàntica, Departament de Física, Universitat Autònoma de Barcelona · ICREA—Institució Catalana de Recerca i Estudis Avançats · Department Mathematik/Informatik—Abteilung Informatik, Universität zu Köln · Institute for Advanced Study, Technische Universität München

quant-ph, math-ph, math.MP

Submitted: 2025-04-07

Updated: 2026-10-05

Comments: 40 pages, 4 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 54/100

The gist: The paper approaches quantum optimal transport by generalizing classical Monge's theory to quantum systems, defining a "state over time" as a Jordan product that integrates an initial density matrix

Key concepts

State Over Time (stote): This is a mathematical construct combining an initial quantum state ($ ho$) and a quantum channel ($J$) using a Jordan product. It serves as the fundamental object that integrates the starting point and the transport process, analogous to coupling in classical transport.
Quantum Optimal Transport Cost: This cost measures the minimum 'cost' of moving from an initial state ($ ho$) to a final state ($ ho o ext{E}( ho)$) through a specific quantum channel (E). It is defined as the minimum value of a bilinear functional involving the state over time and the cost matrix.
Unitary Invariant Cost: This refers to transport costs that remain unchanged even if both the initial and final states are transformed by arbitrary unitary matrices. The paper finds that only certain specific cost matrices, related to $K_0 = d_1 - S$, satisfy this property in the context of states over time.
Limit Behavior ($d o ext{∞}$): As the dimension of the Hilbert space becomes very large, the quantum transport cost converges to a classical cost measure related to entanglement fidelity. For commuting states, this limit simplifies to a classical transport problem based on total variation distance.

Terminology

Optimal Transport — 53× in this paper · explained in 5 episodes
Hilbert Space — 35× in this paper · explained in 2 episodes
Trace — 29× in this paper · explained in 2 episodes
Fidelity — 10× in this paper · explained in 2 episodes
Entanglement — 9× in this paper · explained in 4 episodes
Consistency — 3× in this paper · explained in 3 episodes
Core — 3× in this paper · explained in 2 episodes
Generative Model — 3× in this paper · explained in 2 episodes
Generative Models — 3× in this paper · explained in 3 episodes
Physical Constraints — 3× in this paper · explained in 7 episodes
Regularization — 3× in this paper · explained in 3 episodes
Additivity — 2× in this paper · explained in 1 episode
Matrix Completion — 2× in this paper · explained in 4 episodes
Wasserstein Distance — 2× in this paper · explained in 4 episodes
Admissibility — 1× in this paper · explained in 2 episodes
Aharonov-Bohm Effect — 1× in this paper · explained in 2 episodes
Axioms — 1× in this paper · explained in 2 episodes
Bayesian Inference — 1× in this paper · explained in 5 episodes
Fisher Information — 1× in this paper · explained in 3 episodes
Generalization — 1× in this paper · explained in 13 episodes
Gradient Flow — 1× in this paper · explained in 2 episodes
Hamming Distance — 1× in this paper · explained in 2 episodes
Objective Function — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 5 episodes
Semidefinite Programming (SDP) — 1× in this paper · explained in 3 episodes
Structural Constraints — 1× in this paper · explained in 2 episodes
Temporal Correlations — 1× in this paper · explained in 2 episodes
Trace Distance — 1× in this paper · explained in 3 episodes

Summary

The paper approaches quantum optimal transport by generalizing classical Monge's theory to quantum systems, defining a state over time as a Jordan product that integrates an initial density matrix and a completely positive map (quantum channel), thereby establishing a cost functional bilinear in these two elements. This framework is significant because it allows for the study of quantum transport costs that are qualitatively different from classical ones, moving beyond simple distinguishability measures to incorporate underlying spatial properties of the Hilbert space.

Motivation and Formulation

The work seeks to construct a quantum analogue of classical optimal transport, where the mass distribution is generalized to a density matrix and the transport plan is a completely positive and trace-preserving map. The core idea is that these two data are naturally integrated into their Jordan product, called state over time (stote), and the transport cost is postulated to be a linear function of it. This approach attempts to preserve the feature of classical optimal transport where the cost is bilinear in the initial density and the conditional probability.

Mathematical Preliminaries: States Over Time

The mathematical foundation relies on Choi and Jamiołkowski isomorphisms, which relate completely positive (CPTP) maps to bipartite states. The state over time is formally defined as a Jordan product: Q = (ρ ⊗ 1) ⋆ J, where ρ is the initial state and J is the Jamiołkowski matrix of a quantum channel. This definition has the interpretation of a classical probability coupling, where marginals are the initial and final states, and the map connecting them can be reconstructed from Q. The set of states over time between two states ρ and σ is defined as Q(ρ, σ) = ρ ⋆ J J ∈ J (HA → HB) and TrA [ρJ] = σ, specifying the channels that map ρ to σ.

Quantum Optimal Transport Cost

The quantum transport cost of a state through a channel E with cost matrix K is defined as κ(ρ, E) = Tr [KQρ,E], where Qρ,E is the state over time associated with sending ρ through the channel E. The quantum optimal transport cost between two states ρ and σ is then defined as: K(ρ, σ) = min E Qρ,E∈Q(ρ,σ) κ(ρ, E). This cost explicitly depends on the choice of the Hermitian cost matrix K.

Properties of Cost Matrices

The paper investigates sufficient conditions on the cost matrix K for it to induce desirable properties in the quantum optimal transport cost. Key properties studied include:

Cost 0 for identity channel: A condition is TrB [S ⋆ K] = 0, where S is the swap operator.
Positivity: The requirement that K(ρ, σ) ≥ 0 for all ρ, σ ∈ S(HA, HB), which relates K to the dual cone of states over time.
Triangle Inequality: This is fulfilled if KAB ⊗ 1C + 1A ⊗ KBC − KAC ⊗ 1B ∈ Q(HA: HB: HC)∗, where Q(HA: HB: HC) is the set of states over multiple times, and the dual cone provides a partial order for cost matrices.

Unitary Invariant Cost

The paper focuses on unitary invariant costs, defined by K(ρ, σ) = K(U ρU∗, UσU∗) ∀U ∈ U(d), which is equivalent to the cost matrix being invariant under simultaneous unitary transformations of both input and output states. The only cost matrices belonging to the dual cone of states over time that assign zero cost to the identity channel and commute with unitaries of the form U ⊗ U are positive multiples of K0 = d1 − S. For commuting states, the optimal quantum transport cost is analytically calculated as: K(ρ, σ) = 1/d (d − X ij pi q min[1, qi pi] min[1, qj pj]) in a common diagonal basis.

Limit Behavior and Asymmetry

The study also explores the limit as the ambient Hilbert space dimension d → ∞. The cost in this limit is shown to be K∞(ρ, σ) = 1 − max E: E(ρ)=σ F(ρ, E), where F(ρ, E) is the entanglement fidelity of state ρ under channel E. Furthermore, for commuting states in the limit, the cost simplifies to K∞(ρ, σ) = 1 − X i pi q p(ii), which corresponds to a classical transport problem related to total variation distance. The paper notes that a symmetry gap exists between K(ρ, σ) and K(σ, ρ), and this gap can be discontinuous when the channel maps a non-pure state to a pure state.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements that could be made to Artificial Intelligence (AI) systems, categorized by the aspect of AI they would enhance:

) Can improve AI systems by integrating Quantum Optimal Transport (QOT).

Improving State-to-State Transfer and Mapping:

Find an optimal quantum transport plan between two density matrices (initial state and target state). This is analogous to classical optimal transport, but the cost is defined using the Jordan product of the initial density matrix with a Completely Positive Trace Preserving (CPTP) map.

Developing Quantum-Aware Distinguishability Metrics:

Define metrics for distinguishing quantum states based on energy or information costs (like Hamming distance or entanglement measures) that are explicitly bilinear in both the state and the transport plan, rather than relying solely on standard measures like Total Variation distance.

Designing Unitary Invariant Cost Functions for Robust AI:

Implement cost functions that remain unchanged under global unitary transformations of the input and output systems (Unitary Invariant Cost). This is crucial for creating AI models whose performance metrics are robust against changes in the chosen basis or local rotations, leading to more stable and generalizable learning algorithms.

Creating Quantum-Aware Generative Models:

Develop generative models that learn transport plans (quantum channels) that transform one quantum state into another while minimizing a physically meaningful cost function (e.g., minimizing the energy required for a transformation).

Enhancing Model Interpretability via Cost Analysis:

Use the structure of the optimal transport cost to gain insights into how different quantum operations affect correlations between subsystems, allowing researchers to map specific computational steps to physical resources like entanglement or coherence.

) An improved AI system could perform the following:

Perform high-fidelity state preparation or state transfer in quantum hardware by calculating the minimum required cost (energy/effort) associated with applying a specific quantum channel between two states.
Develop new regularization techniques for neural networks by using state over time formalisms to minimize the cost of evolving an initial state through a sequence of layers, ensuring the evolution respects physical constraints (like CPTP maps).
Design quantum machine learning algorithms that are inherently robust to basis rotations and local transformations by optimizing their parameters based on unitary invariant cost matrices, leading to more stable and generalizable models across different computational bases.
Build advanced quantum classifiers or state discrimination algorithms where the decision boundary is defined not just by state overlap (like fidelity), but by a transport cost that reflects the physical effort required to transform one class of states into another.
Create quantum generative models capable of producing complex, high-fidelity entangled states from simple initial states while minimizing the energy expenditure during the process.

Abstract

We approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where the mass distribution is generalised to a density matrix, and the transport plan to a completely positive and trace preserving map. % These two data are naturally integrated into their Jordan product, which is called state over time (``stote''), and the transport cost is postulated to be a linear function of it. We explore the properties of this transport cost, as well as the optimal transport cost between two given states (simply the minimum cost over all suitable transport plans). After that, we analyse in considerable detail the case of unitary invariant cost, for which we can calculate many costs analytically. These findings suggest that our quantum transport cost is qualitatively different from Monge's classical transport.