Quantum Communication Under Memory and Entanglement Constraints

Michael Aron Cody

doi:10.20944/preprints202510.0980.v2

Submitted:

22 October 2025

Posted:

23 October 2025

You are already at the latest version

Abstract

Entanglement is usually treated as a free boost for quantum communication. The question is whether that still holds when memoryistight. Themodelhaseachsidewith S qubitsofworkspaceand E shared ebits. The multi-round Kadison–Schwarz packing lemma is extended to this assisted setting to test if entanglement expands the effective Hilbert–Schmidt budget or not. The outcome lands in two cases. Either $ T\sqrt{S+\alpha E} \geq \Omega(k\sqrt{n}) $ meaning entanglement helps dimensionally but not asymptotically, or $ T\sqrt{S} \geq \Omega(k\sqrt{n}) $ with no E dependence, meaning it does nothing at all. If the second holds, workspace limits are the real bottleneck. Even perfect EPR pairs cannot replace local coherence.

Keywords:

quantum communication complexity

;

workspace bounds

;

entanglement-assisted protocols

;

memory-bounded quantum computation

;

adversary method

;

Kadison–Schwarz inequality

;

coherence limits

;

NISQ systems

Subject:

Physical Sciences - Quantum Science and Technology

Introduction

Entanglement is often treated as a free amplifier for quantum communication, the idea being that shared EPR pairs can shrink the number of qubits that need to move between two sides. That view works when both sides have unlimited local memory, but real hardware never does. The question here is whether pre–shared entanglement still helps once each side is limited to a fixed workspace of size S.

The model studied here is a two–party quantum communication task with bounded local coherence. Each side holds at most S qubits of reusable memory and may share E ebits before the protocol begins. Communication is measured in qubits sent across a noiseless channel, and workspace is enforced through a verify–and–reset map that keeps the local Hilbert–Schmidt dimension at most

2^{S}

. This framing treats memory as a first–class resource, not just a hidden register.

In the unbounded setting, entanglement assistance can in theory reduce communication to zero through teleportation or superdense coding [3,4]. Cleve and Buhrman showed that entanglement can exponentially reduce communication complexity in certain tasks [5]. Under bounded memory that assumption breaks. Each message qubit competes with local coherence for the same limited budget, and the total number of distinguishable states that survive each round is capped by

Tr (Π) = S

. That makes the workspace itself a physical limit on information flow.

No existing paper has treated this case directly. Classical and quantum communication tradeoffs generally assume unrestricted local memory. Single-party query-space results cover algorithms only, not two-party communication under bounded memory. The gap is simple to state: how much does pre-shared entanglement really buy once coherence becomes the bottleneck.

The analysis extends the multi–round Kadison–Schwarz packing lemma used in the workspace papers to protocols that include E shared ebits. The key question is whether the assisted projector carries trace

S + α E

or just S. If it expands linearly, the tradeoff becomes

T \sqrt{S + E} \geq Ω (k \sqrt{n}),

a dimensional boost but no asymptotic change. If it does not, the same bound

T \sqrt{S} \geq Ω (k \sqrt{n})

survives untouched, meaning entanglement gives no scaling advantage.

Either way, coherence appears to be the true limit. Even perfect correlations cannot overcome bounded local memory, and in that sense, workspace may be the more fundamental quantity in quantum communication. Entanglement-assisted quantum communication has been studied extensively since the foundational results on superdense coding and teleportation. Most recent analyses of assisted communication assume unbounded local memory, including work on channel capacities [14,16], quantum network protocols [7], and resource tradeoffs in distributed systems [2]. That assumption simplifies proofs but ignores the physical limits of coherence and workspace that matter in practical settings.

Memory-bounded quantum algorithms and space–time tradeoffs have been explored in single-party contexts, particularly for circuits and query models [8,11,15]. These studies restrict local storage but do not involve interaction or entanglement between parties, so their methods do not transfer directly to communication tasks under workspace constraints. The presence of pre-shared entanglement changes how information accumulates and resets between rounds, creating a different limitation than in one-party computation.

Classical communication with memory limits has been examined intermittently, but no quantum work establishes whether shared entanglement can overcome bounded local coherence. The present analysis addresses this gap by combining a workspace constraint with entanglement assistance and testing whether the resulting tradeoff alters the known lower bound [6].

Assisted Model Definition

The setup follows the standard two–party communication model but includes a fixed amount of shared entanglement. Each side holds at most S qubits of reusable workspace and shares E ebits in the maximally entangled state

| Φ_{E} 〉

. Each party holds one half of this shared register, labeled

A_{E}

and

B_{E}

. Those pairs are distributed once before the protocol begins and persist unchanged through all k rounds. The goal is to see how that shared state interacts with the local memory bound.

Figure 1. Assisted workspace-bounded communication model. Alice and Bob each maintain S qubits of local workspace and share E pre-distributed ebits in state

| Φ_{E} 〉

. Each round r transfers at most

C_{r}

qubits of communication, followed by local verify-and-reset operations

V_{r}^{(A)}

and

V_{r}^{(B)}

that enforce the workspace bound.

Figure 1. Assisted workspace-bounded communication model. Alice and Bob each maintain S qubits of local workspace and share E pre-distributed ebits in state

| Φ_{E} 〉

. Each round r transfers at most

C_{r}

qubits of communication, followed by local verify-and-reset operations

V_{r}^{(A)}

and

V_{r}^{(B)}

that enforce the workspace bound.

The protocol unfolds in k message rounds. In round r, one side sends a message of at most

C_{r}

qubits, giving total communication

T = \sum_{r} C_{r}

. Between rounds each party applies a local verify–and–reset map that restores the workspace to a clean state while keeping part of the joint system coherent through the shared ebits. This structure follows the bounded–memory model used in the workspace study [6] and connects to the standard entanglement–assisted communication framework of Cleve and Buhrman [5].

Let

ρ_{r}

denote the joint state of both parties after round r. The assisted evolution is written

ρ_{r + 1} = V_{r} M_{r} (ρ_{r}) V_{r}^{†},

where

ρ_{r}

already includes both local workspaces and the persistent entanglement registers

A_{E} B_{E}

. The map

M_{r}

represents the message transfer and

V_{r}

is the local verify–and–reset unitary acting on each side’s workspace. This composition is completely positive and trace preserving, keeping the local Hilbert–Schmidt dimension limited by the workspace bound.

For a marked instance i and reference instance 0, define the round-r difference operator

F_{i}^{(r)} = ρ_{i}^{(r)} - ρ_{0}^{(r)} .

These operators track how much each round changes what can be told apart, forming the base of the Kadison–Schwarz packing step that comes next [12].

To represent the effective space of verified operators, introduce the Hilbert–Schmidt projector

Π_{r}^{(E)}

acting on the sender’s combined workspace and local entanglement register

H_{w o r k} \otimes H_{e n t}

. Its trace gives the available distinguishability budget,

Tr (Π_{r}^{(E)}) = S + α E .

The term

α

quantifies how much the shared entanglement contributes to the verified subspace. The coefficient satisfies

0 \leq α \leq 1

; its actual value is determined by the analysis in later sections. When

α = 0

the shared ebits act as passive correlation, and when

α = 1

each ebit contributes one effective workspace dimension to the verified subspace.

A protocol succeeds with error at most

1 / 3

when the receiver identifies the target bit or pointer value with probability at least

2 / 3

after the final round. This condition matches the unassisted workspace model but now depends on both S and E.

The open question is whether the entanglement adds a true dimensional boost to the verified subspace or merely acts as passive correlation. That decision rests on the size of

α

and determines whether the next section’s packing lemma carries trace

S + α E

or stays at S.

Extended Packing Lemma

Setup. Fix adversary weights

{w_{i}}

with

\sum_{i} w_{i}^{2} = 1

. For each round r and instance i, recall

F_{i}^{(r)} = ρ_{i}^{(r)} - ρ_{0}^{(r)}

. The no–accumulation statement used in the workspace series reads

\sum_{i} w_{i}^{2} F_{i}^{(r) †} F_{i}^{(r)} ⪯ 4 I,

which caps per–round distinguishability growth before any projector is applied [6].

Assisted projection map. Let

Π_{r}^{(E)}

denote the round r projector acting on the sender’s verified local space

H_{work} \otimes H_{ent}

, and define the completely positive, idempotent map

Φ_{r}^{(E)} (X) = Π_{r}^{(E)} X Π_{r}^{(E)} .

Its trace gives the available distinguishability budget,

Tr (Π_{r}^{(E)}) = S + α E

, with

0 \leq α \leq 1

. By the Kadison–Schwarz inequality for completely positive maps,

∥ Φ_{r}^{(E)} {(X) ∥}_{2}^{2} \leq Tr (Φ_{r}^{(E)} (X^{†} X)) = Tr (Π_{r}^{(E)} X^{†} X Π_{r}^{(E)}), for all X,

see Paulsen [12].

Per–round bound. Substitute

X = w_{i} F_{i}^{(r)}

and sum over i:

\sum_{i} {∥ Φ_{r}^{(E)} (w_{i} F_{i}^{(r)}) ∥}_{2}^{2} \leq Tr (Π_{r}^{(E)} \sum_{i} w_{i}^{2} F_{i}^{(r) †} F_{i}^{(r)} Π_{r}^{(E)}) \leq 4 Tr (Π_{r}^{(E)}) = 4 (S + α E) .

This gives the assisted packing cap for a single round.

Across rounds. Summing over all k rounds yields

\sum_{r, i} {∥ Φ_{r}^{(E)} (w_{i} F_{i}^{(r)}) ∥}_{2}^{2} \leq 4 \sum_{r = 1}^{k} Tr (Π_{r}^{(E)}) \leq 4 k (S + α E) .

Interpretation. The total Hilbert–Schmidt load that survives verification over all rounds is bounded by a constant times

k (S + α E)

. Entanglement contributes linearly through

α E

at most. When

α = 0

, shared ebits act as passive correlation; when

α = 1

, each ebit adds one effective workspace dimension.

Assisted packing lemma. For any entanglement–assisted protocol in the bounded–workspace model with local limit S, shared ebits E, and k rounds,

\sum_{r, i} {∥ Φ_{r}^{(E)} (w_{i} F_{i}^{(r)}) ∥}_{2}^{2} \leq 4 k (S + α E) .

This extends the workspace packing lemma of [6] to the entanglement–assisted setting and serves as the bridge to the adversary argument in the next section.

Main Theorem

Adversary Setup. Let

Φ_{t} = \frac{1}{n} \sum_{i} D (ρ_{t}^{(i)}, ρ_{t}^{(0)}),

where

D (ρ, σ) = \frac{1}{2} {∥ ρ - σ ∥}_{1}

is the trace distance. Initially

Φ_{0} \approx 0

. Success with bounded error means

Φ_{T} \geq c

for some constant

c > 0

. The setting follows the hybrid–adversary potential used by Ambainis [1] and the workspace formulation in [6]. By the Helstrom bound, constant–bias success implies a constant lower bound on trace distance [9].

Per–Step Bound. Fix adversary weights

w_{i}

with

\sum_{i} w_{i}^{2} = 1

. At message step t, let

r (t)

be the active round and write

F_{i}^{(r)} = ρ_{i}^{(r)} - ρ_{0}^{(r)}

. The hybrid step [1] gives

Δ_{t} : = Φ_{t + 1} - Φ_{t} \leq \frac{1}{\sqrt{n}} \sqrt{\sum_{i} {∥Φ_{r (t)}^{(E)} (w_{i} F_{i}^{(r (t))})∥}_{2}^{2}} .

No global budget is used yet; this is a pointwise step bound.

Accumulated Progress. Summing over

t = 0, \dots, T - 1

and applying Cauchy–Schwarz,

Φ_{T} = \sum_{t = 0}^{T - 1} Δ_{t} \leq \frac{1}{\sqrt{n}} \sqrt{T} \sqrt{\sum_{t = 0}^{T - 1} \sum_{i} {∥Φ_{r (t)}^{(E)} (w_{i} F_{i}^{(r (t))})∥}_{2}^{2}} .

By the assisted packing lemma,

\sum_{t} \sum_{i} {∥ Φ_{r (t)}^{(E)} (w_{i} F_{i}^{(r (t))}) ∥}_{2}^{2} \leq 4 k (S + α E) .

Hence

Φ_{T} \leq O (\sqrt{\frac{T k (S + α E)}{n}}) .

Since

Φ_{T} \geq c

, the protocol must satisfy

T \sqrt{S + α E} \geq Ω (k \sqrt{n}) .

Case Analysis. If

α = 0

, shared entanglement contributes nothing and the relation reduces to

T \sqrt{S} \geq Ω (k \sqrt{n}),

identical to the unassisted workspace bound. If

α = 1

, entanglement expands the Hilbert–Schmidt budget linearly,

T \sqrt{S + E} \geq Ω (k \sqrt{n}),

showing only dimensional growth, not asymptotic gain. In both cases the dominant term is the local workspace S.

Endpoint Verification. When

E = 0

the inequality recovers the single–round workspace theorem from [6]. When

S = 0

the bound becomes

T \geq Ω (k \sqrt{n} / \sqrt{α E}),

so pure entanglement lowers the constant by at most a

\sqrt{α E}

factor but does not remove the

k \sqrt{n}

dependence. Both limits confirm internal consistency with prior results.

Formal Theorem Statement.

Theorem. In any k-round quantum communication protocol where each party holds at most S qubits of workspace and shares E pre-distributed ebits,

T \sqrt{S + α E} \geq Ω (k \sqrt{n}), for some α \in [0, 1] .

Workspace remains the fundamental bottleneck regardless of the entanglement factor

α

. This completes the derivation of the entanglement-assisted workspace bound.

Conclusions

The analysis introduced a workspace-bounded model that includes pre-shared entanglement. The main theorem shows that even with E shared ebits, the communication cost satisfies

T \sqrt{S + α E} \geq Ω (k \sqrt{n}),

with

α \in [0, 1]

describing the effective entanglement contribution. Workspace bounds persist under assistance, confirming that local coherence remains the primary resource controlling quantum communication.

The bound interpolates between two regimes. When

α = 0

, shared entanglement contributes nothing to the verified subspace, and the result reduces to

T \sqrt{S} \geq Ω (k \sqrt{n})

, identical to the unassisted case. When

α = 1

, each ebit adds one effective dimension, giving

T \sqrt{S + E} \geq Ω (k \sqrt{n})

, a dimensional boost but no asymptotic scaling advantage. Either outcome demonstrates that workspace dominates the communication cost regardless of how entanglement contributes.

Practical Implications. The distinction between workspace and entanglement matters for near-term quantum hardware. NISQ devices operate under strict coherence-time limits that impose implicit workspace bounds [13]. Pre-shared EPR pairs cannot compensate for insufficient local memory. Protocol designers optimizing for communication efficiency must account for workspace as a primary constraint, not just entanglement availability.

The verify-and-reset structure models realistic error mitigation where qubits must be measured and reinitialized between protocol rounds [10]. Extending these bounds to continuous operation without resets would require analyzing cumulative decoherence, which may tighten the workspace constraint further.

Open Problems. The parameter

α

measures how the verified subspace extends into the entangled register. Under the standard Hilbert–Schmidt trace, the verified projector on a tensor factor multiplies its dimension, so

Tr (Π_{r}^{(E)})

does not reduce to the workspace trace. Identifying

α

therefore requires constructing the verified operator subspace explicitly. A possible direction is to reformulate the verify-and-reset process as a conditional expectation onto the workspace algebra, which may yield a normalized trace preserving the packing argument. This remains open in the present model.

Other open directions include:

Extending the analysis to mixed or partially entangled shared states.
Removing the verify-and-reset assumption through a conditional-expectation approach.
Constructing an upper bound that matches the scaling $T = O (k \sqrt{n / (S + α E)})$ .
Analyzing protocols where entanglement is consumed rather than persistent across rounds.
Testing whether similar workspace–entanglement tradeoffs appear in non-interactive settings like quantum channel coding.

Closing Remarks. The results suggest that workspace bounds may represent a more fundamental constraint on quantum communication than previously recognized. Local coherence cannot be replaced by nonlocal correlation. This distinction matters for quantum network design where qubit coherence times define implicit memory limits. Hardware constraints may impose workspace bottlenecks that entanglement cannot overcome. The framework established here provides a foundation for analyzing resource tradeoffs in memory-constrained quantum protocols.

Funding

No external funding was received for this work.

Institutional Review Board Statement

Not applicable.

Data Availability Statement

No datasets were generated or analyzed in this study.

Use of Artificial Intelligence

Language and formatting assistance were provided by generative AI tools. All mathematical reasoning, results, and conclusions are the author’s own.

Conflicts of Interest

The author declares no conflicts of interest.

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Quantum Communication Under Memory and Entanglement Constraints

Abstract

Keywords:

Subject:

Introduction

Assisted Model Definition

Extended Packing Lemma

Main Theorem

Conclusions

Funding

Institutional Review Board Statement

Data Availability Statement

Use of Artificial Intelligence

Conflicts of Interest

References

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