Nayak Bound in Quantum Information

• Nayak bound is an information-theoretic limit, establishing that an (n,m,p)-random access encoding must satisfy m ≥ (1-H_bin(p))n, ensuring a baseline entropy tradeoff.
• It reformulates QRAC optimization as a spectral problem for noncommuting Hermitian operators, yielding refined finite-size bounds and deeper insights into decoding probabilities.
• Beyond encodings, the bound informs one-way quantum communication and QPIR lower bounds, as well as frameworks like the Jain–Klauck–Nayak method in two-way communication.

The expression Nayak bound most commonly denotes the information-theoretic lower bound for random access encodings and, in binary QRAC notation, the corresponding entropy upper bound on decoding probability. In the form used for random access encodings, any $(n,m,p)$-random access encoding satisfies

$m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$

while in the QRAC setting the same tradeoff is written as

$$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$

The term is nevertheless context-sensitive: it also appears in one-way quantum communication via a coding-based upper bound implicit in Nayak’s thesis, in two-way communication via the Jain–Klauck–Nayak subdistribution framework, in rough-surface contact mechanics through the Nayak parameter, and in a quantum-walk conjecture coauthored by A. Nayak that is unrelated to the standard lower bound (Baumeler et al., 2013, Akibue et al., 10 Jun 2026).

1. Random access encodings and the standard information-theoretic form

An $(n,m,p)$-random access encoding is a map

$$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$

such that for every $i\in\{1,\dots,n\}$, there exists a measurement $M_i$ with outcomes $0,1$ satisfying

$$\Pr\left[ M_i\left( f\left( x \right) \right)=x_i \right] \ge p$$

on average over all $x\in\{0,1\}^n$. The standard Nayak lower bound used in this setting is

$m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$0

In the QPIR literature this appears as a theorem on the size of random access encodings, and it functions as the core information-theoretic obstruction behind one-server lower bounds (Baumeler et al., 2013).

The operational meaning of the inequality is that compressing $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$1 classical bits into $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$2 qubits while retaining uniform recoverability of any designated bit with success probability $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$3 forces $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$4 to remain linear in $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$5 unless $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$6 is close to $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$7. This is the form most directly associated with the phrase “the Nayak bound” in quantum information and communication complexity.

A closely related formulation arises for binary QRACs. There one studies an $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$8-QRAC in which Alice receives a uniformly random string $m \ge \left(1-H_{\mathrm{bin}}(p)\right)n,$9, encodes it into an $$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$0-qubit state $$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$1, and Bob attempts to recover coordinate $$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$2 using a binary POVM. The standard entropy constraint is

$$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$3

equivalently

$$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$4

This QRAC form is the one explicitly treated as the “standard general upper bound on the decoding probability” in recent measurement-geometric work (Akibue et al., 10 Jun 2026).

2. QRAC formulation, spectral methods, and finite-size refinements

Recent work reformulates binary QRAC optimization in terms of the decoding measurements rather than the encoding states. For binary observables

$$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$5

define the signed sum

$$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$6

For fixed decoding observables, the optimal average decoding probability is

$$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$7

so the QRAC design problem becomes a spectral problem for noncommuting Hermitian operators. This perspective yields an elementary proof of the Nayak bound via a Chernoff/Laplace-transform argument applied to $$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$8, and it also produces finite-size upper bounds that are stated to be strictly sharper than Nayak’s bound for every finite $$M \ge (1-H_2(p))N \qquad\Longleftrightarrow\qquad p \le H_2^{-1}\!\left(1-\frac{M}{N}\right).$$9 (Akibue et al., 10 Jun 2026).

The moment comparison bound in that analysis is

$(n,m,p)$0

for an analytic $(n,m,p)$1 with nonnegative even Taylor coefficients, applied to

$(n,m,p)$2

Choosing $(n,m,p)$3 recovers the standard entropy bound. Optimizing over a larger class of functions yields the refined finite-size estimate

$(n,m,p)$4

Several explicit specializations are recorded. For $(n,m,p)$5,

$(n,m,p)$6

which reproduces the Mančinska–Storgaard bound. The same work discusses the conjectured stronger finite-size inequality

$(n,m,p)$7

noting that it is known for $(n,m,p)$8 and $(n,m,p)$9, but open in general. It also introduces mutually unbiased projector-valued measurements (MUPVMs) as the measurement geometry suggested by equality conditions, and constructs an $$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$0-QRAC family with

$$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$1

which exactly matches the conjectured bound at $$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$2. This suggests that the entropy bound is universal but not generally optimal in the finite-dimensional regime (Akibue et al., 10 Jun 2026).

3. One-way quantum communication and the Equality function

In one-way quantum communication complexity, “Nayak” is also attached to an upper-bound technique for Equality. For the $$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$3-bit Equality function with error $$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$4, there is a one-way pure-state protocol of cost

$$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$5

and this bound is stated to have been implicitly already shown in Nayak’s 1999 PhD thesis. The construction uses a binary linear code

$$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$6

with pairwise relative distances concentrated near $$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$7, and the transmitted fingerprint state

$$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$8

Bob measures with projectors onto $$f:\{0,1\}^n \to \text{density operators on } m \text{ qubits}$$9, obtaining the upper bound

$i\in\{1,\dots,n\}$0

in the pure-state one-way model (Lalonde et al., 2021).

The same work shows that this pure-state upper bound is nearly tight: $i\in\{1,\dots,n\}$1 for $i\in\{1,\dots,n\}$2. It is explicit that this lower bound is not proved by directly applying Nayak’s random-access-code lower bound. Instead, the proof proceeds through geometric separation of the pure message states, conversion of complex vectors to real vectors, construction of an approximate identity matrix, and an application of Alon’s approximate-rank lower bound. The same paper also gives a mixed-state upper bound

$i\in\{1,\dots,n\}$3

showing that the older Nayak-style pure-state fingerprinting bound is optimal only in the pure-state-restricted one-way model, not in the unrestricted one-way quantum model (Lalonde et al., 2021).

This usage is historically related to, but conceptually distinct from, the standard random-access encoding bound. Here “Nayak” refers primarily to a coding-based one-way upper bound rather than to an entropy lower bound.

4. Quantum private information retrieval and robust linear lower bounds

The standard Nayak lower bound becomes a direct engine for lower bounds in one-server QPIR. In that setting, an $i\in\{1,\dots,n\}$4-round protocol with total communication complexity

$i\in\{1,\dots,n\}$5

is converted into a random access encoding of the database. The reduction works even when the protocol is only $i\in\{1,\dots,n\}$6-correct and only approximately private against purified or specious servers. The resulting lower bound is

$i\in\{1,\dots,n\}$7

qubits of communication for an $i\in\{1,\dots,n\}$8-bit database (Baumeler et al., 2013).

More precisely, the theorem states that if a QPIR protocol is $i\in\{1,\dots,n\}$9-correct and ultimately $M_i$0-private against purified servers, then its communication complexity is at least

$M_i$1

Since purified servers are specious, the same lower bound applies against $M_i$2-specious adversaries. When $M_i$3 and $M_i$4 are negligible, the asymptotic consequence is

$M_i$5

The proof strategy is structurally important. One purifies both parties, fixes a query index, compresses the final client subsystem using a Schmidt-rank bound for interactive protocols, and then uses privacy to show that a client-side unitary can approximately convert the query-1 final state into a query-$M_i$6 final state for any $M_i$7. This yields a random access encoding with decoding probability

$M_i$8

after which the Nayak bound applies verbatim. In this sense, the QPIR results are not a new Nayak bound; they are a robust extension of Nayak’s lower-bound framework to approximate privacy, approximate correctness, and specious-adversary security (Baumeler et al., 2013).

5. Two-way communication, subdistribution methods, and Nayak-style information-cost frameworks

A different object sometimes associated with Nayak is the two-way product subdistribution bound of Jain, Klauck, and Nayak. In the formulation used later, the relative min-entropy version is

$M_i$9

with the Jain–Klauck–Nayak product version obtained when the outer maximization is restricted to product distributions $0,1$0. This is not the standard random-access-code Nayak bound, but it is a lower-bound framework in two-way public-coin communication complexity that explicitly bears Nayak’s name through coauthorship (Jain, 2010).

That framework was generalized to arbitrary, possibly correlated input distributions by the two-way conditional relative entropy bound

$0,1$1

The associated direct product theorem yields, for suitable parameters,

$0,1$2

and specializes back to the earlier Jain–Klauck–Nayak product-subdistribution direct product theorem (Jain, 2010).

A related but distinct Nayak-associated line concerns the Augmented Index problem and quantum streaming lower bounds for $0,1$3. There the key statement is an asymmetric quantum information-cost tradeoff for Augmented Index: $0,1$4 This work explicitly describes itself as a quantum generalization of the argument of Jain and Nayak and uses that to derive the streaming lower bound

$0,1$5

for $0,1$6-pass unidirectional quantum streaming algorithms for $0,1$7 (Nayak et al., 2016).

These results show that “Nayak bound” can also denote a family of information-theoretic and subdistribution-based lower-bound methodologies, rather than a single theorem.

6. Ambiguity of the term: unrelated uses in mechanics and quantum walks

Outside quantum information and communication complexity, the name Nayak appears in technically unrelated settings. In rough-surface contact mechanics, the central quantity is the Nayak parameter

$0,1$8

which determines the breadth of the surface spectrum. In the class of isotropic random rough surfaces considered there, the admissible range is explicitly stated as

$0,1$9

That literature studies how the true contact area decreases logarithmically with $$\Pr\left[ M_i\left( f\left( x \right) \right)=x_i \right] \ge p$$0, and derives closed-form relations such as

$$\Pr\left[ M_i\left( f\left( x \right) \right)=x_i \right] \ge p$$1

but it does not introduce a new “Nayak bound” in the sense of a universal theorem or sharp inequality (Yastrebov et al., 2017).

A different source of confusion is the paper “On the Ambainis-Bach-Nayak-Vishwanath-Watrous Conjecture,” which is not about the standard “Nayak bound” from quantum information theory. It studies a conjecture on absorption probabilities in the finite Hadamard walk,

$$\Pr\left[ M_i\left( f\left( x \right) \right)=x_i \right] \ge p$$2

and refutes it by the counterexample

$$\Pr\left[ M_i\left( f\left( x \right) \right)=x_i \right] \ge p$$3

Here “Nayak” enters only because A. Nayak was one of the coauthors of the conjecture under discussion (Ampadu, 2010).

This ambiguity is substantive rather than merely terminological. In quantum information, “Nayak bound” ordinarily refers to an entropy tradeoff for random access encodings and QRACs. In adjacent literatures, the same surname may label a parameter, a coauthored conjecture, or a broader lower-bound framework. A plausible implication is that the phrase should be interpreted only after the surrounding technical domain—QRACs, QPIR, one-way Equality, two-way communication, contact mechanics, or quantum walks—has been fixed.