Weak Decoupling Duality and Quantum Identification24 October 2001. A preliminary version of this paper was presented as a contributed talk at the 12th QIP workshop, Santa Fe (NM), 12-16 January 2009.PH is with the School of Computer Science, McGill University, Montreal, Canada. He was supported by the Canada Research Chairs program, the Perimeter Institute, CIFAR, FQRNT’s INTRIQ, MITACS, NSERC, ONR through grant N000140811249 and QuantumWorks. Email: patrick@cs.mcgill.ca.AW is with the Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. and the Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117542. He was supported through an Advanced Research Fellowship of the U.K. EPSRC, the EPSRC’s “QIP IRC”, the European Commission IP “QAP”, by a Wolfson Research Merit Award of the Royal Society, a Philip Leverhulme Prize and an ERC Advanced Grant. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme. Email: a.j.winter@bris.ac.uk.

Weak Decoupling Duality and Quantum Identification††thanks: 24 October 2001. A preliminary version of this paper was presented as a contributed talk at the 12th QIP workshop, Santa Fe (NM), 12-16 January 2009. PH is with the School of Computer Science, McGill University, Montreal, Canada. He was supported by the Canada Research Chairs program, the Perimeter Institute, CIFAR, FQRNT’s INTRIQ, MITACS, NSERC, ONR through grant N000140811249 and QuantumWorks. Email: patrick@cs.mcgill.ca. AW is with the Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. and the Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117542. He was supported through an Advanced Research Fellowship of the U.K. EPSRC, the EPSRC’s “QIP IRC”, the European Commission IP “QAP”, by a Wolfson Research Merit Award of the Royal Society, a Philip Leverhulme Prize and an ERC Advanced Grant. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme. Email: a.j.winter@bris.ac.uk.

Patrick Hayden, Member, IEEE and Andreas Winter Dedicated to the memory of Rudolf Ahlswede
24 October 2011
Abstract

If a quantum system is subject to noise, it is possible to perform quantum error correction reversing the action of the noise if and only if no information about the system’s quantum state leaks to the environment. In this article, we develop an analogous duality in the case that the environment approximately forgets the identity of the quantum state, a weaker condition satisfied by -randomizing maps and approximate unitary designs. Specifically, we show that the environment approximately forgets quantum states if and only if the original channel approximately preserves pairwise fidelities of pure inputs, an observation we call weak decoupling duality. Using this tool, we then go on to study the task of using the output of a channel to simulate restricted classes of measurements on a space of input states. The case of simulating measurements that test whether the input state is an arbitrary pure state is known as equality testing or quantum identification. An immediate consequence of weak decoupling duality is that the ability to perform quantum identification cannot be cloned. We furthermore establish that the optimal amortized rate at which quantum states can be identified through a noisy quantum channel is equal to the entanglement-assisted classical capacity of the channel, despite the fact that the task is quantum, not classical, and entanglement-assistance is not allowed. In particular, this rate is strictly positive for every non-constant quantum channel, including classical channels.

I Introduction

Quantum channels in modern quantum information theory [1] are modeled as completely positive and trace-preserving maps between the state spaces of quantum systems with Hilbert spaces and . The requirement of complete positivity means that is not just positive, mapping positive semidefinite operators to positive semidefinite operators, but that is positive for the identity map on any . This distinction plays a central role in the geometry of entanglement because positive but not completely positive maps can be used to identify entangled quantum states [2]. This paper will take as its starting point a similar observation about channel norms.

The Stinespring dilation theorem establishes a fundamental property of quantum channels: for every channel there exists an ancilla space and an isometry such that  [3]. This means that quantum noise can always be interpreted as information loss in an otherwise deterministic evolution. Since and are essentially unique (up to unitary equivalence), each channel also has an associated complementary channel , with , which is uniquely defined up to coordinate changes of .

In quantum Shannon theoretic error correction we try to find two channels and (an encoder and decoder) such that . For now we shall consider the encoding fixed, so that can be treated as a single channel. The central insight of quantum error correction [4, 5, 6, 7] is that the existence of a decoding operation for a channel , i.e.

 ∀ρ∈S(RA)∥∥(id⊗D∘N)ρRA−ρRA∥∥1≤ϵ, (1)

is equivalent to the complementary channel being completely forgetful: for all Hilbert spaces ,

 (2)

with a universal relation between and .

Here we determine a matching duality for the weaker property of the complementary channel being only (approximately) forgetful:

 ∀ρ,σ∈S(A)∥∥Nc(ρA)−Nc(σA)∥∥1≤δ. (3)

That this is a much weaker property was noticed in the contexts of approximate encryption and remote state preparation [8, 9]. The difference between Eqs. (2) and (3) is precisely the difference between two norms on superoperators, the naïve one inherited from the trace norm, and the so-called completely bounded norm [10, 11, 7]. Not surprisingly, Eq. (3) will hold provided the main channel approximately preserves the pairwise fidelities between input pure states, a property we call geometry preservation:

 (4)

In fact, the reverse is also true. Our investigations will revolve around weak decoupling duality, which asserts that a channel is geometry-preserving if and only if its complement is approximately forgetful, with dimension-independent functions relating and . Thus, an isometry with two outputs can preserve geometry to at most one of them. Symmetrically, the isometry can be forgetful to at most one output.

The geometry preservation property, though much weaker than transmission of quantum information, must nonetheless be considered a way of preserving coherence: by virtue of weak decoupling duality, geometry preservation cannot be cloned. Indeed, if a channel has multiple outputs, one of which is geometry-preserving, then the rest must be forgetful.

Via weak decoupling duality, the many known examples of approximately forgetful channels that are not completely forgetful also provide examples of geometry-preserving channels that are not correctable [8, 9, 12, 13, 14, 15, 16, 17]. Most strikingly, it is possible to preserve geometry while almost halving the number of qubits from input to output [18]. In that case, the geometry of the unit sphere in is necessarily encoded into the eigenvectors and eigenvalues of the much smaller output state on . In contrast to quantum error correction, dimension counting reveals the mixedness of the output state to be crucial to preserving the geometry. Some of the geometry of the input state space of pure quantum states is thus faithfully encoded as noise in the output state.

Moreover, the analogy with the quantum error correction duality can be made much stronger. There is a channel communication task very similar to quantum state transmission which is intimately related to geometry preservation: quantum identification [18, 19].

Quantum identification is a cooperative communication game between two parties – conventionally called Alice and Bob – where Alice has a given quantum state that she encodes in some way into the channel, and Bob only wants to simulate measurements consisting of an arbitrary pure state projector and its complement, which can interpreted as performing the experiment asking “Is this the state?” [18]. The idea is that Alice has an encoding channel and Bob has, for every pure state , a POVM such that

 (5)

Such an object is called an -quantum-ID code. (The name is adapted from the classical case [20, 21]. Indeed, in [22, 23] the Ahlswede-Dueck theory of identification is studied in the context of quantum channels; both papers define “quantum identification codes”, which however, in the light of the above definition and [18, 19], are better named “(classical) identification codes via quantum channels”.)

Note that Bob has at his disposal various quantum measurements at the output of the channel, but the quality of the code is measured by how well the statistics of this measurement approximate the statistics of the ideal measurement he wants to perform on the message state. While it may seem that this is an odd way of defining a quantum communication task, normal quantum error correction can also be described this way; namely, Bob wants to be able to simulate all measurements on the message state. Clearly, if he can perform quantum error correction in the usual sense, then he can perform the simulation. But conversely, it follows from the methods of [24, 25, 26] that if he only has two measurements approximating generalized and observables sufficiently well, he can build a quantum error correction procedure . Moreover, a quantum-ID code with is itself a quantum error correcting code; there is no difference between error correction and identification if both tasks are to be performed perfectly. But as we shall see, in the regime of non-zero error, , the two concepts diverge. Even the task of transmitting classical information is conveniently reflected in the framework of simulating measurements: In that case, Bob only wants to simulate the measurement of the generalized observable.

With this, one can define in the usual way a quantum-ID capacity of many uses of the channel as the highest rate at which qubits can be encoded and decoded as in Eq. (5) with vanishing error – see Section III for details. Previously it was only known that for the noiseless qubit channel , , double the value of both the the quantum and classical transmission capacities [18].

While reasoning directly about quantum identification (quantum-ID) codes has proved challenging, the duality between geometry preservation and approximate forgetfulness provides a new approach to studying them. Up to some technical conditions, geometry preservation is equivalent to the existence of a quantum-ID code. It is therefore possible to construct quantum-ID codes by finding approximately forgetful maps. This approach is fruitful because destroying information is a comparatively indiscriminate task. Indeed, the analogous strategy has led to a number of straightforward proofs of the hashing bound on the quantum capacity of a quantum channel [27, 25, 28, 29]. Classical data is not immune to analysis by purification either. The duality between privacy amplification and data compression with quantum side information has recently led to a proof in this spirit [30, 31] of the Holevo-Schumacher-Westmoreland theorem on the classical capacity of a quantum channel [32, 5] .

With weak decoupling duality in hand, it is even possible to establish a simple formula for an amortized version of the quantum identification capacity; it is exactly equal to the entanglement-assisted classical capacity of a quantum channel.

I-a Structure of the paper

Section II contains the formal statement and proof of the weak decoupling duality. The duality is studied in more detail in Section III, where forgetfulness is shown to be nearly equivalent to quantum identification. In that section we provide a simple statement whose proof eliminates many technical difficulties, as well as a more flexible version that we prove from first principles. Section IV uses the flexible version of the equivalence to construct quantum-ID codes for memoryless quantum channels. Section V explores how much side communication is required to achieve the amortized quantum identification capacity, establishing that for some channels, a positive rate is necessary.

I-B Notation

We will restrict our attention throughout to finite dimensional Hilbert spaces. If is a Hilbert space, we write for the set of density operators acting on . Also, if and are two finite dimensional Hilbert spaces, we write for their tensor product. The Hilbert spaces on which linear operators act will be denoted by a superscript. For instance, we write for a density operator on . Partial traces will be abbreviated by omitting superscripts, such as . We use a similar notation for pure states, e.g. , while abbreviating . We will write for the identity map on and for the identity qubit channel. The symbol will be reserved for the identity matrix acting on the Hilbert space and for the maximally mixed state on (where we denote by the dimension of the Hilbert space ).

The trace norm of an operator, is defined to be . The similarity of two density operators and can be measured by the trace distance , which is equal to the maximum over all possible measurements of the variational distance between the outcome probabilities for the two states. The trace distance is zero for identical states and one for perfectly distinguishable states.

A complementary measure is the mixed state fidelity

 F(φ,ψ)=∥∥√φ√ψ∥∥21=(tr√√φψ√φ)2, (6)

defined such that when one of the states is pure, . More generally, the fidelity is equal to one for identical states and zero for perfectly distinguishable states. We will make frequent use of the following fundamental inequality between fidelity and trace distance of states [33, Prop. 5]:

 1−√F(φ,ψ)≤12∥φ−ψ∥1≤√1−F(φ,ψ). (7)

Both measures can be extended to unnormalized states, but Eq. (7) need not hold in that case. Further properties of the distance measures are collected in the Appendix.

Ii Weak decoupling duality

Our investigations will revolve around the duality between geometry preservation and approximate forgetfulness, which we call weak decoupling duality. The rigorous statement is as follows:

Theorem 1 (Weak decoupling duality)

Let be a quantum channel with complementary channel . Approximate geometry preservation on implies approximate forgetfulness for . That is,

 ∀|ψ⟩,|φ⟩∈A ∥φ−ψ∥1−∥N(φ)−N(ψ)∥1≤δ implies ∀|ψ⟩,|φ⟩∈A ∥Nc(φ)−Nc(ψ)∥1≤4√2δ1/4.

Conversely, approximate forgetfulness for implies approximate geometry preservation on :

 ∀|ψ⟩,|φ⟩∈A ∥Nc(φ)−Nc(ψ)∥1≤ϵ implies ∀|ψ⟩,|φ⟩∈A ∥φ−ψ∥1−∥N(φ)−N(ψ)∥1≤4√2ϵ.

Note that we have dropped an absolute value sign as compared to Eq. (4) since holds automatically for all quantum channels . (See, for example, [34].)

The duality is a straightforward consequence of two basic results in quantum information theory. The first is that the ability to transmit classical data in two conjugate bases is equivalent to the ability to transmit entanglement. That observation is the basis for the stabilizer approach to quantum error correcting codes [35]. Here we will use a clean approximate formulation due to Renes [26]. The second result is the continuity of the Stinespring dilation of a quantum channel, established by Kretschmann et al. [7]. Here we only need a corollary, which can be interpreted as a bound on the information-disturbance trade-off. The theorem is stated in terms of the following norms:

Definition 2

For a linear superoperator , let

 ∥Γ∥(k)⋄=max∥X∥1≤1∥∥(idk⊗Γ)X∥∥1,

where maximization is over operators on . Define , the completely bounded trace norm [10] (also known as diamond norm [11]).

Note that the convexity of the trace norm implies that the supremum is achieved on a rank-one operator (if is Hermitian-preserving, then on a pure quantum state). Since any operator on can be “purified” by a system of dimension , it follows that the supremum is achieved when .

Of course, since all our Hilbert spaces are finite-dimensional, all these norms are equivalent – indeed, by Lemma 23 in the Appendix,

 ∥Γ∥(1)⋄≤∥Γ∥(k)⋄≤k∥Γ∥(1)⋄.

Since the factor of cannot be improved, this means that the norms can differ by a factor as large as the dimension of , rendering the norms inequivalent in asymptotic settings, such as will be considered in the following. This can also be seen in the difference between approximately and completely forgetful maps. There, is the difference between a completely positive, trace-preserving map and a constant map (on states); approximate forgetfulness postulates a bound on while complete forgetfulness requires bounding .

Theorem 3 (Information-disturbance [7])

Let be an isometric extension of the channel and let be the complementary channel. Fix a state and let be the channel taking all inputs to . Then

 14infD∥∥D∘N−id∥∥2⋄≤∥∥Nc−R∥∥⋄≤2infD∥∥D∘N−id∥∥1/2⋄.

Both infimums are over all quantum channels.

The proof of weak decoupling duality is a fairly routine matter of combining these results:

Proof of Theorem 1 We begin by assuming approximate geometry preservation. Fix in then set . Suppose that

 ∥N(ω)−N(ξ)∥1≥∥ω−ξ∥1−δ

for all . Then if , we have

 ∥∥N(φ)−N(ψ)∥∥1 ≥2−δand ∥∥N(χ+)−N(χ−)∥∥1 ≥2−δ.

We can therefore transmit data in two conjugate bases through , which implies that entanglement is also faithfully transmitted. In particular [26, Thm. 1] (with “guessing probability” ) implies that there exists a channel such that

 ∥∥(id2⊗D∘N)Φ−Φ∥∥1≤2√δ,

where . But trace norm monotonicity with respect to dephasing the first system then gives

 ∥∥(id2⊗D∘N)Φ−Φ∥∥1 ≥12∥∥|0⟩⟨0|⊗[(D∘N)φ−φ] +|1⟩⟨1|⊗[(D∘N)ψ−ψ]∥∥1 =12∥(D∘N)φ−φ∥1+12∥(D∘N)ψ−ψ∥1.

Therefore, and by changing the choice of dephasing basis, we can conclude that . Combining this with Lemma 23 in the Appendix implies that . The information-disturbance theorem (Theorem 3) applied with the map taking all states to then implies that for all ,

 ∥Nc(φ)−Nc(ω)∥1≤2(8√δ)1/2=4√2δ1/4.

Since is an arbitrary two-dimensional subspace of , however, the inequality must hold for all and in .

For the converse, suppose that, for all states , the inequality holds. Fix and then let be the restriction of to states on . Let be the channel on that always outputs . Then once more by Lemma 23 in the Appendix, . Using this time the lower bound from Theorem 3, there exists a channel such that . In particular, for all ,

 14∥∥(D∘N)ω−ω∥∥21≤2ϵ.

Applying the triangle inequality several more times gives:

 4√2ϵ ≥∥∥(D∘N)φ−φ∥∥1+∥∥(D∘N)ψ−ψ∥∥1 ≥∥φ−ψ∥1−∥∥(D∘N)(φ−ψ)∥∥1 ≥∥φ−ψ∥1−∥∥N(φ−ψ)∥∥1,

where the final inequality used that the quantum channel cannot increase the trace norm. Rearranging the final expression gives the desired inequality. \QED

Iii Quantum identification

Quantum identification allows a sender to transmit arbitrary quantum states but only allows the receiver to perform a restricted set of measurements, namely tests to determine whether the transmitted state consists of an arbitrary target state. The receiver gets to choose the target state after the sender has transmitted, so the code must work for all targets. If the test can be performed perfectly, then quantum identification is easily seen to be equivalent to quantum state transmission, but in the approximate setting, the tasks are not equivalent.

Definition 4

[18] An -quantum-ID code for the channel consists of an encoding map and, for every pure state , a POVM acting on such that

 ∀|ψ⟩,|φ⟩∈S∣∣tr((N∘E)ψ)Dφ−|⟨φ|ψ⟩|2∣∣≤ϵ.

If the receiver had been able to perform the measurement on the input state , then he would have observed outcome with probability . The definition therefore ensures that the receiver can simulate the measurement for all input and target states.

Many variants of the definition have been proposed. In particular, one could imagine drawing a distinction between oblivious ID codes, in which the sender is only given a physical quantum state to send, and visible ID codes, in which the sender knows the identity of the state she is trying to transmit [18]. Entanglement assistance is also interesting and exceptionally powerful in the visible setting [36]. A different task that is nonetheless similar in spirit is to use quantum states as “fingerprints” for identifying classical messages in a model where pairs of messages are to be compared by a referee [37]. For comparing quantum states, however, the simple definition considered here is arguably the most natural.

If we integrate the encoding and noisy channel from Definition 4 into a single map with output and environment , we may think of the code Hilbert space as a subspace of . More formally, if we let be the Stinespring dilation of , then and we can identify the code with a subspace of . This identification simplifies the notation and we will use it for the remainder of the paper.

The main result of this section is a demonstration that a subspace of is a quantum-ID code for iff it is approximately forgetful for . (There is a small technical caveat to the statement: the reduced states on must also obey a regularity condition for the reverse implication to hold, but we will defer discussion of the details.) For the moment, let us begin by considering the relationship between quantum identification and geometry preservation.

Lemma 5

Let be a subspace of a tensor product Hilbert space that is an -quantum-ID code for . In other words, suppose that, for each pure state , there exists an operator on such that for all pure states ,

Then, for all ,

 F(φ,ψ)≤F(φB,ψB)≤F(φ,ψ)+4√ϵ.
Proof.

Consider the measurement and associated channel which acts on . By applying the monotonicity of the fidelity under quantum channels to and , we get

 F(ψ,φ)≤F(ψB,φB)≤F(M(ψB),M(φB))≤(√trψBDφ+√ϵ)2≤F(ψ,φ)+2√ϵ+ϵ+ϵ,

which proves the lemma.

The fidelity is therefore approximately preserved by quantum-ID codes. Geometry preservation is defined in terms of the trace distance, however, not the fidelity. While it is indeed the case that quantum-ID codes preserve geometry, the argument is somewhat more delicate because applying the measurement causes a significant drop in the trace distance even as it leaves the fidelity nearly unchanged. Instead, Theorem 7 will allow us to infer that quantum-ID codes preserve geometry by virtue of the fact that their complementary channels are forgetful.

In order to succeed at quantum identification, the following lemma demonstrates that it is sufficient to be able to identify orthogonal states:

Lemma 6

Let be a subspace of a tensor product Hilbert space such that for there exists acting on satisfying

 trφBDφ≥1−δ% andtrψBDφ<δ

whenever is orthogonal to . Then is a quantum-ID code with error probability .

Proof.

Let be arbitrary and let be orthogonal to in . Write

 |ψ⟩=α|φ⟩+β|φ′⟩.

Expanding shows that is equal to

 |α|2trφBDφ+|β|2trφ′BDφ=+α¯¯¯βtr|φ⟩⟨φ′|(Dφ⊗\openone)+¯¯¯¯αβtr|φ′⟩⟨φ|(Dφ⊗\openone),

which results in

 ∣∣trψBDφ−|α|2∣∣≤|α|2(1−trφBDφ)+|β|2trφ′BDφ======+2|αβ||⟨φ|(Dφ⊗\openone)|φ′⟩|≤|α|2(1−trφBDφ)+|β|2trφ′BDφ======+2|αβ|√⟨φ′|(Dφ⊗\openone)|φ′⟩≤δ+2√δ,

where we have used the Cauchy-Schwarz inequality and the assumption that orthogonal states in can be well discriminated.

Now we are ready to state and prove our main result on the duality between quantum identification and approximate forgetfulness. As with weak decoupling duality, we have chosen to prove the theorem by composing general purpose results for the purpose of pedagogical clarity, which leads to artificially poor scaling of the parameters. Readers concerned with optimizing the parameters should also consult Theorem 8.

Theorem 7 (Identification and forgetfulness)

Quantum-ID codes and forgetfulness are dual in the following quantitative sense. If a subspace is an -quantum-ID code for , then is approximately -forgetful:

 ∀|φ⟩,|ψ⟩∈S12∥∥φE−ψE∥∥1≤δ:=74√ϵ.

Conversely, if is approximately -forgetful, then geometry is approximately preserved on :

 ∀|φ⟩,|ψ⟩∈S∥∥φ−ψ∥∥1−∥∥φB−ψB∥∥1≤ϵ:=4√2δ.

If, in addition, the nonzero eigenvalues of lie in the interval for all , then is an -quantum-ID code for .

Remark  While it would be desirable to eliminate the eigenvalue condition at the end of the theorem, the condition is fairly natural in this context. If the reduced states are very close to a single state for all , then all the are very close to being purifications of , meaning that they differ from one another only by a unitary plus a small perturbation. If is the maximally mixed state or close to it, then the assumption will be satisfied.

Proof.

For the first part, recall that if is a quantum-ID code with error probability , then for each pure state there exists an operator on such that for all pure states ,

Just as in the proof of Theorem 1, the hypothesis implies that data can be transmitted in two conjugate bases with guessing probability . Running exactly the same argument as was made in that proof gives that for all ,

 12∥∥φE−ψE∥∥1≤4√2(2ϵ)1/4≤7ϵ1/4. (8)

The second part is just a restatement of one direction of the weak decoupling duality, but it is a useful step on the way to the third part, which is more challenging since it requires the construction of the decoder, that is, the operators .

Indeed, given , and arbitrary in , we learn from the second part that

 ∥φB−ψB∥1≥2−4√2δ. (9)

By Helstrom’s theorem on the optimal discrimination of and  [38], there exists a projector on such that

 trφBPφ,ψ≥1−2√2δ,trψBPφ,ψ≤2√2δ. (10)

The problem with using as the decoding is that this projector may indeed depend not only on , but also on . Since the goal is to find a single projector that the receiver can use to identify that will work regardless of whether the input is or , that is unacceptable. Still, let us confirm first that if we manage to find one effect operator that can deal with all at once, then by Lemma 6 we’ll be done. Our strategy for doing so will be to first extend Eq. (10) to all mixed states orthogonal to and supported on , and then use a minimax argument to extract a single operator independent of .

Lemma 21 in the Appendix can be used directly to see that for all mixed states supported on and orthogonal to ,

 F(φB,σB)≤λ2μ2maxF(φB,ψB)≤4√2δλ2μ2,

where the maximization is over all orthogonal to and the second inequality is an application of Eq. (7) to Eq. (9). Applying Eq. (7) a second time gives

 12∥∥φB−ψB∥∥1≥1−2(2δ)1/4λμ.

Applying Helstrom’s theorem to and yields a projector with

 trφBPσ−trσBPσ≥1−2(2δ)1/4λμ.

Von Neumann’s minimax theorem then ensures the existence of a saddle point in the following two-player game [39] (see Ky Fan [40] for a more general version). One player selects while the other player selects a state supported on and orthogonal to . The strategy spaces are therefore closed and convex. The payoff function is , which is linear in each argument. Thus, the minimax theorem guarantees that there exists an operator such that for all supported on and orthogonal to ,

 trφBDφ ≥1−2(2δ)1/4λμ, trσBDφ ≤2(2δ)1/4λμ,

and applying Lemma 6 finishes the proof.

Unfortunately, Theorem 7 is not quite strong enough to prove our main result on the quantum identification capacity. To control the ratio of the largest to smallest eigenvalues of the coding states, we need to act on them by typical projectors that cause a slight distortion. To accomodate this complication, we will instead use the following slightly more flexible version of the converse that behaves better with respect to the distortion. In particular, the amount of distortion enters the bound on the quality of the quantum-ID code in a term independent of the eigenvalue constraint. That separation proves to be crucial because the eigenvalues cannot be controlled independently of the distortion.

Theorem 8

Let be a subspace and an operator acting on such that for all . For any state , write . If there exists a state such that

 ∀|φ⟩∈S∥∥~ΩE−~φE∥∥1≤δ

with and, in addition, the nonzero eigenvalues of lie in the interval , then is an -quantum-ID code for .

Proof.

Let and be orthonormal states in . We will begin by showing that and can be effectively distinguished. To this end, consider the states

 |ϑ±⟩ =1√2|φ⟩±1√2|ψ⟩, |χ±⟩ =1√2|φ⟩±i√2|ψ⟩,

which form two orthogonal pairs. Then

 ~ϑE± =12~φE+12~ψE±12(trB|~φ⟩⟨~ψ|+trB|~ψ⟩⟨~φ|), χE± =12~φE+12~ψE∓i2(trB|~φ⟩⟨~ψ|−trB|~ψ⟩⟨~φ|),

and, by assumption,

 12∥~ϑE+−~ϑE−∥1≤δand12∥~χE+−~χE−∥1≤δ.

Combining these relations reveals that , hence by the triangle inequality, . But this gives us, by virtue of Lemma 20,

 F(~φB,~ψB)≤64δ2. (11)

To proceed as in the proof of Theorem 7, we need to show that any and mixed state supported on the orthogonal complement of in can also be distinguished. In order to apply Lemma 21 in the Appendix, we will show that the largest and smallest nonzero eigenvalues of , or equivalently, , are well-behaved modulo a little bit of truncation. Indeed, let and be the eigenvalues of and , respectively, in nonincreasing order. Then

Define the set

 J={j:(1−γ)pj≤Oj≤(1+γ)pj}.

Then

 γ∑j∉Jpj≤∑j∉J|Oj−pj|≤δ,

implying that

 ∑j∈Jpj=∑jpj−∑j∉Jpj≥(1−ϵ)−δ/γ.

Fixing implies that for each , there is a positive semidefinite operator satisfying and whose eigenvalues lie in the interval .

Now let and consider any state whose support lies in the orthogonal complement of in . Since the states are in , the truncation procedure of the previous paragraph can be used to construct operators . Let . Then by Lemma 21,

 F(^φB,^σB) ≤9λ2μ2maxF(^φB,^ψB) ≤9λ2μ2maxF(~φB,~ψB) ≤9λ2μ264δ2=576λ2δ2μ2.

Both maximizations are over states such that . The second inequality follows from the fact that (and likewise for ) along with Lemma 22 while the third arises by substituting in the result of Eq. (11). Introducing one last decoration for our states, let and likewise for . Applying Eq. (7) with attention paid to the fact that and are not normalized gives

 12∥∥¯φB−¯σB∥∥1≥1−24λδμ11−ϵ−2δ≥1−30λδμ,

where the final inequality uses that . Applying Helstrom’s theorem to and implies that there exists a projector such that

 tr¯φBPσ−tr¯σBPσ≥1−30λδμ.

Next we invoke von Neumann’s minimax theorem, just as in the proof of Theorem 7, for the payoff function , with the strategy space of the second player the convex hull of the operators , where ranges over states orthogonal to . (The operators are not normalized but that will not cause any difficulties.) This provides an operator such that

 tr¯φBDφ ≥1−30λδμand (12) tr^σBDφ ≤30λδμ. (13)

But

 ∣∣trφBDφ −tr¯φBDφ∣∣ ≤∥φB−¯φB∥1 ≤∥φB−~φB∥1+∥~φB−^φB∥1+∥^φB−¯φB∥1 ≤2√ϵ+2δ+∣∣1−(1−ϵ−2δ)∣∣ ≤3√ϵ+4δ,

where the fourth line follows from the gentle measurement lemma (Appendix, Lemma 24), the definition of , and the fact that . Similarly, for any a convex combination of states arising from perpendicular to ,

 ∣∣trσBDφ −tr^σBDφ∣∣ ≤∥σB−^σB∥1 ≤∑iqi∥ψBi−^ψBi∥1 ≤∑iqi(∥ψBi−~ψBi∥1+∥~ψBi−^ψBi∥1) ≤2√ϵ+2δ.

Combining these estimates with the outcome of the minimax theorem in Eq. (12) and Lemma 6 completes the proof.

Iv Quantum identification capacity

While it might not be possible to design low error quantum-ID codes for any given channel, the situation becomes more promising if many uses of the channel are allowed. In analogy with classical and quantum data transmission, we can define asymptotic quantum-ID codes as follows.

Definition 9 (Quantum-ID capacity [18])

A rate is said to be achievable for quantum identification over if for all and sufficiently large , there are -quantum-ID codes for with encoding domain of dimension at least . The quantum identification capacity is defined as the supremum of the achievable rates.

The capacity should be interpreted as the number of qubits that can be identified per use of the channel in the limit of many uses of the channel. The only nontrivial channel for which the quantum identification capacity was known prior to this paper was the identity channel: asymptotically, a noiseless qubit channel can be used to identify two qubits. That is,  [18]. As we will see below, the theory of the quantum identification capacity is considerably simpler when the given channel can be used in conjunction with noiseless channels to the receiver. This obviously increases the capacity, so the interesting question is how much the use of increases the quantum identification capacity over what would have been achievable with the noiseless channels alone. When defining the achievable amortized rates it is therefore necessary to subtract off two qubits for every noiseless qubit channel used per copy of .

Definition 10 (Amortized quantum-ID capacity)

A rate is said to be achievable for amortized quantum identification over if for all and sufficiently large , there are -quantum-ID codes for with encoding domain such that . where is the binary logarithm throughout this paper. The amortized quantum identification capacity is defined as the supremum of the achievable rates.

Readers familiar with the identification capacities of classical channels might be surprised to see that the dimension of a quantum-ID code scales only exponentially with the number of channel uses, as opposed to doubly exponentially. The essential difference between the classical and quantum settings is that the number of distinguishable quantum states in dimension already scales exponentially with , which makes quantum identification a much more demanding task. Nonetheless, as we will see below, the amortized quantum identification capacity can be positive for some channels with zero quantum capacity, like the noiseless bit channel. One then finds that the dimension of the quantum-ID code can scale super-exponentially with the number of qubits used to supplement the classical channel.

Weak decoupling duality is a very effective tool for studying the quantum-ID capacities. As a warm-up, the fact that the complements of quantum-ID codes are forgetful supplies a quick answer to an open question from [18]:

Theorem 11

If is an antidegradable channel, that is, if there exists channel such that , then . This is true in particular for the noiseless cbit channel . More generally, if the quantum capacity of the channel vanishes, , then so does the quantum-ID capacity, .

Proof.

Given a quantum-ID code for the channel that encodes as little as one qubit, the channel will be geometry-preserving if is the encoding map. Hence, by weak decoupling duality, the channel complementary to will be approximately forgetful. But if is antidegradable, then so is , meaning that the channel complementary to can simulate . But then the complementary channel would be simultaneously forgetful and geometry-preserving, a contradiction.

For the more general statement, we show the contrapositive: assume , then for all and sufficiently large , has in particular a -dimensional quantum-ID code which is -close to being forgetful for the environment, by Lemma 5. But by Lemma 23 this means that the channel from the code qubit to the environment is arbitrarily close to a constant map in the diamond norm. At this point we can then invoke Theorem 3 on information-disturbance [7] to conclude that the channel from the code qubit to can be arbitrarily well error-corrected. (Note that this argument is following our proof of the weak decoupling duality; in particular, any -dimensional subspace of a quantum-ID code, and in fact any subspace of sufficiently small dimension, is a quantum error-correcting code!) By the Lloyd-Shor-Devetak theorem on the quantum capacity (see [6]), this implies that there exists an input state for which the coherent information is positive, and hence .

As usual, quantitative statements about asymptotically achievable rates and upper bounds on the identification capacities are naturally expressed in terms of entropies. For a bipartite density matrix , we write

 H(A)φ≡H(φA)≡−trφAlogφA

for the von Neumann entropy of . The mutual information of the state is defined to be

 I(A:B)φ=H(A)φ+H(B)φ−H(AB)φ

while the coherent information and the conditional entropy are, respectively,

 I(A⟩B)φ =H(B)φ−H(AB)φ H(A|B)φ =H(AB)φ−H(B)φ.

Our main theorem on the quantum identification capacities includes a concise formula for that eliminates the optimization over multiple channel uses.

Theorem 12 (Quantum identification capacity)

For any quantum channel , its quantum-ID capacity is given by , where

 Q(1)ID(N)=sup|φ⟩{I(A:B)ρ s.t. I(A⟩B)ρ>0},

where is the purification of any input state to and , and where we declare the to be if the set above is empty.

Furthermore, the amortized quantum-ID capacity equals

 QamID(N)=sup|φ⟩I(A:B)ρ=CE(N),

the entanglement-assisted classical capacity of  [41].

Remark  It follows from Theorem 12 that the amortized quantum-ID capacity of a noiseless cbit channel is one. Reconciling this observation with Theorem 11, which asserts this channel’s unamortized quantum-ID capacity is zero, reveals that some amortized noiseless quantum communication is necessary to achieve , without determining how much. In fact, inspection of the proof of Theorem 12 reveals that, for the noiseless cbit channel , a zero rate of noiseless side qubits is sufficient to achieve the maximum value of one. These observations extend to cq-channels, so named because they consist of a destructive measurement resulting in classical information, followed by the preparation of a state conditioned on the measurement outcome. For these channels, the entanglement-assisted capacity is equal to the unassisted classical capacity , also known as the Holevo capacity [32, 42]. As a result, for all such channels even as , the latter strictly positive for all nontrivial channels. The difference in all cases can be traced to a sublinear amount of free quantum communication in the amortized setting.

This effect can be viewed as an instance of (un-)locking since the quantum-ID rate increases from strictly to an arbitrarily large amount by the addition of any positive rate of quantum communication, cf. [43, 44, 24]. Unlike the previously known examples where a certain finite rate is always required, however, here an arbitrarily small rate of extra quantum communication is sufficient to bring about an unbounded increase in the capacity.

The intuition behind the achievability of the rates in Theorem 12 is quite simple. The structure of an amortized code is illustrated in Figure 1. Fix a state purifying any input to the channel and let be , where is the Stinespring extension of . The encoding will embed the input into a random subspace of a typical subspace of tensored with ancillary spaces and , where will consist of the amortized quantum communication and the environment for the encoding. Since the encoding is into a random subspace, it will produce states highly entangled between and