The Quantum-House Effect

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We introduce the quantum-house effect, a non-local phenomenon which apparently does not require quantum discord to be present. It suffices for the effect if neither subsystem of a bipartite system is in a pure state. This way, the quantum-house effect completely fills the gap between trivial correlations and quantum discord. However, we discuss why the situation is more subtle than that, by showing that in a concrete cryptographic setting called "the quantum-house game", the ability to produce quantum discord is in fact necessary for the quantum-house effect to work. Then, we suggest a principle called "quantum detachment" to characterize where quantumness in general departs from classicality, based on the information a physical system contains about itself. The quantum-house effect is demonstrated on SpinQ Gemini, a 2-qubit liquid-state NMR desktop quantum computer.

keywords:
quantum-house effect, quantum discord, quantumness vs. classicality, SpinQ Gemini, nuclear magnetic resonance (NMR) desktop quantum computer, quantum detachment
\jyear
2022

\fnm
Tamás \surVarga

Quantum entanglement was discovered nearly a century ago einstein , and has become a signature effect of quantum mechanics. Schrödinger called entanglement the characteristic trait of quantum mechanics, which alone embodies the difference between quantumness and classicality schrodinger .

In the past decades, entanglement has turned out to be a key resource in quantum information processing as well nielsen ; schumacher . In particular, several authors pointed out that entangled states play an essential role in achieving exponential speed-up in certain quantum-computing algorithms ekert , and others hinted that in the absence of entanglement one should not talk about "true" quantum computation, but rather a simulation thereof braunstein .

It was against this backdrop that in 2001 quantum discord was discovered ollivier . Quantum discord is a measure of the quantumness of correlations between two quantum subsystems: non-zero discord certifies the presence of non-classical correlations. As one would intuitively expect, entangled states always have non-zero discord. Surprisingly, however, there also exist bipartite states which aren’t entangled but still have non-vanishing discord. So entanglement isn’t necessary for the non-classicality of correlations, non-zero discord already suffices.

But can we take this even further? Are there bipartite states with zero discord that exhibit non-classical correlations? The short answer is yes, and in Section 2 we introduce the quantum-house effect, which pushes the territory of non-classical correlations right into the realm of product states, as every bipartite product state where neither subsystem is in a pure state is capable of this non-local effect. However, we also argue that the situation is more subtle than that, and in a cryptographic sense quantum discord is still required for the quantum-house effect to work, even when bipartite states with zero discord are used.

The quantum-house effect can be considered as an extension of locally non-effective unitary operations, first proposed in fu , and further investigated in datta .111The present work came about independently of fu . The core idea we arrived at is basically the same, but our original motivation was educational, to explore quantum effects that can be demonstrated on SpinQ Gemini.

The paper is organized as follows. Sections 2 and 4 present theoretical results, while in Section 3 a demonstration of the quantum-house effect is given using SpinQ Gemini, a 2-qubit liquid-state nuclear magnetic resonance (NMR) desktop quantum computer hou , shown in Fig. 1. Then, cryptographic aspects are touched upon in Section 5. Finally, Section 6 summarizes our findings and suggest a principle to differentiate quantumness from classicality in general.

2 Theory - Part 1

Quantum discord is reviewed briefly in Subsection 2.1, after which the quantum-house effect is introduced in Subsection 2.2.

Throughout this paper, we work with a bipartite system AB𝐴𝐵ABitalic_A italic_B, where subsystem A𝐴Aitalic_A belongs to Alice, and subsystem B𝐵Bitalic_B to Bob. The overall quantum state of AB𝐴𝐵ABitalic_A italic_B is represented by the density matrix ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, while those of A𝐴Aitalic_A and B𝐵Bitalic_B are given by the partial trace formulas ρA=trB(ρAB)subscript𝜌𝐴subscripttr𝐵subscript𝜌𝐴𝐵\rho_A=\mathrmtr_B(\rho_AB)italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) and ρB=trA(ρAB)subscript𝜌𝐵subscripttr𝐴subscript𝜌𝐴𝐵\rho_B=\mathrmtr_A(\rho_AB)italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = roman_tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ), respectively.

2.1 Quantum discord

Quantum discord for ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT is a non-negative number, denoted by δABsubscript𝛿𝐴𝐵\delta_ABitalic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ollivier . In this paper, we are only interested in whether or not the discord is vanishing, which can be characterized as follows ollivier :

Theorem 1 (Non-vanishing discord).

δAB>0subscript𝛿𝐴𝐵0\delta_AB>0italic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT >0 if and only if any complete projective measurement performed on subsystem A𝐴Aitalic_A would perturb ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT for a bystander who is unaware of the measurement.

That is, if Alice secretly performs any complete projective measurement on her subsystem A𝐴Aitalic_A, then for Charlie who is unaware of that measurement, the overall state of system AB𝐴𝐵ABitalic_A italic_B will change to some ρAB′≠ρABsubscriptsuperscript𝜌′𝐴𝐵subscript𝜌𝐴𝐵\rho^\prime_AB eq\rho_ABitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ≠ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, giving him a chance to figure out that Alice did something.

Example 1.

Let’s take the EPR pair ρAB=12(|00⟩+|11⟩)(⟨00|+⟨11|)subscript𝜌𝐴𝐵12ket00ket11bra00bra11\rho_AB=\frac12(|00\rangle+|11\rangle)(\langle 00|+\langle 11|)italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 00 ⟩ + | 11 ⟩ ) ( ⟨ 00 | + ⟨ 11 | ), where Alice owns the first qubit, Bob the second. To keep things simple, let Alice measure the first qubit in the computational basis. For Charlie, unaware of the measurement, the result will be that AB𝐴𝐵ABitalic_A italic_B is either in the state |00⟩ket00|00\rangle| 00 ⟩ or |11⟩ket11|11\rangle| 11 ⟩, with probability 1212\frac12divide start_ARG 1 end_ARG start_ARG 2 end_ARG each. We can thus write for Charlie ρAB′=12|00⟩⟨00|+12|11⟩⟨11|subscriptsuperscript𝜌normal-′𝐴𝐵12ket00quantum-operator-product001211bra11\rho^\prime_AB=\frac12|00\rangle\langle 00|+\frac12|11\rangle% \langle 11|italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 11 ⟩ ⟨ 11 |, which is different from ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT.222Charlie could get ρAB′subscriptsuperscript𝜌normal-′𝐴𝐵\rho^\prime_ABitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT to any precision by quantum-state tomography, if the experiment is repeated enough times. So Charlie may figure out that Alice did something. Later, in Example 4, we’ll see that Alice could have chosen any basis for her measurement, it would always perturb the overall 2-qubit state. That is, the EPR pair has non-zero discord.

On the other hand, vanishing discord δAB=0subscript𝛿𝐴𝐵0\delta_AB=0italic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = 0 implies there is a way for Alice to secretly measure A𝐴Aitalic_A via complete projection such that Charlie surely wouldn’t notice anything. This scenario is what one would expect to be always possible in the classical world. In this sense, vanishing discord captures a notion of "classicality", while non-vanishing discord δAB>0subscript𝛿𝐴𝐵0\delta_AB>0italic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT >0 certifies the presence of non-classical correlations that cannot exist in a classical setting.

Example 2.

Let’s continue where we left off at Example 1, and let this time ρAB=12|00⟩⟨00|+12|11⟩⟨11|subscript𝜌𝐴𝐵12ket00quantum-operator-product001211bra11\rho_AB=\frac12|00\rangle\langle 00|+\frac12|11\rangle\langle 11|italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 11 ⟩ ⟨ 11 |. Now, if Alice measures the first qubit in the computational basis, it won’t change the 2-qubit density matrix for Charlie who is unaware of the measurement, i.e. ρAB′=ρABsubscriptsuperscript𝜌normal-′𝐴𝐵subscript𝜌𝐴𝐵\rho^\prime_AB=\rho_ABitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT will hold. So Charlie won’t have chance to figure out if Alice has measured or not. That is, ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT has zero discord.

It’s easy to see that the presence of discord can be characterized in terms of superposition, which was also noted in ollivier and modi :

Theorem 2 (Discord is superposition).

δAB=0subscript𝛿𝐴𝐵0\delta_AB=0italic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = 0 if and only if ρAB=∑ipi|ai⟩⟨ai|⊗ρBisubscript𝜌𝐴𝐵subscript𝑖tensor-productsubscript𝑝𝑖ketsubscript𝑎𝑖brasubscript𝑎𝑖superscriptsubscript𝜌𝐵𝑖\rho_AB=\sum_ip_i|a_i\rangle\langle a_i|\otimes\rho_B^iitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ ⟨ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT, where pi≥0subscript𝑝𝑖0p_i\geq 0italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0, ∑ipi=1subscript𝑖subscript𝑝𝑖1\sum_ip_i=1∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 and ai⟩ketsubscript𝑎𝑖\a_i\rangle\ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ is an orthonormal basis of A𝐴Aitalic_A.

Thus, vanishing discord means ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT can be produced essentially without superposition in subsystem A𝐴Aitalic_A, relying solely on the orthogonal states of the single basis ketsubscript𝑎𝑖\a_i\rangle\ .333One can arrive at the formula in Theorem 2 from a hardware perspective as well. It can be shown that the pseudo-entangled states hou produced by SpinQ Gemini cannot be expressed by such a formula, i.e. without superposition.

Example 3.

Since 12|00⟩⟨00|+12|11⟩⟨11|=12|0⟩⟨0|⊗|0⟩⟨0|+12|1⟩⟨1|⊗|1⟩⟨1|12ket00quantum-operator-product001211bra11tensor-producttensor-product12ket0bra0ket0quantum-operator-product0121bra1ket1bra1\frac12|00\rangle\langle 00|+\frac12|11\rangle\langle 11|=\frac12|% 0\rangle\langle 0|\otimes|0\rangle\langle 0|+\frac12|1\rangle\langle 1|% \otimes|1\rangle\langle 1|divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 11 ⟩ ⟨ 11 | = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 0 ⟩ ⟨ 0 | ⊗ | 0 ⟩ ⟨ 0 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 1 ⟩ ⟨ 1 | ⊗ | 1 ⟩ ⟨ 1 |, according to Theorem 2 it is an example of zero discord, with p1=p2=12subscript𝑝1subscript𝑝212p_1=p_2=\frac12italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG, |a1⟩=|0⟩ketsubscript𝑎1ket0|a_1\rangle=|0\rangle| italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ = | 0 ⟩, |a2⟩=|1⟩ketsubscript𝑎2ket1|a_2\rangle=|1\rangle| italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩ = | 1 ⟩, ρB1=|0⟩⟨0|superscriptsubscript𝜌𝐵1ket0bra0\rho_B^1=|0\rangle\langle 0|italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = | 0 ⟩ ⟨ 0 | and ρB2=|1⟩⟨1|superscriptsubscript𝜌𝐵2ket1bra1\rho_B^2=|1\rangle\langle 1|italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = | 1 ⟩ ⟨ 1 |.

Example 4.

Since the EPR pair is an entangled state, it cannot be expressed by the separable-state formula in Theorem 2, and thus δAB>0subscript𝛿𝐴𝐵0\delta_AB>0italic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT >0 must hold (see also Example 1).

Finally, if a device is able to produce just two (different) non-orthogonal states of A𝐴Aitalic_A, say, |u⟩ket𝑢|u\rangle| italic_u ⟩ and |v⟩ket𝑣|v\rangle| italic_v ⟩ with 0<|⟨u|v⟩|<10inner-product𝑢𝑣10<|\langle u|v\rangle|<10 <| ⟨ italic_u | italic_v ⟩ | <1, it’s already enough to create quantum discord. E.g. the state ρAB=0.6|u⟩⟨u|⊗|0⟩⟨0|+0.4|v⟩⟨v|⊗|1⟩⟨1|subscript𝜌𝐴𝐵tensor-producttensor-product0.6ket𝑢bra𝑢ket0quantum-operator-product00.4𝑣bra𝑣ket1bra1\rho_AB=0.6|u\rangle\langle u|\otimes|0\rangle\langle 0|+0.4|v\rangle\langle v% |\otimes|1\rangle\langle 1|italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = 0.6 | italic_u ⟩ ⟨ italic_u | ⊗ | 0 ⟩ ⟨ 0 | + 0.4 | italic_v ⟩ ⟨ italic_v | ⊗ | 1 ⟩ ⟨ 1 | has non-vanishing discord.

2.2 The quantum-house effect

Let’s start with the technical definition:

Definition 1 (Quantum-house effect).

The quantum-house effect is the phenomenon when a local operation on subsystem A𝐴Aitalic_A changes the overall state of system AB𝐴𝐵ABitalic_A italic_B, but not that of A𝐴Aitalic_A.

That is, the state of AB𝐴𝐵ABitalic_A italic_B changes to some ρAB′≠ρABsubscriptsuperscript𝜌′𝐴𝐵subscript𝜌𝐴𝐵\rho^\prime_AB eq\rho_ABitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ≠ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, while the state of A𝐴Aitalic_A remains ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. The "local operation" can be anything: measurements, unitaries, or combinations thereof, with or without ancilla system.444When only unitaries are allowed without ancilla, the local operation is called locally non-effective unitary operation, see fu and datta for details. The only (obvious) restriction is that we don’t have access to subsystem B𝐵Bitalic_B.

The quantum-house effect is a non-classical phenomenon, because in the classical world any local operation on A𝐴Aitalic_A which doesn’t change the state of A𝐴Aitalic_A cannot change the state of AB𝐴𝐵ABitalic_A italic_B either. But this is different when AB𝐴𝐵ABitalic_A italic_B is a quantum system!

Example 5.

Imagine this time it is Charlie who secretly measures, in the computational basis, Alice’s qubit of an EPR pair ρAB=12(|00⟩+|11⟩)(⟨00|+⟨11|)subscript𝜌𝐴𝐵12ket00ket11bra00bra11\rho_AB=\frac12(|00\rangle+|11\rangle)(\langle 00|+\langle 11|)italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 00 ⟩ + | 11 ⟩ ) ( ⟨ 00 | + ⟨ 11 | ). Then, since Alice and Bob are unaware of Charlie’s action, the new 2-qubit state for them will be ρAB′=12|00⟩⟨00|+12|11⟩⟨11|subscriptsuperscript𝜌normal-′𝐴𝐵12ket00quantum-operator-product001211bra11\rho^\prime_AB=\frac12|00\rangle\langle 00|+\frac12|11\rangle% \langle 11|italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 11 ⟩ ⟨ 11 |, which is different from ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. That said, the state of Alice’s individual qubit stays the same: ρA′=ρA=12|0⟩⟨0|+12|1⟩⟨1|subscriptsuperscript𝜌normal-′𝐴subscript𝜌𝐴12ket0quantum-operator-product0121bra1\rho^\prime_A=\rho_A=\frac12|0\rangle\langle 0|+\frac12|1\rangle% \langle 1|italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 0 ⟩ ⟨ 0 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 1 ⟩ ⟨ 1 |. So Alice has no chance to figure out by herself that Charlie did something. But together with Bob, they may figure it out!

For a more intuitive understanding, an analogy is shown in Fig. 2. In this analogy, subsystems A𝐴Aitalic_A and B𝐵Bitalic_B are houses of Alice and Bob, respectively. Then, a change made secretly by Charlie on Alice’s house may only be detected by Alice and Bob together, but not by Alice alone examining her own house.

Here is a theorem about the relationship to quantum discord:

Theorem 3 (Discord implies quantum-house).

If δAB>0subscript𝛿𝐴𝐵0\delta_AB>0italic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT >0, then the quantum-house effect can be achieved with ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT.

Proof: Let ρA=∑ipi|ai⟩⟨ai|subscript𝜌𝐴subscript𝑖subscript𝑝𝑖ketsubscript𝑎𝑖brasubscript𝑎𝑖\rho_A=\sum_ip_i|a_i\rangle\langle a_i|italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ ⟨ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | be the spectral decomposition of ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. Now, if Alice’s subsystem A𝐴Aitalic_A gets measured in the ai⟩ketsubscript𝑎𝑖\ basis, then for a bystander unaware of the measurement the state of A𝐴Aitalic_A won’t change, but due to Theorem 1 the state of the overall system AB𝐴𝐵ABitalic_A italic_B will.

The quantum-house effect is also possible with zero discord, as it can be seen in the example below. This way, it extends the notion of non-classicality offered by quantum discord, to a wider range of bipartite quantum systems.

Example 6.

3 Demonstration on SpinQ Gemini

In this section, we’ll showcase the quantum-house effect on the SpinQ Gemini 2-qubit NMR desktop quantum computer hou .

The SpinQ Gemini device comes with the user-interface software SpinQuasar (see Fig. 3), together forming an integrated hardware-software platform for quantum computing education and research. For further technical details, including how the qubits are physically realized, the reader is referred to hou .

We’re going to demonstrate the following example on SpinQ Gemini, with the help of SpinQuasar:

Example 7.

Charlie secretly applies a Pauli-X𝑋Xitalic_X gate on Alice’s qubit of an EPR pair ρAB=12(|00⟩+|11⟩)(⟨00|+⟨11|)subscript𝜌𝐴𝐵12ket00ket11bra00bra11\rho_AB=\frac12(|00\rangle+|11\rangle)(\langle 00|+\langle 11|)italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 00 ⟩ + | 11 ⟩ ) ( ⟨ 00 | + ⟨ 11 | ). Then, the overall 2-qubit state for Alice and Bob changes to ρAB′=12(|10⟩+|01⟩)(⟨10|+⟨01|)subscriptsuperscript𝜌normal-′𝐴𝐵12ket10ket01bra10bra01\rho^\prime_AB=\frac12(|10\rangle+|01\rangle)(\langle 10|+\langle 01|)italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 10 ⟩ + | 01 ⟩ ) ( ⟨ 10 | + ⟨ 01 | ). However, the state of Alice’s individual qubit stays the same: ρA′=ρA=12|0⟩⟨0|+12|1⟩⟨1|subscriptsuperscript𝜌normal-′𝐴subscript𝜌𝐴12ket0quantum-operator-product0121bra1\rho^\prime_A=\rho_A=\frac12|0\rangle\langle 0|+\frac12|1\rangle% \langle 1|italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 0 ⟩ ⟨ 0 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 1 ⟩ ⟨ 1 |. So Alice has no chance to figure out by herself that Charlie did something. But together with Bob, they may figure it out!

The SpinQuasar screenshots in Fig. 4 show how we implemented the EPR pair on SpinQ Gemini, as well as the EPR pair followed by a Pauli-X𝑋Xitalic_X gate on the first qubit. In each case, SpinQuasar displays not only the ideal, i.e. noiseless, 2-qubit density matrix, but also the noisy density matrix which was actually produced by the hardware.555Due to the peculiarities of liquid-state NMR technology, whenever we command SpinQ Gemini to produce a pure n𝑛nitalic_n-qubit state ρ=|ψ⟩⟨ψ|𝜌ket𝜓bra𝜓\rho=|\psi\rangle\langle\psi|italic_ρ = | italic_ψ ⟩ ⟨ italic_ψ |, such as the EPR pair, the hardware will instead produce a so-called pseudo-pure state σ=(1-η)I2n+η|ψ⟩⟨ψ|𝜎1𝜂𝐼superscript2𝑛𝜂ket𝜓bra𝜓\sigma=(1-\eta)\fracI2^n+\eta|\psi\rangle\langle\psi|italic_σ = ( 1 - italic_η ) divide start_ARG italic_I end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG + italic_η | italic_ψ ⟩ ⟨ italic_ψ |, where η∼10-5similar-to𝜂superscript105\eta\sim 10^-5italic_η ∼ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT for n=1,2𝑛12n=1,2italic_n = 1 , 2. This happens under the hood, and as ρ𝜌\rhoitalic_ρ and σ𝜎\sigmaitalic_σ are equivalent in the sense that we can unambiguously calculate one from the other, SpinQuasar only shows us ρ𝜌\rhoitalic_ρ (both ideal and noisy), but not σ𝜎\sigmaitalic_σ. We can clearly see that applying a Pauli-X𝑋Xitalic_X gate on the first qubit changes the overall 2-qubit state.

Then, the screenshots in Fig. 5 show that as opposed to the overall 2-qubit state, the state of the first qubit alone isn’t changed (apart from noise) by applying a Pauli-X𝑋Xitalic_X gate! And this completes the demonstration of the quantum-house effect.

But there is something we shouldn’t overlook here. In fact, due to the noise it would be more accurate to say that we only illustrated the quantum-house effect, rather than implemented it on the hardware level. This is because, from a cryptographic point of view, the noise characteristics of the quantum device might give Alice enough hints to be able to figure out whether or not Charlie has applied a Pauli-X𝑋Xitalic_X gate on the first qubit. So the noise always has to be taken into consideration in a realistic situation.

Therefore, we propose an informal definition for the non-ideal case where noise is present, but won’t pursue it further in this paper.

Definition 2 (Noisy quantum-house effect).

The noisy quantum-house effect is the phenomenon when a local operation on subsystem A𝐴Aitalic_A changes the overall state of system AB𝐴𝐵ABitalic_A italic_B significantly, while causing only insignificant change to the state of A𝐴Aitalic_A.

It’s like a non-local, immediate butterfly effect, with the twist that in the extreme, noiseless case, even no change to subsystem A𝐴Aitalic_A causes a significant change to system AB𝐴𝐵ABitalic_A italic_B.

4 Theory - Part 2

Next, we give a characterization of the ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT states with which the quantum-house effect can be achieved. Surprisingly, we’ll find that the quantum-house effect is possible even for some product states ρAB=ρA⊗ρBsubscript𝜌𝐴𝐵tensor-productsubscript𝜌𝐴subscript𝜌𝐵\rho_AB=\rho_A\otimes\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, provided that neither ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT nor ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT are pure states. Loosely speaking, the quantum-house effect is possible whenever there is a non-trivial correlation initially present between A𝐴Aitalic_A and B𝐵Bitalic_B.666The ensemble of a pure density |ψ⟩⟨ψ|ket𝜓bra𝜓|\psi\rangle\langle\psi|| italic_ψ ⟩ ⟨ italic_ψ | consists only of |ψ⟩ket𝜓|\psi\rangle| italic_ψ ⟩, so there is exactly one way it can be related to any other ensemble. Put it differently, non-trivial correlation is only possible between sets that have multiple elements.

Theorem 4 (Non-product implies quantum-house).

The quantum-house effect can be achieved with any non-product state ρAB≠ρA⊗ρBsubscript𝜌𝐴𝐵tensor-productsubscript𝜌𝐴subscript𝜌𝐵\rho_AB eq\rho_A\otimes\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ≠ italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT.

Proof: If we swap A𝐴Aitalic_A with an independently prepared quantum system in state ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, then the new overall state for Alice and Bob will be ρAB′=ρA⊗ρBsubscriptsuperscript𝜌′𝐴𝐵tensor-productsubscript𝜌𝐴subscript𝜌𝐵\rho^\prime_AB=\rho_A\otimes\rho_Bitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, which is a product state, so clearly ρAB′≠ρABsubscriptsuperscript𝜌′𝐴𝐵subscript𝜌𝐴𝐵\rho^\prime_AB eq\rho_ABitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ≠ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT.777Local operations on A𝐴Aitalic_A never change the quantum state of B𝐵Bitalic_B, due to the no-signaling principle. From this, we can also see that the state of Alice’s subsystem remained ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, and thus we have achieved the quantum-house effect.

Example 8.

Let ρAB=12|00⟩⟨00|+12|11⟩⟨11|subscript𝜌𝐴𝐵12ket00quantum-operator-product001211bra11\rho_AB=\frac12|00\rangle\langle 00|+\frac12|11\rangle\langle 11|italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 11 ⟩ ⟨ 11 |. This is a non-product state with zero discord (see Example 3), and a straightforward calculation reveals that ρA=12|0⟩⟨0|+12|1⟩⟨1|=I2subscript𝜌𝐴12ket0quantum-operator-product0121bra1𝐼2\rho_A=\frac12|0\rangle\langle 0|+\frac12|1\rangle\langle 1|=\fracI% 2italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 0 ⟩ ⟨ 0 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 1 ⟩ ⟨ 1 | = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG, the 1-qubit maximally mixed state. Now, if Charlie replaces Alice’s qubit with an independently prepared qubit in state I2𝐼2\fracI2divide start_ARG italic_I end_ARG start_ARG 2 end_ARG, then the resulting new overall 2-qubit state for Alice and Bob will be ρAB′=12I2⊗|0⟩⟨0|+12I2⊗|1⟩⟨1|=I2⊗I2subscriptsuperscript𝜌normal-′𝐴𝐵tensor-product12𝐼2ket0bra0tensor-product12𝐼2ket1bra1tensor-product𝐼2𝐼2\rho^\prime_AB=\frac12\fracI2\otimes|0\rangle\langle 0|+\frac12% \fracI2\otimes|1\rangle\langle 1|=\fracI2\otimes\fracI2italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG italic_I end_ARG start_ARG 2 end_ARG ⊗ | 0 ⟩ ⟨ 0 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG italic_I end_ARG start_ARG 2 end_ARG ⊗ | 1 ⟩ ⟨ 1 | = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG ⊗ divide start_ARG italic_I end_ARG start_ARG 2 end_ARG, which is a product state and thus different from ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. At the same time, the state of Alice’s qubit stays ρA′=ρA=I2subscriptsuperscript𝜌normal-′𝐴subscript𝜌𝐴𝐼2\rho^\prime_A=\rho_A=\fracI2italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG.

Theorem 5 (Non-trivial correlation implies quantum-house).

The quantum-house effect can be achieved with any product state ρAB=ρA⊗ρBsubscript𝜌𝐴𝐵tensor-productsubscript𝜌𝐴subscript𝜌𝐵\rho_AB=\rho_A\otimes\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT where neither ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT nor ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is pure.

Proof: Let σA′B′subscript𝜎superscript𝐴′superscript𝐵′\sigma_A^\primeB^\primeitalic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT be a non-product state of some bipartite system A′B′superscript𝐴′superscript𝐵′A^\primeB^\primeitalic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT with σA′=ρAsubscript𝜎superscript𝐴′subscript𝜌𝐴\sigma_A^\prime=\rho_Aitalic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and σB′=ρBsubscript𝜎superscript𝐵′subscript𝜌𝐵\sigma_B^\prime=\rho_Bitalic_σ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Such a state can always be produced, because both ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT have support containing more than one element, and thus they can be made classically correlated with each other. We give B′superscript𝐵′B^\primeitalic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT to Bob (i.e. B=B′𝐵superscript𝐵′B=B^\primeitalic_B = italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT), and keep A′superscript𝐴′A^\primeitalic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for ourselves. Additionally, we independently prepare another system A𝐴Aitalic_A in state ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, and give that to Alice. Now, the overall state of the system possessed by Alice and Bob is ρAB=ρA⊗ρBsubscript𝜌𝐴𝐵tensor-productsubscript𝜌𝐴subscript𝜌𝐵\rho_AB=\rho_A\otimes\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Then, if we swap Alice’s subsystem A𝐴Aitalic_A with the A′superscript𝐴′A^\primeitalic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT we kept before, the overall state for Alice and Bob changes to ρAB′=σA′B′subscriptsuperscript𝜌′𝐴𝐵subscript𝜎superscript𝐴′superscript𝐵′\rho^\prime_AB=\sigma_A^\primeB^\primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, which is different from ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. But since the state of Alice’s subsystem remains ρA′=σA′=ρAsubscriptsuperscript𝜌′𝐴subscript𝜎superscript𝐴′subscript𝜌𝐴\rho^\prime_A=\sigma_A^\prime=\rho_Aitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, we have achieved the quantum-house effect.

An important difference between Theorem 4 and Theorem 5 is that the proof of the latter requires that there is side-information available which is correlated with Bob’s subsystem, while for the former theorem it’s enough if we just know ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT.

Example 9.

Let Charlie first prepare σA′B′=12|00⟩⟨00|+12|11⟩⟨11|subscript𝜎superscript𝐴normal-′superscript𝐵normal-′12ket00quantum-operator-product001211bra11\sigma_A^\primeB^\prime=\frac12|00\rangle\langle 00|+\frac12|11% \rangle\langle 11|italic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 11 ⟩ ⟨ 11 |, which is a non-product state with σA′=σB′=I2subscript𝜎superscript𝐴normal-′subscript𝜎superscript𝐵normal-′𝐼2\sigma_A^\prime=\sigma_B^\prime=\fracI2italic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG. Charlie gives the second qubit to Bob, and keeps the first for himself. Then, he prepares a new qubit, independently in state I2𝐼2\fracI2divide start_ARG italic_I end_ARG start_ARG 2 end_ARG, and gives that to Alice. Thus, for Alice and Bob the overall 2-qubit state is ρAB=I2⊗I2subscript𝜌𝐴𝐵tensor-product𝐼2𝐼2\rho_AB=\fracI2\otimes\fracI2italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG ⊗ divide start_ARG italic_I end_ARG start_ARG 2 end_ARG. Now, if Charlie swaps Alice’s qubit with the one he kept before, the overall 2-qubit state for Alice and Bob will change to ρAB′=σA′B′subscriptsuperscript𝜌normal-′𝐴𝐵subscript𝜎superscript𝐴normal-′superscript𝐵normal-′\rho^\prime_AB=\sigma_A^\primeB^\primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, which is different from ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. However, the state of Alice’s qubit remains the same: ρA′=σA′=I2subscriptsuperscript𝜌normal-′𝐴subscript𝜎superscript𝐴normal-′𝐼2\rho^\prime_A=\sigma_A^\prime=\fracI2italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG.

Finally, it’s easy to see that with the previous theorem we’ve reached the limit:

Theorem 6 (Trivial correlation implies no quantum-house).

The quantum-house effect cannot be achieved with any product state ρAB=ρA⊗ρBsubscript𝜌𝐴𝐵tensor-productsubscript𝜌𝐴subscript𝜌𝐵\rho_AB=\rho_A\otimes\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT where either ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT or ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is pure.

5 The quantum-house game

We present a protocol called "the quantum-house game". The game is played by Alice, Bob and Charlie. Alice has to find out if Charlie (secretly) did something to her subsystem A𝐴Aitalic_A. If she succeeds, she scores points, and her goal is to maximize her expected score.

We’ll show that in certain non-classical setups of the game, due to the quantum-house effect Alice is expected to score more points if she asks Bob to help her, while in a classical setup it never improves Alice’s expected score if she joins forces with Bob.

5.1 The protocol

The game is rather open-ended, leaving Charlie a lot of freedom in Steps 1, 2 and 3 to shape what "flavor" to play.

Before going further, we’d like to underline that having a score of "negative infinity" is perfectly fine, as long as we can calculate with it in a mathematically rigorous way. (The score is a logical entity, not physical.) For example, we can use hyperreal numbers keisler , an extension of the real numbers which contains infinite numbers as well as infinitesimals.888On the other hand, we can assume that Alice isn’t able to physically perform e.g. a rotation gate with an infinitesimal angle.

5.2 Classical AB𝐴𝐵ABitalic_A italic_B

The first thing we notice is that whenever AB𝐴𝐵ABitalic_A italic_B is a classical system, Step 5 isn’t needed.

It’s because Alice can, in principle, observe a classical A𝐴Aitalic_A without perturbing either A𝐴Aitalic_A or AB𝐴𝐵ABitalic_A italic_B. So she won’t be caught in Step 2. Thus, she is able to observe the exact state of A𝐴Aitalic_A both before and after Charlie’s action of Step 3.

Now, if Charlie causes in Step 3 a noticeable change to AB𝐴𝐵ABitalic_A italic_B, then it will also be noticeable by examining solely A𝐴Aitalic_A. So Alice won’t need Bob’s help (Step 5), as it wouldn’t improve her expected score.

5.3 Non-classical AB𝐴𝐵ABitalic_A italic_B

As for non-classical AB𝐴𝐵ABitalic_A italic_B, we analyze a concrete example (i.e. "flavor") of the game.

Step 1. Charlie prepares for Alice and Bob a 2-qubit system AB𝐴𝐵ABitalic_A italic_B in the following state:

ρAB=13|00⟩⟨00|+16|01⟩⟨01|+16|10⟩⟨10|+13|11⟩⟨11|subscript𝜌𝐴𝐵13ket00quantum-operator-product001601quantum-operator-product011610quantum-operator-product101311bra11\rho_AB=\frac13|00\rangle\langle 00|+\frac16|01\rangle\langle 01|\\ +\frac16|10\rangle\langle 10|+\frac13|11\rangle\langle 11|start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 3 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | 01 ⟩ ⟨ 01 | end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | 10 ⟩ ⟨ 10 | + divide start_ARG 1 end_ARG start_ARG 3 end_ARG | 11 ⟩ ⟨ 11 | end_CELL end_ROW (1)

Due to Theorem 2, ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT has zero discord.999Since it can be written as ρAB=12|0⟩⟨0|⊗(23|0⟩⟨0|+13|1⟩⟨1|)+12|1⟩⟨1|⊗(13|0⟩⟨0|+23|1⟩⟨1|)subscript𝜌𝐴𝐵tensor-product12ket0bra023ket0quantum-operator-product0131bra1tensor-product12ket1bra113ket0quantum-operator-product0231bra1\rho_AB=\frac12|0\rangle\langle 0|\otimes(\frac23|0\rangle\langle 0|% +\frac13|1\rangle\langle 1|)+\frac12|1\rangle\langle 1|\otimes(\frac1% 3|0\rangle\langle 0|+\frac23|1\rangle\langle 1|)italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 0 ⟩ ⟨ 0 | ⊗ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG | 0 ⟩ ⟨ 0 | + divide start_ARG 1 end_ARG start_ARG 3 end_ARG | 1 ⟩ ⟨ 1 | ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 1 ⟩ ⟨ 1 | ⊗ ( divide start_ARG 1 end_ARG start_ARG 3 end_ARG | 0 ⟩ ⟨ 0 | + divide start_ARG 2 end_ARG start_ARG 3 end_ARG | 1 ⟩ ⟨ 1 | ). In general, a mixed quantum state can be prepared using different ensembles. In this example, Charlie prepares AB𝐴𝐵ABitalic_A italic_B using the ensemble given by:

ρAB=16|00⟩⟨00|+16|11⟩⟨11|+16|+0⟩⟨+0|+16|-0⟩⟨-0|+16|+i1⟩⟨+i1|+16|-i1⟩⟨-i1|subscript𝜌𝐴𝐵16ket00quantum-operator-product001611quantum-operator-product11160quantum-operator-product0160quantum-operator-product016𝑖1quantum-operator-product𝑖116𝑖1bra𝑖1\rho_AB=\frac16|00\rangle\langle 00|+\frac16|11\rangle\langle 11|\\ +\frac16|+0\rangle\langle+0|+\frac16|-0\rangle\langle-0|\\ +\frac16|+i1\rangle\langle+i1|+\frac16|-i1\rangle\langle-i1|start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 6 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | 11 ⟩ ⟨ 11 | end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | + 0 ⟩ ⟨ + 0 | + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | - 0 ⟩ ⟨ - 0 | end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | + italic_i 1 ⟩ ⟨ + italic_i 1 | + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | - italic_i 1 ⟩ ⟨ - italic_i 1 | end_CELL end_ROW (2)

Let Charlie inform Alice that he used the ensemble in Equation 2 to produce ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. It’s important to emphasize here that from Alice’s perspective, the density matrix of her qubit isn’t |ψ⟩⟨ψ|ket𝜓bra𝜓|\psi\rangle\langle\psi|| italic_ψ ⟩ ⟨ italic_ψ |, but ρA=trB(ρAB)=I2subscript𝜌𝐴subscripttr𝐵subscript𝜌𝐴𝐵𝐼2\rho_A=\mathrmtr_B(\rho_AB)=\fracI2italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG, the 1-qubit maximally mixed state.

Step 2. Charlie does the check by measuring Alice’s qubit in a basis which does not to perturb the state of the qubit. E.g. if Charlie gave Alice the |+⟩ket|+\rangle| + ⟩ state, he will do the check by measuring it in the Hadamard basis. Then, it’s easy to see the following:

Theorem 7.

Unless Alice makes sure |ψ⟩ket𝜓|\psi\rangle| italic_ψ ⟩ isn’t perturbed, she will have a non-zero probability of being caught in Step 2, and thus an expected score of negative infinity.

Given that just by random guessing Alice would have an expected score of 50 points, she won’t attempt to gain more information about the state of her qubit before Step 2, because that would entail a non-zero chance of perturbing |ψ⟩ket𝜓|\psi\rangle| italic_ψ ⟩, as the states in S𝑆Sitalic_S are not pairwise orthogonal, they include conjugate bases wiesner . This is a key difference compared to the classical case of Subsection 5.2.

Step 3. Let the pre-agreed operation, which Charlie may or may not secretly perform, be a Pauli-X𝑋Xitalic_X gate on Alice’s qubit. Based on Equation 1, a Pauli-X𝑋Xitalic_X gate brings ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT to:

ρAB′=13|10⟩⟨10|+16|11⟩⟨11|+16|00⟩⟨00|+13|01⟩⟨01|subscriptsuperscript𝜌′𝐴𝐵13ket10quantum-operator-product101611quantum-operator-product111600quantum-operator-product001301bra01\rho^\prime_AB=\frac13|10\rangle\langle 10|+\frac16|11\rangle% \langle 11|\\ +\frac16|00\rangle\langle 00|+\frac13|01\rangle\langle 01|start_ROW start_CELL italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 3 end_ARG | 10 ⟩ ⟨ 10 | + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | 11 ⟩ ⟨ 11 | end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 6 end_ARG | 00 ⟩ ⟨ 00 | + divide start_ARG 1 end_ARG start_ARG 3 end_ARG | 01 ⟩ ⟨ 01 | end_CELL end_ROW (3)

Clearly, ρAB≠ρAB′subscript𝜌𝐴𝐵subscriptsuperscript𝜌′𝐴𝐵\rho_AB eq\rho^\prime_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ≠ italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. However, it doesn’t change the state of Alice’s qubit: ρA′=trB(ρAB′)=I2subscriptsuperscript𝜌′𝐴subscripttr𝐵subscriptsuperscript𝜌′𝐴𝐵𝐼2\rho^\prime_A=\mathrmtr_B(\rho^\prime_AB)=\fracI2italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = roman_tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = divide start_ARG italic_I end_ARG start_ARG 2 end_ARG, same as ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT.

Step 4. As we’ve just seen, even if Charlie decides to secretly apply a Pauli-X𝑋Xitalic_X gate, it only changes the state of AB𝐴𝐵ABitalic_A italic_B, but not that of A𝐴Aitalic_A. Thus, if Alice ends the game here, all she can do is random guessing, resulting in an expected score of 50 points. But we’ll see she can do better if she joins forces with Bob. So she goes to Step 5.

Step 5. Alice and Bob measure AB𝐴𝐵ABitalic_A italic_B in the computational basis. If the result is 00000000 or 11111111, Alice guesses that Charlie did nothing. On the other hand, if the result is 01010101 or 10101010, Alice guesses that Charlie has applied the Pauli-X𝑋Xitalic_X gate. This way, Alice will be correct with probability 2323\frac23divide start_ARG 2 end_ARG start_ARG 3 end_ARG, giving an expected score of 60 points. So it’s worth asking Bob for help.

5.4 Discussion: implicit discord

We saw in Subsection 5.3 that in a quantum setting, Alice could make use of the quantum-house effect to achieve a higher expected score by joining forces with Bob. GAMING NEWS explained in Subsection 5.2 why the same can never happen in a classical setting. Furthermore, in the example we analyzed, ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT had zero discord. So, apparently, non-classical correlation was possible without quantum discord. But was it really the case? We’ll argue that the ability of Charlie’s device to create quantum discord was in fact necessary.

To start with, imagine Alice knows in the quantum-house game that Charlie’s device isn’t able to produce non-vanishing discord, i.e. ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT such that δAB>0subscript𝛿𝐴𝐵0\delta_AB>0italic_δ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT >0. Based on Subsection 2.1, this means that the device cannot create two non-orthogonal states of A𝐴Aitalic_A. Thus, Charlie can only prepare A𝐴Aitalic_A in one of the states of a fixed orthonormal basis ketsubscript𝑎𝑖\a_i\rangle\ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ . If Alice is aware of that, she can simply measure A𝐴Aitalic_A in that basis, gaining information without being caught.

Now, in particular, let Alice know in the example of Subsection 5.3 that Charlie’s device is only capable of producing the ket0ket1\ basis states of A𝐴Aitalic_A. So if she measures her qubit in this basis before Step 2, then she can find out already in Step 4 with certainty (by measuring again), whether or not Charlie has applied the Pauli-X𝑋Xitalic_X gate in Step 3. Thus, without the implicit presence (ability) of quantum discord, the quantum-house effect wouldn’t make a difference here, because Alice would never need Step 5.

In general, if in a physical experiment it’s only possible to prepare A𝐴Aitalic_A in itself, in one of the pairwise orthogonal states ai⟩ketsubscript𝑎𝑖\ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ , then for all intents and purposes A𝐴Aitalic_A can be considered as classical. In this sense, we can say that (the possibility of) non-vanishing quantum discord is necessary for quantumness. Or, based on Theorem 2, it’s eventually the possibility of superposition that is necessary.

6 Quantum detachment

In this paper, we introduced the quantum-house effect, a non-local phenomenon which can be exhibited even with bipartite product states, provided that neither subsystem is in a pure state. The effect was demonstrated (with some inevitable noise) on the SpinQ Gemini 2-qubit liquid-state NMR desktop quantum computer.

We also argued that although the effect apparently doesn’t require quantum discord to be present, the ability to create discord is necessary in a cryptographic (as well as physical) sense, so that the quantum-house effect can make a difference compared to classicality.111111In the mathematical sense, if we take the formal definition of quantum state at face value, no implicit quantum discord is needed, that’s what we saw in Sections 2 and 4. This isn’t actually surprising if we consider that implicit discord is basically the possibility of superposition.

To go beyond the quantum-house effect, we can view quantumness from a slightly different angle. Let’s take the BB84 protocol bennett as an example. It merely uses single qubits in pure states and achieves something that is classically unattainable: unconditionally secure communication. What makes this possible in BB84 is that when Alice sends a qubit to Bob, she holds back relevant information about the qubit’s state (she can do that because her device is capable of creating non-orthogonal qubit states), which renders the eavesdropper’s task impossible. We suggest calling this phenomenon quantum detachment, a principle which roughly means that relevant information about the state of a physical system is kept separate from the system itself.121212”Relevant information” is such that it could influence what outcome an experimenter can achieve with the system in a local lab.

In our opinion, one way to characterize the point where quantumness departs from classicality is the presence of quantum detachment. The idea can be conveyed as follows: when the (locally unavailable) information is somewhere else, we have a mixed state; and when it’s nowhere else, we have superposition. The latter case can be considered as the ultimate quantum detachment, because the missing information doesn’t even exist in the universe.

We can contrast quantum detachment with the classical world where no relevant information about the state of a physical system can be held back, as all of it can be found out locally in the lab, in principle. Put it differently, in the classical world a physical system contains all the relevant information about itself.

As for future work, the quantum-house game might be turned into a protocol by which Charlie could securely cast a "yes/no" vote, locally in Alice’s lab.

Declarations

At the time of writing, Papafut Quantum is an exclusive agent of SpinQ in Switzerland.

Public Last updated: 2022-06-28 03:04:04 AM